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# Radiation Reaction on Moving Particles in General Relativity

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RADIA TION REA CTION ON MO VING P AR TICLES IN GENERAL RELA TIVITY By EIRINI MESSARIT AKI A DISSER T A TION PRESENTED TO THE GRADUA TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORID A 2003

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A CKNO WLEDGMENTS First and foremost I w ould lik e to thank Dr. Stev e Det w eiler, Professor of Ph ysics at the Univ ersit y of Florida, who w as m y advisor during m y graduate studies. He has alw a ys b een v ery supp ortiv e of m y w ork, v ery willing to help and extremely patien t with me. I feel v ery luc ky to ha v e w ork ed with him during the past 5 y ears. I w ould also lik e to thank Dr. Bernard Whiting for his help on writing this dissertation and for his con tributions to the researc h. Man y thanks go to the other mem b ers of m y committee: Dr. Jim F ry Dr. Andrew Rinzler and Dr. Vic ki Sara jedini, as w ell as to Dr. Ric hard W o o dard and Dr. Luz Diaz, for their useful commen ts on the dissertation. I w ould also lik e to thank m y brothers, m y family and m y close friends. Their help and supp ort has alw a ys b een v ery imp ortan t and I could not ha v e come this far without them. My researc h w as partly supp orted b y the Institute of F undamen tal Theory at the Univ ersit y of Florida and I am grateful for that. ii

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T ABLE OF CONTENTS A CKNO WLEDGMENTS . . . . . . . . . . . . ii ABSTRA CT . . . . . . . . . . . . . . . v CHAPTER 1 INTR ODUCTION . . . . . . . . . . . . . 1 2 RADIA TION REA CTION IN FLA T SP A CETIME . . . . . . 4 2.1 Defnitions of the Fields . . . . . . . . . . . . . 4 2.2 Deriv ation of the Equations of Motion . . . . . . . . . 7 2.3 Radiation Reaction . . . . . . . . . . . . . . 12 3 RADIA TION REA CTION IN CUR VED SP A CETIME . . . . 14 3.1 Nonlo cal Quan tities . . . . . . . . . . . . . . 14 3.2 Green's F unctions . . . . . . . . . . . . . . . 18 3.2.1 Green's F unctions for a Scalar Field . . . . . . . . 18 3.2.2 Green's F unctions for a V ector Field . . . . . . . 23 3.2.3 Green's F unctions for a Gra vitational Field . . . . . . 27 3.3 Motion of a Charged P article in Curv ed Spacetime . . . . . 32 3.3.1 Equations of Motion . . . . . . . . . . . . 32 3.3.2 Calculation of the Fields . . . . . . . . . . . 34 3.3.3 Conserv ation of Energy and Momen tum . . . . . . 37 3.3.4 Radiation Reaction . . . . . . . . . . . . 40 4 SELF-F OR CE . . . . . . . . . . . . . 41 4.1 Scalar Self-F orce . . . . . . . . . . . . . . . 41 4.1.1 Direct and T ail Fields . . . . . . . . . . . 42 4.1.2 The S-Field and the R-Field . . . . . . . . . . 43 4.2 Electromagnetic Self-F orce . . . . . . . . . . . . 48 4.3 Gra vitational Self-F orce . . . . . . . . . . . . . 50 5 SINGULAR FIELD F OR SCHW ARZSCHILD GEODESICS . . . 53 5.1 Thorne-Hartle-Zhang Co ordinates . . . . . . . . . . 54 5.2 Scalar Field of a Charged P article . . . . . . . . . . 60 5.3 Scalar Field of a Dip ole . . . . . . . . . . . . . 61 5.4 Electromagnetic P oten tial of a Charged P article . . . . . . 63 5.5 Electromagnetic P oten tial of an Electric Dip ole . . . . . . 66 5.6 Electromagnetic P oten tial of a Magnetic Dip ole . . . . . . 69 5.7 Gra vitational Field of a Spinning P article . . . . . . . . 72 iii

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5.8 Gr tensor Co de . . . . . . . . . . . . . . . 77 6 REGULARIZA TION P ARAMETERS F OR THE SCALAR FIELD . 80 6.1 Regularization Pro cedure . . . . . . . . . . . . . 81 6.2 Order Calculation of the Scalar Field . . . . . . . . . 84 6.3 Scalar Monop ole Field . . . . . . . . . . . . . 88 7 CALCULA TION OF THE RET ARDED FIELD . . . . . . 102 7.1 Analytical W ork . . . . . . . . . . . . . . . 102 7.2 Numerical Co de . . . . . . . . . . . . . . . 110 8 APPLICA TIONS AND CONCLUSIONS . . . . . . . 111 8.1 Equations of Motion . . . . . . . . . . . . . . 111 8.2 Eects of the Scalar Self-F orce . . . . . . . . . . . 112 8.2.1 Calculation of the Self-F orce . . . . . . . . . . 115 8.2.2 Change in Orbital F requency . . . . . . . . . 116 8.3 Eects of the Gra vitational Self-F orce . . . . . . . . . 117 A GR TENSOR CODE F OR THE SINGULAR FIELDS . . . . 119 B DECOMPOSITION OF ~ P )Tj/T1_93 7.9701 Tf12.1219 Tc 6.5992 0 Td(Q . . . . . . . . . 134 C NUMERICAL CODE F OR THE RET ARDED FIELD . . . . 143 REFERENCES . . . . . . . . . . . . . . 163 BIOGRAPHICAL SKETCH . . . . . . . . . . . 166 iv

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Abstract of Dissertation Presen ted to the Graduate Sc ho ol of the Univ ersit y of Florida in P artial F ulllmen t of the Requiremen ts for the Degree of Do ctor of Philosoph y RADIA TION REA CTION ON MO VING P AR TICLES IN GENERAL RELA TIVITY By Eirini Messaritaki August 2003 Chairman: Stev en L. Det w eiler Ma jor Departmen t: Ph ysics A particle in the vicinit y of a Sc h w arzsc hild blac k hole is kno wn to trace a geo desic of the Sc h w arzsc hild bac kground, to a rst appro ximation. If the in teraction of the particle with its o wn eld (scalar, electromagnetic or gra vitational) is tak en in to accoun t, the path is no longer a bac kground geo desic and the self-force that the particle exp eriences needs to b e tak en in to accoun t. In this dissertation, a recen tly prop osed metho d for the calculation of the selfforce is implemen ted. According to this metho d the self-force comes from the in teraction of the particle with the eld R = ret S for a scalar particle; with the electromagnetic p oten tial A Ra = A reta A Sa for a particle creating an electromagnetic eld; or with the metric p erturbation h Rab = h retab h Sab for a particle creating a gra vitational eld. First, the singular elds S ; A Sa and h Sab are calculated for dieren t sources mo ving in a Sc h w arzsc hild bac kground. F or that, the Thorne-Hartle-Zhang co ordinates in the vicinit y of the mo ving source are used. Then a mo de-sum regularization metho d initially prop osed for the direct scalar eld is follo w ed, and the regularization parameters for the singular part of the scalar eld and for the rst radial deriv ativ e v

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of the singular part of the self-force are calculated. Also, the n umerical calculation of the retarded scalar eld for a particle mo ving on a circular geo desic in a Sc h w arzsc hild spacetime is presen ted. Finally the self-force for a scalar particle mo ving on a circular Sc h w arzsc hild orbit is calculated and some results ab out the eects of the self-force on the orbital frequency of the circular orbit are presen ted. vi

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CHAPTER 1 INTR ODUCTION The General Theory of Relativit y dev elop ed b y Alb ert Einstein at the b eginning of the t w en tieth cen tury has b een successfully used b y scien tists for man y y ears to explain ph ysical phenomena and to mak e predictions ab out ph ysical systems. One of the most exciting predictions of General Relativit y is the existence of gra vitational radiation, whic h can b e though t of as a w a v e-lik e distortion of spacetime. It is exp ected that gra vitational radiation from remote astroph ysical systems will yield v ery imp ortan t information ab out those systems, th us answ ering man y questions that scien tists curren tly ha v e. Ev en though the detection of gra vitational w a v es has pro v ed to b e a c hallenging task, recen t dev elopmen ts in tec hnology ha v e made scien tists conden t that gra vitational w a v es will indeed b e detected in the near future. Both earth-based detectors (suc h as LIGO and VIR GO) and space-based detectors (suc h as LISA) are exp ected to start op erating in the next decade or t w o. F or the data collected b y these detectors to b e useful, accurate information ab out the sources of gra vitational radiation is required. Only if scien tists kno w what the gra vitational w a v es emitted b y sp ecic systems should lo ok lik e, can they compare them with the patterns observ ed and dra w conclusions ab out the ph ysical c haracteristics of those systems. That is wh y there has b een a lot of in terest lately in predicting the gra vitational radiation emitted b y dieren t astroph ysical systems. One astroph ysical system for whic h gra vitational w a v es are exp ected to b e detected, sp ecically b y space-based detectors, is the binary system of a small neutron star or blac k hole (of mass equal to a few times the mass of the Sun) and a sup ermassiv e blac k hole (of mass equal to a few million times the mass of the Sun) [1]. Sup ermassiv e blac k holes are b eliev ed to exist at the cen ters of man y galaxies, in1

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2 cluding our o wn. It is also b eliev ed that their strong gra vitational eld can capture smaller stars and blac k holes, whic h then mo v e to w ard the sup ermassiv e blac k hole, un til they are absorb ed b y it. The exact ev olution of the system with time can giv e the pattern of the gra vitational radiation that the system is exp ected to emit. Kno wing that pattern is crucial in distinguishing whether the gra vitational w a v es measured b y the detector come from suc h a system. It can also help extract information ab out man y c haracteristics of the system, suc h as the masses and the angular momen ta of the t w o comp onen ts of the binary system. T o determine ho w the system ev olv es, the path of the smaller star or blac k hole needs to b e calculated. That path is largely determined b y the inruence of the gra vitational eld of the sup ermassiv e blac k hole on the small star or blac k hole; it is also aected b y the in teraction of the small star or blac k hole with its o wn gra vitational eld. This latter in teraction, commonly referred to as the self-force, is resp onsible for the radiation reaction eects, whic h cause the deca y of the small star's or blac k hole's orbit to w ard the sup ermassiv e blac k hole. Ev en though m y motiv ation for studying the self-force eects stems from the need to kno w the ev olution of this system, radiation reaction eects are presen t in other systems that are easier to deal with mathematically One can think, for example, of a particle of a certain scalar c harge, whic h mo v es in a Sc h w arzsc hild bac kground spacetime and creates its o wn scalar eld. There is also the case of a particle that carries an electric c harge and creates an electromagnetic eld as it mo v es in spacetime. The scalar eld of the particle in the rst case and the electromagnetic eld of the electric c harge in the second case will aect the motion of eac h particle, causing its w orldline to dier from what it w ould b e if radiation reaction eects w ere not presen t. It is useful to predict the ev olution of suc h systems, mainly b ecause they can giv e an idea of ho w the more dicult system can b e handled successfully and not b ecause they are realistic systems, since they are not exp ected to b e observ ed in nature.

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3 In the past, v arious eorts ha v e b een made to dev elop a concrete sc heme for including the radiation reaction eects for scalar, electromagnetic and gra vitational elds in the equations of motion. One approac h that has b een used extensiv ely inv olv es calculating the rux, b oth at innit y and at the ev en t horizon of the cen tral sup ermassiv e blac k hole, of quan tities (suc h as the energy and momen tum of the particle) that are constan ts of the motion when radiation reaction is not presen t [2{9]. That rux is then asso ciated with the rate of c hange of those quan tities at the lo cation of the particle and the ev olution of the particle's motion can b e predicted. Ho w ev er, this approac h is generally not applicable in the case in whic h the cen tral blac k hole is a rotating blac k hole, except under some sp ecial circumstances [10]. One reason is that the ev olution of the Carter constan t [11], whic h is a constan t of the motion that is not deriv ed b y a Killing v ector, cannot b e calculated using the rux at innit y and at the ev en t horizon. In addition, this approac h do es not tak e in to accoun t the v ery signican t non-dissipativ e eects of the self-force that the particle's eld exerts on the particle [12]. That is wh y it is imp ortan t for the self-force, and not just its radiation reaction eects, to b e calculated. The calculation of the self-force is the sub ject of this dissertation. In Chapters 2 and 3 of this dissertation, I presen t the inno v ativ e w ork of Dirac [13] for the radiation reaction eects on an electron mo ving in rat spacetime and the theoretical generalization of it to the case of curv ed spacetime b y DeWitt and Brehme [14]. In Chapter 4, I presen t dieren t metho ds for practical calculations of the self-force that ha v e b een prop osed in the past, with emphasis on one in particular. In Chapters 5-8 of this dissertation, I presen t an implemen tation of that particular metho d for calculating the radiation reaction eects for a scalar, an electromagnetic and a gra vitational eld.

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CHAPTER 2 RADIA TION REA CTION IN FLA T SP A CETIME The rst successful attempt to pro vide an expression for the radiation reaction eects on a particle in sp ecial relativit y w as made b y Dirac in 1938 [13]. In his famous pap er, he studied the motion of an electron mo ving in rat spacetime b y using the concept of conserv ation of energy and momen tum along the electron's w orldline. In this c hapter I presen t the decomp osition of the electromagnetic eld and the deriv ation of the equations of motion, whic h include the radiation damping eects on the electron. The sc heme for calculating the radiation reaction eects in curv ed spacetime presen ted in this dissertation extends Dirac's w ork. F or the equations to lo ok as simple as p ossible, the sp eed of ligh t is set equal to 1 in this c hapter. It is noted that the metric signature used b y Dirac is (+ ) and that is what is used in this c hapter, in order to k eep the results iden tical to those deriv ed b y Dirac. In all subsequen t c hapters, ho w ev er, the signature is c hanged to ( + ++), in order to adhere to the curren tly widely used con v en tion. Th us, some of the equations giv en in this c hapter c hange in subsequen t c hapters. Since none of the results describ ed in this c hapter are used in the c hapters that follo w, that should not cause an y confusion at all. 2.1 Denitions of the Fields Dirac's analysis starts b y assuming that an electron of electric c harge q is mo ving in rat spacetime, so that the bac kground metric is the Mink o wski metric The 4

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5 w orldline of the electron is denoted b y z = z ( s ) (2.1) where s is the prop er time along the w orldline. The electromagnetic 4-v ector p oten tial that the electron creates is assumed to satisfy the Loren tz gauge condition r A = 0 (2.2) and ob eys Maxw ell's equations r 2 A = 4 j : (2.3) It is w ell-kno wn that the Loren tz gauge condition lea v es some arbitrariness for the electromagnetic p oten tial. In Equation (2.3), j is the c harge-curren t densit y v ector of the electron, namely j = q Z dz ds ( x 0 z 0 ) ( x 1 z 1 ) ( x 2 z 2 ) ( x 3 z 3 ) ds: (2.4) The electromagnetic eld asso ciated with a 4-v ector p oten tial A is, in general, giv en b y the equation F = r A r A : (2.5) It is clear that, unless the b oundary conditions of the problem are sp ecied, Equations (2.2) and (2.3) do not ha v e a unique solution. In fact, adding an y solution of the homogeneous equation r 2 A = 0 (2.6)

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8 rst step is to surround the w orldline of the electron b y a thin cylindrical tub e of constan t radius. This radius is assumed to b e v ery small, sp ecically smaller than an y length of ph ysical signicance in the problem. The ob jectiv e is to calculate the ro w of energy and momen tum across the three-dimensional surface of the tub e, using the stress-energy tensor T calculated from the actual electromagnetic eld. The stress-energy tensor is giv en b y 4 T = F act F act + 1 4 F act F act : (2.13) The ro w of energy and momen tum out of the surface of the tub e is equal to the dierence in the energy and momen tum at the t w o ends of the tub e. In the follo wing, dots o v er quan tities denote dieren tiation with resp ect to the prop er time s T o simplify the notation, ( dz =ds ) is set equal to v One then obtains the equations v v = 1 ; (2.14) v v = 0 ; (2.15) v v + v v = 0 : (2.16) T o calculate the stress-energy tensor, the electromagnetic 4-v ector p oten tial and the electromagnetic eld need to b e calculated rst. The retarded p oten tial at a p oin t x generated b y an electron mo ving on the w orldline z ( s ) is giv en b y A ret = q z z ( x z ) (2.17) calculated at the retarded prop er time, whic h is the v alue of s that solv es the equation ( x z ) ( x z ) = 0 : (2.18)

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9 An equiv alen t expression for the retarded p oten tial, obtained from Equation (2.17) b y using elemen tary prop erties of the function, is A ret = 2 q Z in t 1 z [( x z ) ( x z )] ds (2.19) where in t is a prop er time b et w een the retarded and adv anced prop er times. By dieren tiating Equation (2.19) and using Equation (2.5), the retarded electromagnetic eld is obtained F ret = 2 q Z in t 1 d ds z ( x z ) z ( x z ) z ( x z ) [( x z ) ( x z )] ds: (2.20) By using again some of the elemen tary prop erties of the function, an equiv alen t expression for the retarded eld is deriv ed F ret = q z ( x z ) d ds z ( x z ) z ( x z ) z ( x z ) : (2.21) Again, all quan tities are calculated at the retarded prop er time. Since the goal is to calculate the stress-energy tensor at the t w o ends of the w orld-tub e surrounding the w orldline of the electron, it can b e assumed that the p oin t x is v ery close to the w orldline. Sp ecically x = z ( s 0 ) + r (2.22) where the r 's are v ery small. Then, the elds can b e T a ylor-expanded in the r 's. In the expansions that follo w, all the co ecien ts are tak en at the prop er time s 0 It can b e assumed that the retarded prop er time is s 0 where it is reasonable that is a small p ositiv e quan tit y of the same order as r

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10 The detailed calculations needed to obtain the T a ylor expansions of the electromagnetic elds are tedious but straigh tforw ard. They can b e found in Dirac's pap er; here I presen t only the results of that calculation. Since r is a space-lik e v ector, it can b e assumed that r r = 2 where is a p ositiv e n um b er. T a ylor-expanding the retarded eld and k eeping only the terms that do not v anish in the limit 0 giv es F ret = q (1 r v ) 1 2 3 ( v r v r ) 1 2 1 ( v v v v ) (1 r v ) + 1 8 1 v v ( v r v r ) + 1 2 1 ( v r v r ) + 2 3 ( v v v v ) : (2.23) The adv anced eld can b e obtained b y c hanging to and c hanging the sign of the whole expression. By using the retarded and adv anced elds in Equation (2.10) the radiation eld on the w orldline is obtained F rad = 4 q 3 ( v v v v ) : (2.24) The actual eld then b ecomes F act = f + q [1 r v ] 1 2 ( 3 1 8 1 v v )( r v r v ) + 1 2 1 (1 + r v )( v v v v + 1 2 1 ( v r v r ) : (2.25) T o calculate the energy and momen tum ro w out of the tub e, the stress-energy tensor needs to b e calculated. In fact, only its comp onen t along the direction of r is

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11 necessary Substituting Equation (2.25) in to Equation (2.13): 4 T r = F act F act r 1 4 F act F act r = q 2 (1 r v ) 1 ( 1 2 4 + 1 2 2 v v ) r 1 2 2 (1 + 3 2 r v ) v + q 1 v f : (2.26) The ro w of energy and momen tum out of the surface of the tub e is giv en b y in tegrating this comp onen t of the stress-energy tensor o v er the surface of the tub e. The result is Z ( 1 2 q 2 1 v q v f ) ds (2.27) where the in tegration is o v er the length of the tub e. As men tioned earlier, this ro w of energy-momen tum dep ends only on the conditions at the t w o ends of the tub e. That means that the in tegrand m ust b e a p erfect dieren tial, namely 1 2 q 2 1 v q v f = B (2.28) for some B Equation (2.15) and the fact that f is an tisymmetric in its indices put a restriction on B sp ecically v B = 1 2 q 2 1 v v q v v f = 0 : (2.29) Th us, the simplest acceptable expression for B is B = k v (2.30)

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12 for some k indep enden t of s F rom Equation (2.28) one obtains k = 1 2 q 2 1 m (2.31) where m m ust b e a constan t indep enden t of in order for Equations (2.28) and (2.29) to ha v e a w ell-dened b eha vior in the limit 0. Finally the equations of motion for the electron are m v = q v f (2.32) and m pla ys the part of the rest mass of the electron. This result is used in Section (2.3) to deriv e an expression for the radiation damping eects on the mo ving electron. 2.3 Radiation Reaction The eld f dened in Section (2.1) allo ws the equations of motion (2.32) to b e expressed in a simple form. Ho w ev er, in practical applications one w ould prefer to ha v e the inciden t radiation eld F in in the equations of motion, as F in is usually giv en. By substituting Equations (2.7) and (2.10) in to Equation (2.11), an expression for f is obtained, that in v olv es the inciden t eld and the radiation eld: f = F in + 1 2 F rad : (2.33) Then, the equations of motion b ecome m v = q v F in + 1 2 q v F rad : (2.34) The rst term of the righ t-hand side of Equation (2.34) in v olv es the inciden t radiation eld and giv es the w ork done b y that eld on the electron. The second term of the righ t-hand side in v olv es the radiation eld emitted b y the electron and is the term of particular in terest when one considers radiation reaction. This term

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13 giv es the eect of the electron radiation on itself and is presen t ev en if there is no external radiation eld, namely ev en if F in = 0. The fact that the radiation reaction eects on the electron can b e describ ed b y using the electromagnetic eld F rad is one of the imp ortan t results of Dirac's w ork and is the idea used when motion of particles in curv ed spacetime is considered, to calculate radiation reaction eects on those particles. It is w orth writing the equations of motion in terms of the c haracteristics of the w orldline of the electron and the external radiation eld, whic h can b e ac hiev ed b y using Equation (2.24) for the radiation eld of the electron m v = q v F in + 2 3 q 2 v + 2 3 q 2 v v v : (2.35) It is an imp ortan t and a v ery in teresting feature of Equation (2.35) that, in addition to the rst deriv ativ e of v its second deriv ativ e sho ws up as w ell. A discussion of that fact and of some of its implications is presen ted in [13], but that is b ey ond the scop e of this dissertation.

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CHAPTER 3 RADIA TION REA CTION IN CUR VED SP A CETIME Dirac's study of the radiation reaction eects on an electron ga v e a relativ ely simple result, b ecause the electron w as assumed to b e mo ving in rat spacetime. The analysis b ecomes signican tly more complicated for particles mo ving in curv ed spacetime. DeWitt and Brehme [14] w ere the rst to study the motion of an electrically c harged particle in a curv ed spacetime using Green's functions. Later, Mino, Sasaki and T anak a [15] generalized that analysis for the case of a particle creating a gra vitational eld while mo ving in curv ed spacetime. Since Green's functions are used extensiv ely in subsequen t c hapters to determine the self-force eects on dieren t particles, it is appropriate to presen t the analyses of DeWitt and Brehme and of Mino, Sasaki and T anak a b efore pro ceeding. In this and all subsequen t c hapters, geometrized units are used, meaning that Newton's gra vitational constan t and the sp eed of ligh t are b oth set equal to 1. Note also that the signature of the metric is c hanged to ( + ++) from no w on, so minor c hanges in some of the equations previously men tioned should not b e surprising. 3.1 Nonlo cal Quan tities One of the main c haracteristics of an y Green's function is that it connects t w o p oin ts in spacetime. It do es that b y propagating the eect of the source, from the p oin t where that source is lo cated (source p oin t) to the p oin t where the eld needs to b e calculated (eld p oin t). Since Green's functions are inheren tly nonlo cal quan tities, the discussion ab out them can b e facilitated if some more elemen tary nonlo cal quan tities are in tro duced rst. 14

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15 The most general class of nonlo cal quan tities is the class of bitensors, whic h are simply tensors whose indices refer to t w o p oin ts in spacetime. In the follo wing, x denotes the eld p oin t and z denotes the source p oin t. In order for the indices for eac h p oin t to b e easily iden tiable, all unprimed indices refer to the eld p oin t x and all primed indices refer to the source p oin t z A v ery imp ortan t quan tit y for the study of nonlo cal prop erties of spacetime is the biscalar of geo desic in terv al s ( x; z ). It is the magnitude of the in v arian t distance b et w een the p oin ts x and z as measured along a geo desic that joins them and is a non-negativ e quan tit y It is dened b y the equations r s r s = r 0 s r 0 s = 1 ; (3.1) lim x z s = 0 ; (3.2) where r denotes co v arian t dieren tiation with resp ect to the bac kground metric g It is ob vious that s m ust b e symmetric under in terc hange of its t w o argumen ts, namely s ( x; z ) = s ( z ; x ) : (3.3) With the signature of the metric b eing ( + ++), the in terv al s is spacelik e when the + sign holds in Equation (3.1) and timelik e when the sign holds. The p oin ts x for whic h s = 0 dene the n ull cone of z It is more con v enien t to use a dieren t quan tit y to measure the in v arian t distance b et w een the source p oin t and the eld p oin t, a quan tit y that is, ho w ev er, related to s ( x; z ). That quan tit y is Synge's [16] w orld function and is dened b y ( x; z ) 1 2 s 2 ( x; z ) : (3.4)

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16 By using the dening equations for s one can deduce that has the prop erties 1 2 r r = 1 2 r 0 r 0 = ; (3.5) lim x z = 0 : (3.6) Also, is p ositiv e for spacelik e in terv als and negativ e for timelik e in terv als. Another v ery signican t nonlo cal quan tit y is the biv ector of geo desic parallel displacemen t, denoted b y g 0 ( x; z ). The dening equations for it are r g 0 r = 0 ; r 0 g 0 r 0 = 0 (3.7) lim x z g 0 = 0 : (3.8) Equation (3.7) signies that the co v arian t deriv ativ es of g 0 are equal to zero in the directions tangen t to the geo desic joining x and z Equation (3.8) expresses the fact that g 0 is equal to the Kronec k erwhen x = z The biv ector of geo desic parallel displacemen t also has the prop ert y that g 0 ( x; z ) = g 0 ( z ; x ) : (3.9) In the follo wing, the determinan t of g 0 is denoted b y g = j g 0 j ; (3.10) and = j g 0 j : (3.11) The eect of applying the biv ector of geo desic parallel displacemen t to a lo cal v ector A 0 at z is a parallel transp ort of that v ector from p oin t z to p oin t x along the

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17 geo desic that connects the t w o p oin ts. The result is a lo cal v ector A at p oin t x In general, g 0 can b e used to transform an y bitensor that has indices refering to the t w o p oin ts x and z to a tensor whose indices refer to one p oin t only for example g 0 g 0 T 0 0 r = T r : (3.12) Ho w useful the biv ector of geo desic parallel displacemen t is in studying nonlo cal prorerties of spacetime can b e understo o d if expansions of a bitensor ab out one p oin t are considered. In order for a bitensor to b e expanded ab out a certain p oin t in spacetime, all its indices m ust refer to that sp ecic p oin t. F or bitensors for whic h that is not the case, their indices m ust rst b e homogenized b y applying g 0 and then the expansion ab out the sp ecic p oin t can b e tak en. A biv ector that is v ery useful for the Hadamard [17] expansion of the Green's functions is dened b y D 0 ( x; z ) = r r 0 ( x; z ) (3.13) and a biscalar relating to it is its determinan t D = j D 0 j : (3.14) DeWitt and Brehme pro v ed that lim x z D 0 ( x; z ) = g 0 ( z ) (3.15) whic h sho ws that the biscalar D is non v anishing, at least when x and z are close to eac h other. In fact, D is the Jacobian of the transformation from the set of v ariables f z 0 ; x g whic h sp ecify the geo desic b et w een x and z in terms of its t w o end p oin ts, to the set of v ariables f z 0 ; r 0 g whic h sp ecify the geo desic in terms of one of its

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18 end p oin ts and the tangen t to the geo desic at that end p oin t. DeWitt and Brehme also sho w ed that D ob eys the dieren tial equation D 1 r ( D r ) = 4 : (3.16) Instead of the biscalar D a dieren t biscalar is used in the Hadamard expansions of the Green's functions. That biscalar is denoted b y and is dened as ( x; z ) = [ g ( x; z )] 1 D ( x; z ) : (3.17) DeWitt and Brehme pro v ed that can b e expanded in terms of deriv ativ es of and that the expansion is = 1 + 1 6 R r r + O ( s 3 ) ; for x z (3.18) where R is the bac kground Ricci tensor. 3.2 Green's F unctions The goal of the w ork of DeWitt and Brehme w as to study the radiation damping eects on a particle of a giv en electric c harge that mo v es in curv ed spacetime. T o do that it is necessary to study the v ector eld that represen ts the electromagnetic p oten tial created b y the particle. Ho w ev er, the equations for the scalar eld are less complicated and th us easier to deal with. Also, the lac k of indices mak es the results more transparen t. F or those reasons the scalar Green's functions and their corresp onding elds are discussed rst. 3.2.1 Green's F unctions for a Scalar Field A p oin t particle of scalar c harge q whic h is mo ving on a w orldline : z 0 ( ), where is the prop er time along the geo desic, creates a scalar eld whic h ob eys

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19 P oisson's equation: r 2 = 4 %: (3.19) In Equation (3.19), % is the source function for the p oin t particle. Sp ecically % ( y ) = q Z ( g ) 1 2 4 ( y z ( )) d (3.20) where y is some p oin t in spacetime. It is desired to express the dieren t solutions of Equation (3.19) as in tegrals con taining Green's functions, namely ( x ) = 4 Z spacetime ( g ) 1 2 G ( x; y ) % ( y ) dy 4 = 4 q Z G [ x; z ( )] d ; (3.21) so the prop erties of the v arious Green's functions are emphasized in this section. One function of imp ortance is the symmetric Green's function, G sym ( x; z ), whic h satises the inhomogeneous dieren tial equation r 2 G sym ( x; z ) = ( g ) 1 2 4 ( x z ) : (3.22) The Hadamard form [17] of this function is G sym ( x; z ) = 1 8 [ U ( x; z ) ( ) V ( x; z ) ( )] (3.23) where is the step function that equals 1 if the argumen t is greater than zero and equals 0 otherwise. U ( x; z ) and V ( x; z ) are biscalars that are free of singularities and symmetric under the in terc hange of x and z namely U ( x; z ) = U ( z ; x ) (3.24)

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20 and V ( x; z ) = V ( z ; x ) : (3.25) They can b e determined b y expanding the solution of Equation (3.22) in p o w ers of in the vicinit y of the geo desic The biscalar U ( x; z ) satises the dieren tial equation U 1 ( x; z ) r U ( x; z ) = 1 2 1 ( x; z ) r ( x; z ) (3.26) and the b oundary condition lim x z U ( x; z ) = 1 (3.27) and is giv en b y U ( x; z ) = [( x; z )] 1 2 = 1 + 1 12 R r r + O ( s 3 ) ; x (3.28) where s is the prop er distance from the p oin t x to measured along the spatial geo desic whic h is orthogonal to The biscalar V ( x; z ) satises the homogeneous dieren tial equation r 2 V ( x; z ) = 0 (3.29) and is giv en b y V ( x; z ) = 1 12 R ( z ) + O ( s ) ; x : (3.30) It is notew orth y that the symmetric Green's function v anishes for > 0, that is for spacelik e separation of the p oin ts x and z Also, b ecause b oth U ( x; z ) and V ( x; z ) are symmetric under the in terc hange of x and z so is G sym : G sym ( x; z ) = G sym ( z ; x ) : (3.31)

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21 If = d ( x; z ( )) =d the scalar eld sym ( x ) asso ciated with the symmetric Green's function is sym ( x ) = q 2 Z [ U ( x; z ) ( ) V ( x; z ) ( )] d = h q U ( x; z ) 2 i ret + h q U ( x; z ) 2 i adv q 2 Z ret 1 + Z + 1 adv V ( x; z ) d (3.32) and consists of t w o parts. The rst part is the one that con tains the biscalar U ( x; z ) and the -function ( ) and is referred to as the dir e ct part. This is the term that corresp onds to = 0, whic h is the part of the eld that comes from the retarded and adv anced prop er times ( ret and adv resp ectiv ely), namely the prop er times that corresp ond to the in tersection of the geo desic with the past and future n ull cone of the p oin t x In other w ords, the direct part has the same singularit y on the n ull cone that the symmetric scalar eld has in rat spacetime. The second part is the one that con tains the biscalar V ( x; z ) and the step-function ( ) and is referred to as the tail part. This term giv es the part of the eld that corresp onds to < 0, whic h is the part that comes from the in terior of the past and future n ull cone of x It is the part of the eld that is due to the curv ature of spacetime and v anishes in rat spacetime. The symmetric Green's function can b e separated in to the retarded and adv anced parts, whic h constitute t w o v ery imp ortan t Green's functions themselv es. Sp ecically it can b e written as G sym ( x; z ) = 1 2 [ G ret ( x; z ) + G adv ( x; z )] (3.33) where the retarded and adv anced Green's functions are giv en b y G ret ( x; z ) = 2 [( x ) ; z ] G sym ( x; z ) ; (3.34) G adv ( x; z ) = 2 [ z ; ( x )] G sym ( x; z ) : (3.35)

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24 Maxw ell's equations for it b ecome r 2 A R A = 4 J ; (3.46) and, again, the goal is to express the v ector p oten tial as an in tegral that con tains a Green's function and the source J sp ecically A ( x ) = 4 Z spacetime ( g ) 1 2 G ( x; y ) J ( y ) d 4 y : (3.47) The symmetric Green's function for a v ector eld ob eys the equation r 2 G sym 0 ( x; z ) R G sym 0 ( x; z ) = g 1 2 g 0 4 ( x; z ) (3.48) and the Hadamard form of it is G sym 0 ( x; z ) = 1 8 [ U 0 ( x; z ) ( ) V 0 ( x; z ) ( )] : (3.49) The biscalar U 0 is giv en b y the dieren tial equation (2 r U 0 + U 0 1 r ) r = 0 (3.50) with the b oundary condition lim x z U 0 ( x; z ) = g 0 ( z ) : (3.51) As sho wn in [14], the solution is unique U 0 ( x; z ) = p ( x; z ) g 0 ( x; z ) = [1 + 1 12 R 0 r 0 r 0 r r 0 + O ( s 3 )] g 0 ; for x z (3.52)

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25 where Equation (3.18) w as used to deriv e the nal expression. F or the biv ector V 0 DeWitt and Brehme pro v e that lim x z V 0 ( x; z ) = 1 2 g 0 ( R 0 0 1 6 g 0 0 R ) + O ( s ) ; for x : (3.53) The symmetric electromagnetic v ector p oten tial is calculated b y A sym ( x ) = 4 Z ( g ) 1 2 G sym ( x; y ) J ( y ) d 4 y = 1 2 Z ( g ) 1 2 n [( x; y )] 1 2 g ( x; y ) ( ) V ( x; y ) ( ) o J ( y ) d 4 y : (3.54) Just as in the case of the symmetric scalar eld, the symmetric electromagnetic p otential also consists of the direct part, that con tains the -function and comes from the retarded and adv anced prop er times, and the tail part, that con tains the -function and is the con tribution from within the past and future n ull cone. An in teresting feature is the app earance of the biv ector of geo desic parallel displacemen t in the direct part. It signies that the electromagnetic radiation is parallel propagated along the n ull geo desic that connects the p oin ts x and z The symmetric Green's function can again b e separated in to the retarded and adv anced parts G sym 0 ( x; z ) = 1 2 [ G ret 0 ( x; z ) + G adv 0 ( x; z )] (3.55) whic h are giv en b y the equations G ret 0 ( x; z ) = 2 [( x ) ; z ] G sym 0 ( x; z ) (3.56) G adv 0 ( x; z ) = 2 [ z ; ( x )] G sym 0 ( x; z ) : (3.57)

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28 Assuming that a massiv e p oin tlik e particle is mo ving on a geo desic : z 0 ( ) and is inducing a p erturbation h ( x ) on the bac kground, the trace-rev ersed metric p erturbation is h ( x ) = h ( x ) 1 2 g ( x ) h ( x ) (3.70) where h ( x ) and h ( x ) are the traces of h ( x ) and of h ( x ) resp ectiv ely It is assumed that the trace-rev ersed metric p erturbation ob eys the harmonic gauge condition r h ( x ) = 0 : (3.71) In this gauge, the linearized Einstein equations [18] b ecome: r 2 h + 2 R r h r = 16 T (3.72) to rst order in the metric p erturbation. The Green's functions of in terest in this case are bitensors and are used to express the trace-rev ersed metric p erturbation in in tegral form h ( x ) = 16 Z spacetime ( g ) 1 2 G r ( x; y ) T r ( y ) d 4 y : (3.73) The symmetric Green's function G r 0 0 ( x; z ) satises the dieren tial equation r 2 G r 0 0 sym ( x; z ) + 2 R ( x ) G r 0 0 sym ( x; z ) = 2 g r 0 ( ( x; z ) g ) 0 ( x; z ) 4 ( x z ) ( g ) 1 2 (3.74) and the Hadamard form for it is G r 0 0 sym ( x; z ) = 1 8 [ U r 0 0 ( x; z ) ( ) V r 0 0 ( x; z )( )] : (3.75)

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29 The bitensor U r 0 0 ( x; z ) is the solution of the homogeneous dieren tial equation [2 r U r 0 0 ( x; z ) + r U r 0 0 ( x; z )] r = 0 (3.76) with the b oundary condition lim x z U r 0 0 ( x; z ) = lim x z 2 g r 0 ( ( x; z ) g ) 0 ( x; z ) : (3.77) Mino, Sasaki and T anak a [15] pro v e that the solution is U r 0 0 ( x; z ) = 2 g r 0 ( ( x; z ) g ) 0 ( x; z ) p ( x; z ) : (3.78) The bitensor V r 0 0 ( x; z ) is div ergence-free r V r 0 0 ( x; z ) = 0 (3.79) and satises the homogeneous dieren tial equation r 2 V r 0 0 ( x; z ) + 2 R V r 0 0 ( x; z ) = 0 (3.80) with the b oundary condition lim 0 V r 0 0 ( x; z ) = 0 : (3.81) It is pro v en in [15] that the solution is V r 0 0 ( x; z ) = g 0 g 0 R r 0 0 0 0 ( z ) + O ( r ) ; for x : (3.82)

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30 The symmetric gra vitational eld is obtained b y h sym ( x ) = 16 Z spacetime ( g ) 1 2 G r sym ( x; y ) T r ( y ) d 4 y = 2 Z spacetime ( g ) 1 2 [ U r ( x; y ) ( ) V r ( x; y )( )] T r ( y ) d 4 y : (3.83) Its direct part is the part that con tains the -function and giv es the con tribution from the n ull cone of p oin t x Its tail part is the part that con tains the -function and giv es the con tribution that comes from within the n ull cone of p oin t x The symmetric Green's function can b e separated to the retarded and adv anced Green's functions G r 0 0 sym ( x; z ) = 1 2 [ G r 0 0 ret ( x; z ) + G r 0 0 adv ( x; z )] (3.84) whic h are giv en b y the equations G r 0 0 ret ( x; z ) = 2 [( x ) ; z ] G r 0 0 sym ( x; z ) (3.85) G r 0 0 adv ( x; z ) = 2 [ z ; ( x )] G r 0 0 sym ( x; z ) : (3.86) They are solutions of the inhomogeneous dieren tial equation r 2 G r 0 0 ret ( x; z ) + 2 R ( x ) G r 0 0 ret ( x; z ) = = r 2 G r 0 0 adv ( x; z ) + 2 R ( x ) G r 0 0 adv ( x; z ) = 2 g r 0 ( ( x; z ) g ) 0 ( x; z ) 4 ( x z ) ( g ) 1 2 (3.87) and ha v e the prop ert y that G r 0 0 ret ( x; z ) = G r 0 0 adv ( z ; x ) : (3.88)

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32 and is a solution of the homogeneous dieren tial equation r 2 h rad ( x ) + 2 R h rad ( x ) = 0 : (3.95) 3.3 Motion of a Charged P article in Curv ed Spacetime F ollo wing DeWitt and Brehme's analysis, I presen t the equations of motion and the radiation reaction eects for a particle carrying an electric c harge e Studying the v ector case is useful in comparing the result deriv ed in curv ed spacetime with Dirac's result for rat spacetime. Similar analyses can b e done for the scalar case and the gra vitational case. Those analyses can b e found in a recen t review of the sub ject b y P oisson [19]. 3.3.1 Equations of Motion Let's assume that a particle of bare mass m 0 and electric c harge e is mo ving in a curv ed spacetime of metric g on a w orldline describ ed b y : z ( ), where is the prop er time. In the follo wing, the notation u 0 = ( dz 0 =d ) = z 0 is used. Also, the sym b ol ( D =d ) is used to denote absolute co v arian t dieren tiation with resp ect to the prop er time along the w orldline. The particle generates an electromagnetic v ector p oten tial A ( x ) and an electromagnetic eld F ( x ) giv en b y Equation (3.67). The action of the system con tains three terms; the rst term comes from the particle, the second from the electromagnetic eld and the third from the in teraction b et w een the t w o: S = S par ticl e + S f iel d + S inter action ; (3.96) where S par ticl e = m 0 Z [ g 0 0 ( z ) u 0 u 0 ] 1 2 d (3.97)

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33 S f iel d = 1 16 Z F F p g d 4 x (3.98) S inter action = e Z A 0 ( z ) u 0 d : (3.99) In determining the equations of motion, the stationary action principle can b e applied to S for v ariations in the p oten tial A and for v ariations in the w orldline z 0 separately V arying A giv es r F = 4 J (3.100) where J is the curren t densit y dened as J ( y ) = e Z g 0 ( y ; z ) u 0 4 ( y ; z ) d : (3.101) The electromagnetic eld F is in v arian t under a gauge transformation of the form A A + r ( x ), where ( x ) is an y scalar function of x F or con v enience, ( x ) can b e c hosen in suc h a w a y that the electromagnetic p oten tial satises the Loren tz gauge condition r A = 0 (3.102) in whic h case Equation (3.100) for the electromagnetic eld b ecomes r 2 A ( x ) R A ( x ) = 4 J ( x ) : (3.103) Equation (3.103) allo ws one to determine the electromagnetic eld, once the w orldline of the c harged particle is kno wn. On the other hand, v arying the w orldline z 0 giv es m 0 D u 0 d = eF 0 0 ( z ) u 0 : (3.104)

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34 Equation (3.104) allo ws one to calculate the w orldline of the particle, once the electromagnetic eld is sp ecied. As is w ell-kno wn, the electromagnetic eld giv en b y Equation (3.103) is div ergen t at the lo cation of the particle and cannot b e used in Equation (3.104). An alternativ e metho d for studying the motion of the particle is to tak e adv an tage of the conserv ation of energy and momen tum for the particle and the eld, the same metho d that w as used b y Dirac in his analysis of the motion of an electron in rat spacetime. 3.3.2 Calculation of the Fields It b ecomes ob vious in Section (3.3.3) that in order for the conserv ation of energy and momen tum to b e implemen ted along the w orldline of the c harged particle, it is necessary for the retarded and adv anced elds generated b y the particle to b e calculated. A brief description of the calculation is giv en here; the details can b e found in [14]. In the follo wing, ret and adv are the prop er times along the w orldline of the particle, where the past and future n ull cones of x resp ectiv ely in tersect that w orldline. They are referred to as retarded and adv anced prop er times. Also, is the prop er time at the p oin t where the h yp ersurface ( x ) in tersects the w orldline of the particle. All these p oin ts are sho wn in Figure (3.1). null coneS x t t (x) adv S t z ret Figure 3.1: Retarded and adv anced prop er times

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35 Starting with Equation (3.60) for the retarded p oten tial, and using the Hadamard expansion for the Green's function, a simple expression for A ret can b e obtained A ret ( x ) = 4 Z d 4 y ( g ) 1 2 G ret ( x; y ) J ( y ) = 4 e Z d 4 y ( g ) 1 2 2 [( x ) ; y ] [ U ( x; y ) ( ) V ( x; y ) ( )] 8 Z d g 0 ( y ; z ) u 0 4 ( y ; z ) = e Z 1 U 0 ( x; z ) ( ) u 0 d e Z 1 V 0 ( x; z ) ( ) u 0 d = e h U 0 u 0 r 0 u 0 i ret e Z ret 1 V 0 ( x; z ) u 0 d : (3.105) By a similar calculation, starting with Equation (3.61), the adv anced p oten tial can b e calculated A adv ( x ) = e h U 0 u 0 r 0 u 0 i adv e Z + 1 adv V 0 ( x; z ) u 0 d : (3.106) Equations (3.105) and (3.106) giv e the co v arian t Lienard-Wiec hert p oten tials. The retarded and adv anced electromagnetic elds can b e calculated b y Equation (3.67) if the resp ectiv e p oten tials are dieren tiated. The result is F ret/adv = e ( U 0 r U 0 r ) u 0 ( r 0 r 0 u 0 u r 0 + r 0 D u 0 d ) ( r 0 u 0 ) 3 [ r 0 ( U 0 r U 0 r ) u 0 u 0 + ( U 0 r U 0 r ) D u 0 d ] ( r r 0 u r 0 ) 2 + ( r U 0 r U 0 + V 0 r V 0 r ) u 0 ( r 0 u 0 ) 1 ret = adv e Z 1 ret = adv ( r V 0 r V 0 ) u 0 d : (3.107)

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36 In Equation (3.107), the lo w er of the t w o signs corresp onds to the retarded eld and the upp er sign corresp onds to the adv anced eld. This is an imp ortan t expression b ecause it giv es the electromagnetic eld at p oin t x in terms of the c haracteristics of the w orldline of the c harged particle. The signicance of this b ecomes clear when the nal equations of motion of the c harged particle are written do wn. As b ecomes ob vious in Section (3.3.3), when calculating the balance of energy and momen tum b et w een the particle and the eld along the w orldline, the v alues of the elds v ery close to the w orldline need to b e kno wn. Expanding the retarded and adv anced elds in p o w ers of and its deriv ativ es is a tedious but straigh tforw ard calculation, whic h yields a complicated expression for the t w o elds, Equation (5.12) of [14]. F ollo wing Dirac's ideas, the total eld in the vicinit y of the c harged particle can b e written as the sum of the retarded eld created b y the particle plus the external incoming eld F = F ret + F in (3.108) whic h, b y using Equations (3.68) and (3.69), can b e written in terms of the radiation and symmetric elds F = F in + 1 2 F rad + F sym : (3.109) This breaking up of the total eld is v ery imp ortan t b ecause, after implemen ting the conserv ation of energy and momen tum along the w orldline of the particle, only the radiation eld F rad and the external incoming eld F in sho w up in the equations of motion of the c harged particle. Since the radiation eld is the dierence b et w een the retarded and adv anced elds, only t w o terms of Equation (3.107) are of in terest here. Sp ecically it is

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37 sucien t to write F ret/adv = e ( g 0 g 0 g 0 g 0 ) 2 3 D 2 d 2 2 D u 0 d u 0 1 2 Z 1 f 0 0 r 0 ( z ( ) ; z ( 0 )) u r 0 d 0 # + F extra + (higher order terms) (3.110) where, again, the lo w er sign corresp onds to the retarded eld and the upp er sign corresp onds to the adv anced eld. In Equation (3.110), the notation f 0 ( x; z ) = r V 0 ( x; z ) r V 0 ( x; z ) (3.111) is used. The term F extra is, in fact, a sum of terms and is the same for the retarded and the adv anced elds, so it v anishes when the dierence of the t w o is tak en. 3.3.3 Conserv ation of Energy and Momen tum The rst step, when considering the conserv ation of energy and momen tum, is the construction of a cylindrical surface that surrounds the w orldline of the c harged particle. That surface is referred to as the w orld-tub e from no w on. As in Dirac's analysis, the radius of the w orld-tub e is v ery small, b ecause the balance of energy and momen tum b et w een the particle and the eld needs to b e calculated in the vicinit y of the particle's tra jectory DeWitt and Brehme [14] explain the pro cedure for the construction of the w orld-tub e in great detail. F or the needs of the presen t calculation, the w orld-tub e can b e though t of as the three-dimensional surface that is generated b y the particle if the particle is surrounded b y a sphere of a v ery small radius and is then left to mo v e along its w orldline, starting at prop er time 1 and ending at prop er time 2 In the follo wing, the cylindrical surface is denoted b y its t w o end caps are denoted b y 1 (corresp onding to the earliest prop er time 1 ) and 2 (corresp onding

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38 to the latest prop er time 2 ). The total v olume enclosed b y the w orld-tub e is denoted b y V t The stress-energy tensor T of the system consists of one part that comes from the particle and one part that comes from the eld T = T P + T F (3.112) where T P = m 0 Z 1 2 g 0 g 0 u 0 u 0 4 d (3.113) T F = 1 4 ( F r F r 1 4 g F r F r ) (3.114) where F is the total eld in the vicinit y of the particle. It is the sum of the eld generated b y the particle plus an y external incoming eld that is presen t. The conserv ation of energy and momen tum for the system can b e expressed as r T = 0 : (3.115) When in tegrating Equation (3.115) o v er the w orld-tub e, sp ecial atten tion should b e giv en to the fact that the in tegral ( R r T d 4 x ) is not a v ector in the curv ed spacetime that is b eing considered here. That means that Gauss's theorem cannot b e used as usual, to con v ert the in tegral o v er the v olume V t to an in tegral o v er the surface of the w orld-tub e. The natural w a y to o v ercome this dicult y is to consider the in tegral R g 0 r T d 4 x instead. In this in tegral, the biv ector of geo desic parallel displacemen t is used, so that an y con tributions to the in tegral b y the p oin t x are referred bac k to the xed p oin t z whic h is assumed to corresp ond to prop er time That in tegral is,

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39 then, a con tra v arian t v ector at p oin t z so Gauss's theorem can b e used. That giv es Z + Z 1 + Z 2 g 0 T d Z V t r g 0 T d 4 x = 0 (3.116) where d is the surface elemen t along the surface of the w orld-tub e. The limit of the radius of the w orld-tub e going to zero is tak en next. Also the in tegral o v er the surface of the w orld-tub e is expressed as a double in tegral, o v er the prop er time and o v er the solid angle. In addition, the prop er times 1 and 2 are let to approac h If their innitesimal dierence is denoted b y d Equation (3.116) b ecomes m 0 D u 0 d d + lim 0 Z 4 g 0 T d = 0 : (3.117) It is in the calculation of the in tegral of the second term that the symmetric eld and the radiation eld men tioned in Equations (3.68), (3.69) and (3.109) are v ery useful. As w as men tioned earlier, the result con tains only the incoming external eld, the radiation eld and the tail eld and is m D u 0 d = eF in 0 0 u 0 + 1 2 eF rad 0 0 u 0 + 1 2 e 2 u 0 Z 1 f 0 0 r 0 ( z ( ) ; z ( 0 )) u r 0 ( 0 ) d 0 (3.118) where m is the renormalized observ ed mass of the c harged particle m = m 0 + lim 0 1 2 e 2 1 : (3.119) An equiv alen t expression is deriv ed b y using Equation (3.110) and the denition of the radiation eld giv en in Equation (3.69), and it is m D u 0 d = eF in 0 0 u 0 + 2 3 e 2 ( D 2 u 0 d 2 u 0 D u 0 d D u 0 d )+ + 1 2 e 2 u 0 Z 1 f 0 0 r 0 ( z ( ) ; z ( 0 )) u r 0 ( 0 ) d 0 : (3.120)

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40 Equation (3.120) con tains only the ph ysical c haracteristics of the w orldline of the c harged particle and the incoming external eld. 3.3.4 Radiation Reaction A short discussion ab out Equations (3.118) and (3.120) is in order. First it is signican t to notice that, except for the last term that con tains the quan tit y f 0 these equations are iden tical to Equations (2.34) and (2.35) deriv ed b y Dirac for rat spacetime. Since that term con tains the biscalar V 0 it comes from the tail part of the retarded electromagnetic eld. Also, it is in teresting that the in tegral that sho ws up in these t w o equations needs to b e calculated from 1 to the prop er time meaning that kno wledge of the en tire past history of the particle is required. That is something that renders the use of Equation (3.120) impractical and is a p oin t that is also discussed in the follo wing c hapters.

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CHAPTER 4 SELF-F OR CE In general, the self-force can b e due to a scalar, an electromagnetic or a gra vitational eld. Tw o dieren t metho ds for the self-force calculation are presen ted in this c hapter. The analysis is giv en in detail for the scalar self-force, due to the simplicit y of the notation. The results are also giv en for the electromagnetic and gra vitational self-force. The rst metho d, in whic h the direct and tail parts of the retarded elds are used, w as describ ed b y DeWitt and Brehme in [14] for the scalar and electromagnetic elds and b y Mino, Sasaki and T anak a in [15] for the gra vitational eld. An axiomatic approac h to this metho d w as also presen ted b y Quinn [20] for the scalar eld and b y Quinn and W ald [21] for the electromagnetic and gra vitational elds. The second metho d, in whic h the singular-source part and the regular remainder of the retarded elds are used, w as prop osed b y Det w eiler and Whiting in [22], where they describ ed it for the scalar, electromagnetic and gra vitational elds. As in Chapter 3, z denotes a p oin t along the w orldline of the mo ving particle and x an y p oin t in spacetime. The limit of x is the coincidence limit. The retarded and adv anced prop er times, ret and adv are the prop er times at the p oin ts where the n ull cone of x in tersects the w orldline (see Figure (3.1)). Primed indices refer to z and unprimed ones refer to x 4.1 Scalar Self-F orce A particle of scalar c harge q is assumed to b e mo ving in a bac kground spacetime, describ ed b y the bac kground metric g ab F or simplicit y it can b e assumed that there is no external scalar eld and consequen tly if the scalar c harge q is small, the lo w est order 41

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42 appro ximation for the w orldline : z a 0 ( ) of the particle is a bac kground geo desic, where is the prop er time. But the scalar eld generated b y the particle in teracts with the particle, inducing a self-force on it. The self-force giv es an acceleration, so, to higher order, the particle do es not trace a bac kground geo desic. 4.1.1 Direct and T ail Fields The discussion of the scalar eld self-force in [14] and [20] sho ws that the selfforce on the particle can b e calculated b y F a = q h r a self i particle (4.1) where self is the scalar eld that in teracts with the particle and is giv en b y self ( x ) = h q U ( x; z ) 2 i adv ret q Z ret 1 V ( x; z ( )) d : (4.2) It can b e inferred from the discussion of the scalar Green's functions of Section (3.2.1) that the rst term of the righ t-hand side of Equation (4.2) comes from the direct part of the retarded eld generated b y the mo ving particle. This rst term is nite and dieren tiable at the lo cation of the particle and its con tribution to the self-force is the curv ed-spacetime generalization of the Abraham-Loren tz-Dirac force of rat spacetime. This con tribution results from the mo ving particle's acceleration, if the w orldline is not a geo desic, and can b e expressed in terms of the acceleration of and comp onen ts of the Riemann tensor [14, 15, 20, 21]. The in tegral term of the righ t-hand side of Equation (4.2) comes from the tail part of the retarded eld of the mo ving particle. Its con tribution to the self-force represen ts the result of the scattering of the retarded eld of the particle, due to the curv ature of spacetime. T aking the deriv ativ e of the tail part of self giv es one term that comes from the implicit dep endence of the retarded prop er time on x and one

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43 term that comes from the dep endence of the in tegrand on x as w as calculated b y Quinn in [20]: r a n q Z ret 1 V ( x; z ( )) d o = q V ( x; z ( ret )) [ r a ret ( x )] q Z ret 1 r a V ( x; z ( )) d = q h V 1 r a i ret q Z ret 1 r a V ( x; z ( )) d = h q R ( x ) 12 r ( x a z a ) i x + O ( r ) q Z ret 1 r a V ( x; z ( )) d : (4.3) In Equation (4.3), r is the prop er distance from x to measured along the spatial geo desic that is orthogonal to The spatial part of the rst term of the deriv ativ e calculated in Equation (4.3) is not w ell dened when x is on unless R ( x ) = 0 there. That mak es this approac h problematic since, for the self-force to b e calculated, the deriv ativ e m ust b e calculated at the lo cation of the particle. T o o v ercome this dicult y r a self can b e a v eraged o v er a small spatial 2-sphere surrounding the particle, th us remo ving the spatial part. Then the limit of the radius of the 2-sphere going to zero can b e tak en and a nite con tribution to the self-force is obtained. Another complication comes from the fact that, in order for the con tribution of the tail term to the self-force to b e calculated, kno wledge of the en tire past history of the mo ving particle is necessary as is clear from the in tegral term in Equation (4.3). It is also imp ortan t to note that the eld self just lik e the eld tail is not a ph ysically w ell-understo o d eld. That is b ecause it is not a solution to an y particular dieren tial equation. 4.1.2 The S-Field and the R-Field In order to pro vide an alternativ e scalar eld that giv es the self-force on the mo ving particle, Det w eiler and Whiting [22] to ok adv an tage of the fact that adding a

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44 homogeneous solution of a dieren tial equation to an inhomogeneous solution of the same dieren tial equation, giv es a new inhomogeneous solution. Sp ecically since the biscalar V ( x; z ) is a symmetric homogeneous solution of the scalar dieren tial Equation (3.29), it can b e added to the symmetric Green's function G sym ( x; z ) to giv e a new symmetric inhomogeneous solution G S ( x; z ) of Equation (3.22) G S ( x; z ) = G sym ( x; z ) + 1 8 V ( x; z ) = 1 8 [ U ( x; z ) ( ) V ( x; z )( ) + V ( x; z )] = 1 8 [ U ( x; z ) ( ) + V ( x; z )( )] ; (4.4) whic h ob eys r 2 G S ( x; z ) = ( g ) 1 2 4 ( x z ) : (4.5) The sup erscript S is used to indicate that this Green's function ob eys the dieren tial equation that con tains the Source term. This new symmetric Green's function has supp ort on the n ull cone, coming from the term that con tains the -function, and outside of the n ull cone, at spacelik e separated p oin ts, coming from the term that con tains ( ). It has no supp ort within the n ull cone, as sho wn in Figure (4.1). The lo cal expansion of the biscalar V ( x; z ) giv en in Equation (3.30) is sucien t for the purp oses of this dissertation, since the singular Green's function is only used for p oin ts x close to the w orldline of the particle. The Regular-Remainder Green's function is dened in terms of the retarded and singular Green's functions as G R ( x; z ) = G ret ( x; z ) G S ( x; z ) = = 1 8 n 2[( x ) ; z ][ U ( x; z ) ( ) V ( x; z )( )] [ U ( x; z ) ( ) + V ( x; z )( )] o ; (4.6)

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45 null conex tadvt zretGtailGGdirectSand Figure 4.1: Supp ort of G direct G tail and G S and it is ob vious that, b y construction, it ob eys the homogeneous dieren tial equation r 2 G R ( x; z ) = 0 : (4.7) Clearly G R ( x; z ) has no supp ort inside the future n ull cone of x Using the new Green's functions, the elds S and R can b e dened. First, the singular eld is S ( x ) = 4 q Z G S [ x; z ( )] d = h q U ( x; z ) 2 i ret + h q U ( x; z ) 2 i adv + q 2 Z adv ret V ( x; z ) d ; (4.8) and ob eys P oisson's equation r 2 S ( x ) = 4 %: (4.9)

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46 It is notew orth y that the singular eld do es not dep end on the en tire past history of the mo ving particle, but only on its motion b et w een the retarded and adv anced prop er times. The regular-remainder is dened as R ( x ) = ret ( x ) S ( x ) = h q U ( x; z ) 2 i adv ret q Z ret 1 + 1 2 Z adv ret # V ( x; z ) d ; (4.10) and b y denition ob eys the homogeneous dieren tial equation r 2 R ( x ) = 0 : (4.11) Since the eld R is a source-free eld, it is smo oth and dieren tiable at an y p oin t in spacetime and consequen tly at an y p oin t along the w orldline of the particle as w ell. That is the most signican t prop ert y of it, as far as the calculation of the self-force is concerned. T o compare the scalar elds self and R the dierence of the t w o is calculated from Equations (4.10) and (4.2) R ( x ) self ( x ) = q 2 Z adv ret V ( x; z ) d : (4.12) Using Equation (3.30) for the biscalar V ( x; z ) to expand the ab o v e in tegrand in the coincidence limit, the dierence of the t w o elds b ecomes R ( x ) self ( x ) = q 2 Z adv ret 1 12 R ( x ) + O ( r ) d = 1 12 q r R ( x ) + O ( r 2 ), for x (4.13)

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47 where the fact that the dierence b et w een the adv anced and the retarded prop er times is equal to adv ret = 2 r + O ( r 2 ) ; for x (4.14) w as used. T aking the deriv ativ e giv es r a R = r a self + q 12 r a [ r R ( x )] + (terms that v anish as x ) : (4.15) The second term of the righ t-hand side of Equation (4.15) giv es an out w ardly p oin ting spatial unit v ector and cancels the rst term of the righ t side of Equation (4.3). Consequen tly the elds R and self giv e the same self-force. In addition, the a v eraging pro cedure that w as necessary for the calculation of the self-force using self is not required when R is used. A self-force calculation that has b een demonstrated in [23] sho w ed an additional b enet of calculating the self-force using the dierence of the retarded and singular elds. The retarded eld can b e computed n umerically and, since the singular eld dep ends only on the motion of the particle b et w een the retarded and adv anced prop er times, the kno wledge of the en tire past history of the particle is not required. That is a signican t adv an tage of this metho d compared to the one that in v olv es the direct and tail elds. Additionally it is imp ortan t that all the scalar elds used in this analysis are sp ecic solutions of the homogeneous or inhomogeneous Equation (3.19). That mak es them w ell-dened ph ysical elds, a prop ert y that the direct and tail scalar elds do not ha v e. F or those reasons, in this dissertation the self-force is calculated using F a = q r a R : (4.16)

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48 4.2 Electromagnetic Self-F orce A detailed analysis for the self-force in terms of the direct and tail electromagnetic p oten tials w as done in [14] and w as presen ted in Chapter 3. In this section, the calculation of the self-force using the singular-source and the regular remainder elds is presen ted. It is assumed that a particle of electric c harge q is mo ving on a w orldline : z a 0 ( ) in a bac kground describ ed b y the metric g ab F or the purp oses of this section it can b e assumed that there is no external electromagnetic eld and the lo w est order appro ximation to the particle's motion is a bac kground geo desic. The particle creates an electromagnetic p oten tial A a and an electromagnetic eld F ab whic h in teract with the particle, causing a self-force to act on it and making its w orldline to deviate from a bac kground geo desic to order q 2 In the Loren tz gauge r a A a = 0 ; (4.17) the electromagnetic p oten tial generated b y the particle can b e calculated b y using Maxw ell's equations r 2 A a R a b A b = 4 J a (4.18) and the dieren t solutions of in terest w ere describ ed b y the Green's functions of Section (3.2.2). If a p oten tial A selfa analogous to self is used to calculate the self-force in this case, its tail part is A self (tail) a ( x ) = q Z ret 1 V ab 0 ( x; z ( )) u b 0 d : (4.19)

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49 T aking the deriv ativ e of it giv es r c A self (tail) a = q h V ab 0 u b 0 r c ret i x q Z ret 1 r c ( V ab 0 u b 0 ) d = q h 1 2 g d 0 a ( R b 0 d 0 1 6 g b 0 d 0 R ) u b 0 x c z c r i x q Z ret 1 r c [ V ab 0 u b 0 ] d ; (4.20) where Equation (3.53) for the biv ector V ab 0 w as used to deriv e the last expression. It is ob vious that the same problems that w ere encoun tered in the scalar case are encoun tered here as w ell. The rst term is not w ell-dened unless ( R a 0 b 0 1 6 g a 0 b 0 R ) u b 0 is zero at the particle's w orldline, the en tire past history of the mo ving particle needs to b e kno wn and the p oten tial used in the self-force calculation is not a solution of Maxw ell's equations. T o o v ercome those diculties, the singular Green's function in the neigh b orho o d of the particle is dened analogously to the scalar case G Saa 0 ( x; z ) = G symaa 0 ( x; z ) + 1 8 V aa 0 ( x; z ) = 1 8 [ U aa 0 ( x; z ) ( ) + V aa 0 ( x; z )( )] : (4.21) It ob eys the inhomogeneous dieren tial equation r 2 G Saa 0 ( x; z ) R b a G Sba 0 ( x; z ) = g 1 2 g aa 0 4 ( x; z ) (4.22) and giv es the singular electromagnetic p oten tial A Sa = 4 Z ( g ) 1 2 G Sab ( x; y ) J b ( y ) d 4 y ; (4.23) whic h is a solution of the inhomogeneous Maxw ell's Equations (4.18). By denition, b oth G Sab 0 and A Sa ha v e no supp ort inside the past and future n ull cone of x The

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50 regular remainder is dened b y A Ra ( x ) = A reta ( x ) A Sa ( x ) : (4.24) It has no supp ort within the future n ull cone of x and it ob eys the homogeneous Maxw ell's equations. Consequen tly it is smo oth and dieren tiable ev erywhere in space, including ev ery p oin t along the w orldline of the particle. The electromagnetic self-force can b e calculated b y the equation F a = q g ac ( r c A Rb r b A Rc ) z b (4.25) and is w ell-dened at the particle's lo cation. It is stressed, again, that b oth A Sa and A Ra are w ell-dened solutions of Maxw ell's equations. 4.3 Gra vitational Self-F orce It is no w assumed that a particle of mass m is mo ving on a w orldline : z a 0 ( ), in a bac kground describ ed b y the metric g ab The particle causes a metric p erturbation h ab on the bac kground metric. This p erturbation ob eys the harmonic gauge r a h ab = 0 (4.26) where h ab is the trace-rev ersed v ersion of the metric p erturbation h ab = h ab 1 2 g ab h c c : (4.27) The linearized Einstein equations for it are r 2 h ab + 2 R c d a b h cd = 16 T ab : (4.28)

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51 This metric p erturbation in teracts with the particle, causing a self-force to act on it and forcing its w orldline to deviate from a bac kground geo desic. If h selfab whic h is not a solution of Einstein's equations, is used to calculate the self-force, its tail part con tains the bitensor V abc 0 d 0 giv en in Equation (3.82). The con tribution of this tail term to the deriv ativ e comes partly from the dep endence of the prop er time ret on the p oin t x Sp ecically r f h self (tail) ab = m h V abc 0 d 0 u c 0 u d 0 r f ret i x m Z ret 1 r f V abc 0 d 0 u c 0 u d 0 d = m h g a 0 a g b 0 b R c 0 a 0 d 0 b 0 ( x f z f ) r i x m Z ret 1 r f V abc 0 d 0 u c 0 u d 0 d : (4.29) Just lik e b efore, the rst term is not w ell-dened at the particle's w orldline, unless R c 0 a 0 d 0 b 0 u c 0 u d 0 equals zero along the w orldline. Also, kno wledge of the en tire past history of the particle is required in order to calculate the in tegral term. The singular Green's function G abc 0 d 0 ( x; z ) in the neigh b orho o d of the particle can b e dened as G Sabc 0 d 0 ( x; z ) = G symabc 0 d 0 ( x; z ) + 1 8 V abc 0 d 0 ( x; z ) = 1 8 [ U abc 0 d 0 ( x; z ) ( ) + V abc 0 d 0 ( x; z )( )] : (4.30) It giv es the singular eld h Sab whic h is an inhomogeneous solution of Equation (4.28) and has no supp ort inside the past or future n ull cone of x h Sab = 16 Z ( g ) 1 2 G Sabcd ( x; y ) T cd ( y ) d 4 y : (4.31) The regular remainder is dened as h Rab ( x ) = h retab ( x ) h Sab ( x ) : (4.32)

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52 It ob eys the homogeneous dieren tial equation r 2 h Rab + 2 R c d a b h Rcd = 0 ; (4.33) whic h means that it is smo oth and dieren tiable ev erywhere in spacetime. Also, it has no supp ort within the future n ull cone of x Then the self-force can b e calculated b y F a = m ( g ab + z a z b ) z c z d ( r c h Rdb 1 2 r b h Rcd ) (4.34) whic h is w ell-dened at the lo cation of the particle. It is imp ortan t that b oth h Rab and h Sab are solutions of sp ecic dieren tial equations. A signican t conclusion can b e dra wn at this p oin t. As b ets the problem, the self-force is causing the particle to mo v e on a geo desic of the metric ( g ab + h Rab ), to order m 2 This metric ( g ab + h Rab ) is, itself, a homogeneous solution of the Einstein equations. An extended discussion of that p oin t is giv en in [22] and [24] and is also presen ted in Chapter 8 of this dissertation.

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CHAPTER 5 SINGULAR FIELD F OR SCHW ARZSCHILD GEODESICS It w as seen in Chapter 4 that the singular eld is imp ortan t in calculating the self-force. It is the part that m ust b e subtracted from the retarded eld to giv e the regular remainder, whic h is then dieren tiated to giv e the self-force. It is also clear that, since the singular eld ob eys the inhomogeneous P oisson, Maxw ell or Einstein equations, it dep ends on the w orldline of the mo ving particle and on the bac kground spacetime. The co ordinates that are used in this c hapter to caclulate the singular eld are the Thorne-Hartle-Zhang co ordinates, abbreviated from this p oin t on as THZ co ordinates. Those co ordinates w ere initially in tro duced b y Thorne and Hartle [25] and later extended b y Zhang [26]. A short discussion on them is presen ted in Section (5.1) of this c hapter. The detailed calculation of the singular eld of a scalar c harge q for geo desics in a Sc h w arzsc hild bac kground w as presen ted in [23]. A brief discussion ab out that eld is giv en here for completeness, since the regularization parameters asso ciated with it are calculated in Chapter 6. The singular eld is also calculated for geo desics in a Sc h w arzsc hild bac kground, for a dip ole generating a scalar eld, for an electric c harge, an electric dip ole and a magnetic dip ole eac h generating its o wn electromagnetic eld and for a spinning particle generating a gra vitational eld. F or all these calculations, Maple and Gr tensor w ere used extensiv ely The Gr tensor co de is explained in Section (5.8) and presen ted in App endix A. In Sections (5.3) through (5.7), only the results that the co de giv es are presen ted. 53

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54 5.1 Thorne-Hartle-Zhang Co ordinates The P oisson, Maxw ell and Einstein equations for the singular eld assume a relativ ely simple form when they are written in a co ordinate system in whic h the bac kground spacetime lo oks as rat as p ossible. In the follo wing, it is assumed that the particle is mo ving on a geo desic in a v acuum bac kground describ ed b y the metric g ab Also, R is a represen tativ e length scale of the bac kground geometry the smallest of the radius of curv ature, the scale of inhomogeneities of the bac kground and the time scale of curv ature c hanges along A normal co ordinate system can alw a ys b e found so that, on the geo desic the metric and its rst deriv ativ es coincide with the Mink o wski metric [18]. Suc h a normal co ordinate system is not unique. The one used here is the THZ co ordinate system and is used only lo cally close to the w orldline of the particle. Sp ecically it is assumed that the bac kground metric close to the w orldline of the particle can b e written as g ab = ab + H ab = ab + 2 H ab + 3 H ab + O ( 4 R 4 ) ; (5.1) where ab is the rat Mink o wski metric in the THZ co ordinates ( t; x; y ; z ) and 2 = x 2 + y 2 + z 2 : (5.2) Also 2 H ab dx a dx b = E ij x i x j ( dt 2 + k l dx k dx l ) + 4 3 k pq B q i x p x i dtdx k 20 21 h E ij x i x j x k 2 5 2 E ik x i i dtdx k + 5 21 h x i j pq B q k x p x k 1 5 2 pq i B q j x p i dx i dx j ; (5.3)

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55 3 H ab dx a dx b = O 3 R 3 ab dx a dx b + O 4 R 4 ij dx i dx j : (5.4) In this and the follo wing, a b c and d denote spacetime indices. The indices i; j; k ; l ; n; p and q are spatial indices and, to the order up to whic h the calculations are p erformed, they are raised and lo w ered b y the 3-dimensional rat space metric ij The dot denotes dieren tiation with resp ect to the time t along the geo desic. Also, ij k is the 3-dimensional rat space an tisymmetric Levi-Civita tensor. If H ab consists of the terms giv en in Equation (5.3), the co ordinates are secondorder THZ co ordinates and are w ell dened up to the addition of arbitrary functions of O ( 4 R 3 ). If H ab also includes the terms giv en in Equation (5.4), the co ordinates are third-order THZ co ordinates and are w ell dened up to the addition of arbitrary functions of O ( 5 R 4 ). The tensors E and B are spatial, symmetric and trace-free and their comp onen ts are related to the Riemann tensor on the geo desic b y E ij = R titj (5.5) B ij = 1 2 pq i R pq j t : (5.6) They are of O ( 1 R 2 ) and their time deriv ativ es are of O ( 1 R 3 ). F or the calculation of the singular elds that follo ws, only the rst t w o terms of Equation (5.3) are included in g ab The remaining t w o terms are of O ( 3 R 3 ) and m ust b e included in a higher-order calculation whic h m ust also tak e in to accoun t terms coming from the 3 H ab part of the metric. The `gothic' form of the metric is also dened as g ab = p g g ab (5.7)

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56 and its dierence from the Mink o wski metric is H ab = ab g ab : (5.8) F or the lo w est order terms of H ab it can b e found that H ab = 2 H ab + 3 H ab + O 4 R 4 (5.9) where 2 H tt = 2 E ij x i x j (5.10) 2 H tk = 2 3 k pq B q i x p x i + 10 21 h E ij x i x j x k 2 5 E k i x i 2 i (5.11) 2 H ij = 5 21 h x ( i j ) pq B q k x p x k 1 5 pq ( i B j ) q x p 2 i (5.12) 3 H ta = O 3 R 3 ; 3 H ij = O 4 R 4 : (5.13) Using the simple symmetry prop erties of the tensors E and B the follo wing relationships can b e sho wn for their comp onen ts and the spatial THZ co ordinates ( x; y ; z ): ij k E k l + ik l E k j j k l E k i = 0 (5.14) ij k B k l + ik l B k j j k l B k i = 0 (5.15) and ( ij k E k n x l + ij k E l n x k ik l E k n x j + j k l E k n x i ) x n x l = 0 (5.16) ( ij k B k n x l + ij k B l n x k ik l B k n x j + j k l B k n x i ) x n x l = 0 : (5.17)

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57 These relationships are used in the follo wing sections when calculating the singular elds for dieren t sources to simplify the expressions for those elds. The calculation of the regularization parameters sho wn in Chapter 6 requires one to kno w the exact relationship b et w een the THZ co ordinates and the bac kground Sc h w arzsc hild co ordinates. A general pro cedure for nding that relationship for a giv en geo desic is to try to satisfy the three basic prop erties of the second order THZ co ordinates: On : the co ordinate t measures the prop er time along the geo desic and the spatial co ordinates x; y and z are equal to 0. Also, g ab = ab and the rst deriv ativ es of the metric v anish there. A t linear, stationary order: H ij = O ( x 3 ). The co ordinates satisfy the harmonic gauge: @ a g ab = O ( x 2 ). The details of this pro cedure are describ ed in App endix A of [23]. F or circular orbits in a Sc h w arzsc hild bac kground of mass M the THZ co ordinates ha v e b een calculated in App endix B of [23] using sp ecic prop erties of the spherically symmetric bac kground, whic h is m uc h simpler than follo wing the pro cedure just describ ed. Their functional relationships with the Sc h w arzsc hild co ordinates are giv en here in order to facilitate the discussion of the calculation of the singular elds and the regularization parameters. The Sc h w arzsc hild co ordinates are ( t s ; r ; ; ) and the Sc h w arzsc hild metric is ds 2 = (1 2 M r ) dt 2s + (1 2 M r ) 1 dr 2 + r 2 d 2 + r 2 sin 2 d 2 : (5.18) The circular orbit giv en b y = n t s has orbital frequency equal to n = ( M r 3 o ) 1 2 at Sc h w arzsc hild radius r o

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58 First, the functions X ; Y ; Z whic h are Lie deriv ed b y the Killing v ector a = @ @ t s + n @ @ are c hosen X r r o (1 2 M r o ) 1 2 (5.19) Y r o sin( ) sin ( n t s ) r o 2 M r o 3 M 1 2 (5.20) Z r o cos( ) : (5.21) Then the functions ~ x ~ y z also Lie deriv ed b y the Killing v ector a and the function t are dened in terms of X ; Y ; Z ~ x = r sin cos ( n t s ) r o (1 2 M r o ) 1 2 + M r 2 o (1 2 M r o ) 1 2 h X 2 2 + Y 2 r o 3 M r o 2 M + Z 2 i + M X 2 r 3 o ( r o 2 M )( r o 3 M ) M 2 X 2 + Y 2 ( r o 3 M )(3 r o 8 M ) + 3 Z 2 ( r o 2 M ) 2 + M r 5 o (1 2 M r o ) 1 2 ( r o 3 M ) h M X 4 ( r 2 o r o M + 3 M 2 ) 8( r o 2 M ) + X 2 Y 2 28 (28 r 2 o 114 r o M + 123 M 2 ) + X 2 Z 2 14 (14 r 2 o 48 r o M + 33 M 2 ) + M Y 4 56( r o 2 M ) 2 (3 r 3 o 74 r 2 o M + 337 r o M 2 430 M 3 ) M 2 Y 2 Z 2 (7 r o 18 M ) 4( r o 2 M ) M Z 4 56 (3 r o + 22 M ) i ; (5.22) ~ y = r sin sin ( n t s ) r o 2 M r o 3 M 1 2 + M Y 2 r 3 o h 2 X 2 + Y 2 r o 3 M r o 2 M + Z 2 i + M X Y 14 r 5 o (1 2 M r o ) 1 2 ( r o 3 M ) 2 M X 2 (4 r o 15 M ) + Y 2 (14 r 2 o 69 r o M + 89 M 2 ) + 2 Z 2 ( r o 2 M )(7 r o 24 M )] ; (5.23)

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59 z = r cos + M Z 2 r 3 o ( r o 3 M ) [ X 2 (2 r o 3 M ) + Y 2 ( r o 3 M ) + Z 2 ( r o 2 M )] + M X Z 14 r 5 o (1 2 M r o ) 1 2 ( r o 3 M ) [ M X 2 (13 r o 19 M ) + Y 2 (14 r 2 o 36 r o M + 9 M 2 ) + Z 2 ( r o 2 M )(14 r o 15 M )] ; (5.24) t = t s (1 3 M r o ) 1 2 r n Y (1 2 M r o ) 1 2 + n M Y r 2 o (1 2 M r o ) 1 2 ( r o 3 M ) h X 2 2 ( r o M ) + M Y 2 r o 3 M 3( r o 2 M ) + M Z 2 i + n M X Y 14 r 3 o ( r o 2 M )( r o 3 M ) X 2 ( r 2 o 11 r o M + 11 M 2 ) + Y 2 (13 r 2 o 45 r o M + 31 M 2 ) + Z 2 (13 r o 5 M )( r o 2 M ) : (5.25) Finally x = ~ x cos (n y t s ) ~ y sin (n y t s ) y = ~ x sin (n y t s ) + ~ y cos (n y t s ) (5.26) where n y = n q 1 3 M r o There are t w o co ordinate systems of in terest. The rst system is ( t; ~ x; ~ y ; z ) whic h is a non-inertial co ordinate system that co-rotates with the particle, meaning that the ~ x axis alw a ys lines up the cen ter of the blac k hole and the cen ter of the particle. The ~ y axis is alw a ys tangen t to the spatially circular orbit and the z axis is alw a ys orthogonal to the orbital plane. As w as already men tioned, for the spatial co ordinates of this system it holds that: L ~ x = L ~ y = L z = 0. The second system is ( t; x; y ; z ) and is a lo cally inertial and non-rotating system in the vicinit y of Ho w ev er, when view ed far a w a y from these co ordinates app ear to b e rotating due to Thomas precession, as is clear from the terms in v olving the sine and the cosine of (n y t s ) in Equations (5.26). It is also noted that: L 2 = L z = L r a t = 0, but L x L y and L t are not equal to zero. This second system is used

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60 to calculate the singular elds. In order to a v oid confusion of the eld p oin t x and the source p oin t z with the THZ co ordinates, the notation needs to b e c hanged. The eld p oin t is denoted as p and its co ordinates as x ap and the source p oin t is denoted as p 0 and its co ordinates as x ap 0 5.2 Scalar Field of a Charged P article The singular eld generated b y a particle that carries a scalar c harge q and is mo ving on a geo desic : p 0 ( ) (where, as usual, is the prop er time) of the Sc h w arzsc hild bac kground m ust ob ey P oisson's equation r 2 S = 4 % (5.27) where the r 2 is written in the THZ co ordinates and the source term is % = q Z ( g ) 1 2 4 ( p p 0 ( )) d : (5.28) That singular eld is deriv ed in [23] and is equal to S = q + O ( 3 R 4 ) : (5.29) Here, instead of follo wing the exact calculation of the singular eld, whic h is p erformed in detail in [23] follo wing a similar deriv ation in [27], I pro v e that it do es indeed satisfy the scalar eld equation to the order sp ecied in Equation (5.29). The dieren tial op erator of the scalar eld equation b ecomes, in THZ co ordinates: p g r a r a S = @ a ( ab @ b S ) @ a ( H ab @ b S ) = ab @ a @ b S H ij @ i @ j S 2 H it @ ( i @ t ) S H tt @ t @ t S : (5.30)

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61 If the eld S = ( q = ) is substituted in to this equation, the rst term giv es the exp ected -function singularit y and the last t w o terms v anish since do es not dep end on the time t An explicit calculation sho ws that for the second term, 2 H ij @ i @ j is equal to zero. The remainder 3 H ij giv es a term that scales as O ( R 4 ). So p g r a r a q = 4 q 3 ( x i ) + O ( R 4 ), for R 0 : (5.31) Consequen tly for the remainder to b e remo v ed a term of O ( 3 R 4 ) m ust b e added to q = So S = q = + O ( 3 R 4 ) is an inhomogeneous solution of the scalar w a v e equation and the error in this appro ximation is C 2 5.3 Scalar Field of a Dip ole The calculation of the singular scalar eld generated b y a dip ole mo ving on a geo desic in a Sc h w arzsc hild bac kground is presen ted in this section. The dip ole momen t is assumed to ha v e a random orien tation and its THZ comp onen ts are denoted as K a = (0 ; K x ; K y ; K z ). In this and the follo wing sections, the subscripts (or sup erscripts) (0), (1) and (2) are used to indicate the order of signicance of eac h term or comp onen t. The subscript (0) refers to the most dominan t con tribution, the subscript (1) refers to the next most signican t correction, whic h is calculated for the singular elds, and the subscript (2) refers to the next correction, the order of whic h is predicted for the singular elds. It is imp ortan t to realize that m ultiplying t w o terms of order (1) do es not necessarily giv e a term of order (2), b ecause of the fact that there is no O ( R ) correction to the metric (see Equation (5.1)) and the fact that the rst correction to the metric is of O ( 2 R 2 ) while the second is of O ( 3 R 3 ). The scalar eld of a dip ole can b e though t of as ha ving the form S = S(0) + S(1) + S(2) (5.32)

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62 and it ob eys the dieren tial equation r 2(0+1+2) S = r 2(0+1+2) ( S(0) + S(1) + S(2) ) = 4 % (5.33) where the source term is giv en b y % = Z K i r i 4 ( p p 0 ( )) p g d : (5.34) The zeroth-order term is the scalar eld generated b y a dip ole that is stationary at the origin of a Cartesian co ordinate system and it ob eys the lo w est order dieren tial equation r 2(0) S(0) = 4 %: (5.35) It is equal to S(0) = K i x i 3 = K x x + K y y + K z z ( x 2 + y 2 + z 2 ) 3 = 2 : (5.36) The rst-order term ob eys the dieren tial equation deriv ed from Equation (5.33) r 2(0) S(1) + r 2(1) S(0) = 0 ) r 2(0) S(1) = r 2(1) S(0) (5.37) whic h means that the r 2(1) of the zeroth-order part of the eld is the source term in the scalar dieren tial equation for the rst-order part of the eld. In general, that source term is exp ected to con tain the E 's and the B 's and to giv e the rst-order correction coming from the dip ole's motion on the Sc h w arzsc hild geo desic. In this case, Equation (5.37) giv es that @ 2 @ x 2 + @ 2 @ y 2 + @ 2 @ z 2 S(1) = 0 (5.38) so the rst order correction to the eld can b e set equal to zero.

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63 The source for the next order correction comes from the part of the r 2 that is of O ( 3 R 3 ) acting on the zeroth-order scalar eld S(0) The dieren tial equation is of the form @ i @ j S(2) = O ( 3 R 3 ) @ i @ j S(0) O ( 1 R 3 ) : (5.39) That means that the next order term m ust b e of O ( R 3 ). Finally the singular scalar eld of a dip ole mo ving on a Sc h w arzsc hild geo desic is equal to S = K x x + K y y + K z z 3 + O ( R 3 ) : (5.40) 5.4 Electromagnetic P oten tial of a Charged P article In this section, the singular electromagnetic p oten tial generated b y a c harge q mo ving on a Sc h w arzc hild geo desic is calculated. Its general form is A aS = A aS(0) + A aS(1) + A aS (2) : (5.41) Since in the v acuum bac kground the Ricci tensor is R ab = 0 ; (5.42) this electromagnetic p oten tial, as w ell as all those calculated in this c hapter, ob ey the v acuum Maxw ell's equations in curv ed spacetime r 2(0+1+2) A aS(0+1+2) = 4 J a : (5.43) The zeroth-order term ob eys the dieren tial equation r 2(0) A aS (0) = 4 J a (5.44)

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64 the source term b eing J a = q Z ( g ) 1 2 4 [ p p 0 ( )] d ; 0 ; 0 ; 0 : (5.45) The solution of this dieren tial equation is the w ell-kno wn Coulom b electromagnetic p oten tial A aS(0) = q ; 0 ; 0 ; 0 : (5.46) The rst-order correction to this electromagnetic p oten tial ob eys the dieren tial equation deriv ed from Equation (5.43) r 2(0) A aS(1) = r 2(1) A aS(0) : (5.47) The rst-order part of r 2 (whic h con tains the E 's and the B 's) acting on the zerothorder electromagnetic p oten tial is the source term for the rst-order correction. Substituting the THZ comp onen ts of A aS (1) A aS(1) = ( A tS(1) ; A xS(1) ; A yS(1) ; A zS(1) ) (5.48) in to the dieren tial equation results in four dieren tial equations, one for eac h one of these comp onen ts. Eac h equation relates a sp ecic sum of second deriv ativ es of a comp onen t to a sum of terms of the form: q E :: x : x : x : x : 5 for the t -comp onen t, q B :: x : x : 3 for the spatial comp onen ts ; (5.49) where the dots denote appropriate indices.

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65 Solving these four equations is straigh tforw ard, once one notices that the solution should ha v e the form q E ij x i x j for the t -comp onen t, q p ij B i k x k x j for the p-spatial comp onen t, (5.50) eac h term m ultiplied b y an appropriate algebraic factor. Substituting these expressions in to the dieren tial equations giv es a set of simple algebraic equations for these factors, whic h can b e easily solv ed to giv e the nal expression for the rst-order correction: A aS (1) = q 2 E ij x i x j ; q 2 x ij B i k x j x k ; q 2 y ij B i k x j x k ; q 2 z ij B i k x j x k : (5.51) The next order correction comes from the part of the r 2 that is of O ( 3 R 3 ) acting on the zeroth-order electomagnetic p othen tial. It giv es a dieren tial equation for eac h comp onen t of A aS(2) of the form @ i @ j A aS (2) = O ( 3 R 3 ) @ i @ j A aS(0) O ( 1 R 3 ) ; (5.52) whic h indicates that the next order correction m ust b e of O ( 2 R 3 ). It is notew orth y that the rst-order correction A aS (1) do es not app ear in the equation for the second-order correction. That is b ecause it only sho ws up in terms that in v olv e the O ( 2 R 2 ) part of the metric, whic h are of the form O ( 2 R 2 ) @ i @ j A aS (1) O ( R 4 ) (5.53) and m ust b e included in a higher-order calculation.

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66 Finally the singular electromagnetic p oten tial of a c harge q that is mo ving on a geo desic in a Sc h w arzsc hild bac kground is equal to A aS = q [1 1 2 E ij x i x j ] ; 1 2 x ij B i k x j x k ; 1 2 yij B i k x j x k ; 1 2 z ij B i k x j x k + O ( 2 R 3 ) : (5.54) 5.5 Electromagnetic P oten tial of an Electric Dip ole The calculation of the singular electromagnetic p oten tial of an electric dip ole is presen ted in this section. The dip ole momen t is assumed to p oin t at some random direction and its THZ comp onen ts are q a = (0 ; q x ; q y ; q z ). The singular electromagnetic p oten tial can b e written as A aS = A aS (0) + A aS (1) + A aS(2) (5.55) and ob eys the v acuum Maxw ell's equations r 2(0+1+2) A aS (0+1+2) = 4 J a (5.56) where the source is J a = Z q i r i 4 ( p p 0 ( )) p g d ; 0 ; 0 ; 0 : (5.57) The zeroth-order term is the electromagnetic p oten tial generated b y an electric dip ole that is stationary at the origin of the Cartesian co ordinate system, so it ob eys the dieren tial equation r 2(0) A aS(0) = 4 J a : (5.58)

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67 The solution to this equation is w ell-kno wn and has only a t -comp onen t A aS (0) = q i x i 3 ; 0 ; 0 ; 0 : (5.59) The rst-order correction to the electromagnetic p oten tial ob eys the dieren tial equation r 2(0) A aS(1) = r 2(1) A aS(0) : (5.60) If its THZ comp onen ts are assumed to b e A aS(1) = ( A tS(1) ; A xS(1) ; A yS(1) ; A zS(1) ) (5.61) and are substituted in to Equation (5.60), the dieren tial equation for A aS(1) b ecomes a set of four second-order dieren tial equations for these comp onen ts. Eac h dieren tial equation relates a sum of second deriv ativ es of a comp onen t to a sum of terms of the form q : E :: x : x : x : x : x : 7 for the t comp onen t ; q : B :: x : x : x : x : x : 7 for the spatial comp onen ts ; (5.62) where, again, the dots denote appropriate indices. These dieren tial equations indicate that the solution should b e equal to a sum of the terms q i E j k x i x j x k 3 and q i E ij x j x k x k 3 (5.63) for the t -comp onen t and a sum of the terms p ij q i B j l x l x k x k 3 ; p ij q i B k l x j x k x l 3 ; p ij q k B i k x j x l x l 3 ; p ij q k B i l x j x k x l 3 ; ij k q i B pj x k x l x l 3 ; ij k q i B j l x p x k x l 3 ; (5.64)

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68 for the p -spatial comp onen t, eac h term m ultiplied b y an appropriate n umerical coecien t. In fact, using Equations (5.15) and (5.17), the last t w o terms that are exp ected to sho w up in the solution for the p -comp onen t can b e eliminated in fa v or of the remaining four. Substituting these expressions in to the dieren tial equations giv es simple abgebraic equations for the co ecien ts. The nal expressions for the comp onen ts of A aS (1) are A tS(1) = 1 2 q i E j k x i x j x k 3 ; A pS(1) = p ij q i B j l x l x k x k 2 3 + p ij q k B i k x j x l x l 2 3 + p ij q k B i l x j x k x l 2 3 : (5.65) The order of the next term in the expansion of the electromagnetic p oten tial can b e predicted. It is the solution of the dieren tial equation whose source term comes from the O ( 3 R 3 ) part of the r 2 acting on A aS(0) That dieren tial equation has the form @ i @ j A aS(2) = O ( 3 R 3 ) @ i @ j A aS(0) O ( 1 R 3 ) : (5.66) Consequen tly the second-order correction m ust b e of O ( R 3 ). The terms that in v olv e the rst-order correction A aS(1) do not con tribute to the equation for the second-order correction, b ecause they in v olv e the O ( 2 R 2 ) part of the metric and that results in terms of O ( 1 R 4 ). Finally the singular electromagnetic p oten tial for an electric dip ole mo ving on a Sc h w arzsc hild geo desic is equal to A tS = 1 3 q i x i 1 2 q i E j k x i x j x k + O ( R 3 ) ; A pS = 1 2 3 p ij q i B j l x l x k x k + p ij q k B i k x j x l x l + p ij q k B i l x j x k x l + O ( R 3 ) : (5.67)

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69 5.6 Electromagnetic P oten tial of a Magnetic Dip ole In this section, the singular electromagnetic p oten tial of a magnetic dip ole mo ving on a geo desic of the Sc h w arzsc hild bac kground is calculated. The magnetization m a is assumed to p oin t at some random direction and its THZ comp onen ts are m a = (0 ; m x ; m y ; m z ). The singular electromagnetic p oten tial can b e written as A aS = A aS (0) + A aS (1) + A aS(2) (5.68) and ob eys the v acuum Einstein equations r 2(0+1+2) A aS (0+1+2) = 4 J a (5.69) where the source term is J 0 = 0 ; J q = Z q ij @ i h m j 4 ( p p 0 ( )) i p g d : (5.70) The zeroth-order term is the electromagnetic p oten tial generated b y a magnetic dip ole that is stationary at the origin of a Cartesian co ordinate system and is the solution of the dieren tial equation r 2(0) A aS(0) = 4 J a : (5.71) Its THZ comp onen ts are A aS (0) = (0 ; x ij m i x j 3 ; y ij m i x j 3 ; z ij m i x j 3 ) : (5.72)

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70 The rst-order term ob eys the dieren tial equation deriv ed from Maxw ell's Equations (5.69) r 2(0) A aS(1) = r 2(1) A aS (0) (5.73) whic h indicates that the source term for A aS (1) is the r 2(1) of the zeroth-order part of the p oten tial. The THZ comp onen ts of A aS (1) are A aS(1) = ( A tS(1) ; A xS(1) ; A yS(1) ; A zS(1) ) (5.74) and when substituted in to Equation (5.73) the result is a set of four second-order dieren tial equations, one for eac h one of those four comp onen ts. Eac h equation relates a sum of second deriv ativ es of one comp onen t to the source term whic h consists of terms of the form: m : B :: x : x : x : 5 for the t -comp onen t ; m : E :: x : x : x : x : x : 7 for the spatial comp onen ts ; (5.75) where the dots denote the appropriate indices. Solving these dieren tial equations is tedious but not dicult, b ecause eac h equation in v olv es only one comp onen t of A aS(1) and only one of the t w o tensors E and B A careful lo ok at the equations indicates that the solution should b e a sum of the terms m i B i j x j x k x k 3 and m i B j k x i x j x k 3 (5.76) for the t -comp onen t and a sum of the terms p ij m i E j k x k x l x l 3 ; p ij m i E k l x j x k x l 3 ; p ij m k E i k x j x l x l 3 p ij m k E i l x j x k x l 3 ; ij k m i E pj x k x l x l 3 ; ij k m i E j l x k x l x p 3 (5.77)

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71 for the p -spatial comp onen t, with appropriate n umerical factors in fron t of eac h term so that the equations are satised. Using Equations (5.15) and (5.17), the rst and last terms exp ected to sho w up in the nal expression for the p -comp onen t can b e eliminated, since they can b e expressed as linear com binations of the remaining four terms. Substituting these expressions in to the dieren tial equations giv es a system of four algebraic equations for those factors. Solving these algebraic equations is trivial. The result is that the rst-order comp onen ts of the electromagnetic p oten tial are: A tS (1) = 1 3 1 6 m i B j k x i x j x k 2 3 m i B ij x j x k x k A pS (1) = 1 3 p ij m i E k l x j x k x l 1 2 p ij m k E i l x j x k x l 1 2 ij k m i E pj x k x l x l : (5.78) The order of the next correction to the singular electromagnetic p oten tial can b e predicted. It is the solution of the dieren tial equation that has the O ( 3 R 3 ) part of the r 2 acting on the zeroth-order electromagnetic p oten tial as the source term. Sp ecically it lo oks lik e @ i @ j A aS (2) = O ( 3 R 3 ) @ i @ j A aS (0) O ( 1 R 3 ) ; (5.79) meaning that the A aS(2) correction is of O ( R 3 ). As in the previously studied cases, the terms that in v olv e the rst-order correction A aS(1) do not con tribute to this equation, since they in v olv e the O ( 2 R 2 ) part of the metric whic h results in terms of O ( 1 R 4 ). Finally the singular electromagnetic p oten tial for a magnetic dip ole mo ving on a Sc h w arzsc hild geo desic is equal to A tS = 1 3 1 6 m i B j k x i x j x k 2 3 m i B ij x j x k x k + O ( R 3 ) A pS = p ij m i x j 3 + p ij m i E k l x j x k x l 3 1 2 p ij m k E i l x j x k x l 3 1 2 ij k m i E pj x k x l x l 3 + O ( R 3 ) : (5.80)

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72 5.7 Gra vitational Field of a Spinning P article The calculation of the singular gra vitational eld of a spinning particle mo ving on a Sc h w arzsc hild geo desic is presen ted in this section. The particle is assumed to ha v e a small angular momen tum p oin ting at some random direction A a = (0 ; A x ; A y ; A z ) in THZ co ordinates. The singular gra vitational eld can b e written as h S ab = h (0)S ab + h (1)S ab + h (2)S ab (5.81) and ob eys the linearized Einstein equations r 2(0+1+2) h (0+1+2)S ab + 2 R c d (0+1+2) a b h (0+1+2)S cd = 16 T ab (5.82) where h S ab = h S ab 1 2 g ab h c S c (5.83) is the trace-rev ersed v ersion of h S ab The zeroth-order part of the singular eld is the gra vitational eld generated b y a particle with angular momen tum A a that is stationary at the origin of a Cartesian co ordinate system. F or the angular momen tum p oin ting along the z-axis, that is the w ell-kno wn Kerr solution with the mass set equal to zero. Since the angular momen tum is assumed to b e small and the eects of the mass of the particle are not tak en in to accoun t, only the terms of the Kerr metric that are linear in the angular momen tum need to b e considered. In the Bo y er-Lindquist co ordinates ( t BL ; r ; ; ) around the spinning particle, h ZS ab (where the sup erscript Z denotes that this is the part of the zeroth-order gra vitational eld coming only from the z -comp onen t of the angular momen tum) is

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73 equal to h ZS ab = 2 r A z sin 2 0BBBBBBB@ 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1CCCCCCCA : (5.84) Using Gr tensor this expression can b e easily con v erted in to the equiv alen t expression in THZ co ordinates h ZS ab = 2 A z 1 3 0BBBBBBB@ 0 y x 0 y 0 0 0 x 0 0 0 0 0 0 0 1CCCCCCCA : (5.85) The relationships b et w een the Bo y er-Lindquist and the THZ co ordinates used for this con v ersion are the usual relationships b et w een the spherical and the Cartesian co ordinates, namely t = t BL r = p x 2 + y 2 + z 2 = arctan p x 2 + y 2 z = arctan y x : (5.86) This is sucien t b ecause the corrections to these relationships that in v olv e the angular momen tum w ould giv e terms of higher order in the angular momen tum and m ust b e ignored in this analysis, since only rst-order terms in the angular momen tum are k ept. The comp onen ts of the angular momen tum along the x and y axes m ust b e treated separately b ecause of the axial symmetry of the Kerr metric. The analyses

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74 for the angular momen tum b eing along the x and y axes are v ery similar to that for the angular momen tum b eing along the z axis and the only c hange comes from the dieren t form that the relationships (5.86) ha v e. Sp ecically when the angular momen tum p oin ts along the x axis, the axial symmetry is around the x axis and the relationships used are t = t BL r = p x 2 + y 2 + z 2 = arctan p y 2 + z 2 x = arctan z y : (5.87) F or the angular momen tum p oin ting along the y axis, the axial symmetry is around the y axis and the relationships are t = t BL r = p x 2 + y 2 + z 2 = arctan p x 2 + z 2 y = arctan x z : (5.88) Adding all the con tributions that result from this analysis, the zeroth-order singular gra vitational eld b ecomes h (0)S ab = 2 3 0BBBBBBB@ 0 ( A y z + A z y ) ( A x z A z x ) ( A x y + A y x ) ( A y z + A z y ) 0 0 0 ( A x z A z x ) 0 0 0 ( A x y + A y x ) 0 0 0 1CCCCCCCA : (5.89)

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75 The rst-order correction to this gra vitational eld ob eys the dieren tial equation deriv ed from Equation (5.82): r 2(0) h (1)S ab + 2 R c d (0) a b h (1)S cd = r 2(1) h (0)S ab 2 R c d (1) a b h (0)S cd (5.90) so the rst-order r 2 and the rst-order Riemann tensor acting on the zeroth-order solution giv e the source for the rst-order correction. The rst-order correction m ust b e a symmetric tensor so it is assumed to b e equal to h (1)S ab = 0BBBBBBB@ h (1)tt h (1)tx h (1)ty h (1)tz h (1)tx h (1)xx h (1)xy h (1)xz h (1)ty h (1)xy h (1)y y h (1)y z h (1)tz h (1)xz h (1)y z h (1)z z 1CCCCCCCA : (5.91) Substituting it in to the rst-order equation results in 10 dieren tial equations. There is one set of four dieren tial equations for the four diagonal comp onen ts, eac h equation con taining all four diagonal comp onen ts. There is also one dieren tial equation for eac h one of the t i comp onen ts and one dieren tial equation for eac h one of the i j comp onen ts, for i 6 = j In eac h equation, a sum of second deriv ativ es of comp onen ts is related to a sum of terms of the form A : B :: x : x : x : x : x : 7 ; for the t t and p q comp onen ts ; A : E :: x : x : x : x : x : 7 ; for the t p comp onen ts ; (5.92) where the dots denote the appropriate indices for eac h term. Solving the dieren tial equations in this case is sligh tly more complicated than in the previous cases, mainly b ecause of the fact that four of them in v olv e all diagonal comp onen ts rather than only one of them. Still, the pro cess b ecomes signican tly easier if one notices that the t t comp onen t m ust b e an appropriate sum of terms

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76 of the form A i B j k x i x j x k 3 and A i B ij x j x k x k 3 ; (5.93) eac h t p comp onen t m ust b e a sum of terms of the form pij A i E j l x l x k x k 3 ; pij A i E l k x j x l x k 3 ; pij A l E i l x j x k x k 3 ; pij A l E i k x j x l x k 3 ; ij l A i E j p x l x k x k 3 ; ij l A i E j k x p x l x k 3 ; (5.94) and eac h p q spatial comp onen t m ust b e a sum of terms of the form A p B q k x k x l x l 3 ; A q B pk x k x l x l 3 ; A p B k l x q x k x l 3 ; A q B k l x p x k x l 3 ; A k B pq x k x l x l 3 ; A k B k l x p x q x l 3 ; pq A i B j k x i x j x k 3 ; pq A i B ij x j x k x k 3 ; A k B pk x q x l x l 3 ; A k B q k x p x l x l 3 ; A k B pl x q x k x l 3 ; A k B q l x p x k x l 3 ; (5.95) with appropriate n umerical co ecien ts in fron t of eac h term. Equations (5.15) and (5.17) can again b e used to eliminate the rst and last terms in fa v or of the remaining four, for the expression for the t p comp onen ts. Substituting these sums in to the dieren tial equations giv es algebraic equations for the co ecien ts, whic h are fairly easy to solv e. The result is that the rst-order correction to the gra vitational eld has comp onen ts h (1)S tt = 2 A i B j k x i x j x k 3 h (1)S tp = 1 3 h pij A l E i k x j x l x k pij A i E l k x j x l x k + 3 ij l A i E j p x l x k x k i h (1)S pq = 1 3 h 2 A i B pq x i x j x j + 2 3 A i B ij x p x q x j + 2 A i B i ( p x q ) x j x j 2 3 A i B j ( p x q ) x i x j 10 3 A ( p B q ) i x i x j x j 2 3 A ( p x q ) B ij x i x j + 2 3 pq A i B j k x i x j x k A i B ij x j x k x k i : (5.96)

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77 The order of the next correction can b e predicted. That correction is the solution of the dieren tial equation r 2(0) h (2)S ab + 2 R c d (0) a b h (2)S cd = r 2(2) h (0)S ab 2 R c d (2) a b h (0)S cd (5.97) where r 2(2) and R c d (2) a b come from the O ( 3 R 3 ) part of the metric. So the equations lo ok lik e @ i @ j h (2)S ab = O ( 3 R 3 ) @ i @ j h (0)S ab O ( 1 R 3 ) (5.98) and the second correction to the gra vitational eld is of O ( R 3 ). Again, h (1)S ab do es not sho w up in the equations for the second correction, since it relates to the O ( 2 R 2 ) part of the metric and results in terms of O ( 1 R 4 ). Finally the singular gra vitational eld due to a spinning particle mo ving on a Sc h w arzsc hild geo desic is equal to h S tt = 2 A i B j k x i x j x k 3 + O ( R 3 ) h S tp = 2 pij A i x j 3 + 1 3 h pij A l E i k x j x l x k pij A i E l k x j x l x k + 3 ij l A i E j p x l x k x k i + O ( R 3 ) h S pq = 1 3 h 2 A i B pq x i x j x j + 2 3 A i B ij x p x q x j + 2 A i B i ( p x q ) x j x j 2 3 A i B j ( p x q ) x i x j 10 3 A ( p B q ) i x i x j x j 2 3 A ( p x q ) B ij x i x j + 2 3 pq A i B j k x i x j x k A i B ij x j x k x k i + O ( R 3 ) : (5.99) 5.8 Grtensor Co de The Gr tensor co de (running under Maple ) used to deriv e the dieren tial equations for the rst order correction to the singular eld is giv en in App endix A.

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78 Since the case of the gra vitational eld is the most complicated one, the analysis is presen ted for the gra vitational eld generated b y a spinning particle with the angular momen tum p oin ting along the THZ z -axis. The analyses for the scalar elds and the electromagnetic p oten tials are v ery similar, and can b e easily deduced from that for the gra vitational eld. An eort w as made to k eep the sym b ols in the co de in accordance with the ones used in this c hapter for the v arious quan tities. In the situations where that is not the case, the commen ts in the co de should mak e the notation clear enough for the reader to follo w. The parameter e is used to k eep trac k of the order of eac h term in the comp onen ts of the tensors E and B and is set equal to 1 at the end. Throughout the calculation, only rst order terms are k ept. Sp ecically the Christoel sym b ols and the comp onen ts of the Riemann and Ricci tensors are calculated rst and all their terms that are of order higher than 1 are set equal to zero. Doing that mak es the subsequen t analysis signican tly simpler and the running time of the co de signican tly shorter. After the v arious quan tities asso ciated with the problem are calculated, the test tensor hbartest(a,b) whose exact dep endence on the THZ co ordinates is not sp ecied, is used as a trial solution in the linearized Einstein equations and the source term coming from the zeroth-order solution h0(a,b) is examined. That helps iden tify the terms that should b e exp ected to sho w up in eac h comp onen t of the solution hbar(a,b) Sp ecically it helps determine whic h tensor's comp onen ts, E (denoted as EE in the co de) or B (denoted as BB ), should sho w up in eac h comp onen t of hbar(a,b) and giv es an idea of ho w they should b e con tracted to the spatial THZ co ordinates x; y ; z The terms that result from this analysis are m ultiplied b y algebraic factors and the appropriate sum is substituted in to the linearized Einstein equations. The

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79 result is a simple system of algebraic equations whic h can b e easily solv ed to giv e the v alues of the algebraic factors. That completes the solution. The last comp onen t of this analysis is a simple conrmation p erformed for the gra vitational eld that w as calculated, that the algebraic co ecien ts obtained do, indeed, giv e the required solution. The conrmation is simply done b y replacing the initially unkno wn algebraic co ecien ts with their exact v alues in the expression for the solution and substituting that expression in to the Einstein equations. Despite the fact that it w as not explicitly men tioned in Sections (5.3)-(5.7), that conrmation w as p erformed for all singular elds and p oten tials that w ere calculated.

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CHAPTER 6 REGULARIZA TION P ARAMETERS F OR THE SCALAR FIELD A mo de-sum regularizarion pro cedure for the scalar singular eld is presen ted and implemen ted in this c hapter. This regularization pro cedure w as rst prop osed b y Barac k and Ori in [28], where they describ ed the calculation of the regularization parameters for the direct part of the self-force on a particle carrying a scalar c harge. The pro cedure w as later implemen ted b y dieren t groups for the calculation of the regularization parameters for the direct part of the self-force on a scalar c harge on dieren t geo desics [28{33] and also for non-geo desic motion [31, 33] around a Sc h w arzsc hild blac k hole. The calculation of the regularization parameters has also b een p erformed for the direct part of the electromagnetic self-force [34] and for the direct part of the gra vitational self-force [34{36], for arbitrary geo desics around a Sc h w arzsc hild blac k hole. Ev en though the regularization pro cedure w as initially describ ed for the con tribution of the direct part of the scalar eld to the self-force, it can b e used equally successfully for the con tribution of the singular scalar eld to the self-force, as w as demonstrated in [23]. In that pap er, the regularization parameters for the self-force on a scalar c harge in circular orbit around a Sc h w arzsc hild blac k hole w ere calculated and the self-force results ended up b eing in excellen t agreemen t with the results that w ere deriv ed using the direct scalar eld [28{31]. This c hapter b egins with an outline of the regularization pro cedure for the scalar self-force. The description closely follo ws that giv en b y Barac k and Ori [28] for the direct self-force but is presen ted here for the singular self-force instead. Then, the regularization parameters are calculated for the singular scalar eld (rather than the scalar self-force) of a c harged particle that mo v es on an equatorial circular orbit in 80

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81 a Sc h w arzsc hild bac kground and the results of [23] for the scalar self-force are reproduced. Finally the regularization parameters for the rst deriv ativ e of the singular part of the self-force are also calculated. 6.1 Regularization Pro cedure As w as sho wn in Chapter 4, the self-force on a particle that carries a scalar c harge q can b e calculated from the equation F R a = q lim p p 0 r a R = q lim p p 0 r a ret S (6.1) where p 0 is the p oin t along the w orldline of the c harged particle on whic h the self-force needs to b e calculated and p is a p oin t in the vicinit y of p 0 It is assumed that the c harge q is mo ving in a Sc h w arzsc hild bac kground of mass M and the Sc h w arzsc hild co ordinates are ( t s ; r ; ; ). F or the calculation of the retarded eld, the source term in P oisson's Equation (3.19) can b e decomp osed in terms of spherical harmonics and the retarded eld can b e written as ret ( t s ; r ; ; ) = 1 X l =0 l X m = l ret l m ( r ; t s ) Y l m ( ; ) : (6.2) Then, the l m -comp onen ts of ret can b e calculated n umerically That calculation is discussed in great detail in Chapter 7. Here it is just noted that the imp ortan t prop ert y of the ret l m 's and of their rst r -deriv ativ es is that they are nite at the lo cation of the particle, ev en though ret is singular there. If the spherical harmonic decomp osition of the singular eld is also considered, Equation (6.1) b ecomes F R a = q lim p p 0 r a 1 X l =0 l X m = l ret l m S l m Y l m = q lim p p 0 r a X l ;m ret l m S l m Y l m : (6.3)

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82 It is helpful to dene the m ultip ole l -mo des of the t w o con tributions to the self-force, whic h result after p erforming the m -summation of eac h term individually in Equation (6.3). Sp ecically F ret l a = q r a X m ret l m Y l m ; F S l a = q r a X m S l m Y l m ; (6.4) whic h giv es for the self-force F R a = lim p p 0 X l F ret l a F S l a : (6.5) In Equation (6.5), the dierence in the m ultip ole l -mo des m ust b e tak en b efore the summation o v er l is p erformed. F rom this p oin t on, the discussion of the regularization pro cedure b ecomes sp ecic to the problem of the scalar eld ( q = ), since more detailed results are a v ailable for this case. Ho w ev er, a similar analysis can b e done for an y other scalar eld. The goal is to nd a function h l a suc h that the series X l ( F ret l a h l a ) (6.6) con v erges. When suc h a function is found, the self-force can b e written as F R a = X l ( F ret l a h l a ) E a (6.7) where E a lim p p 0 X l ( F S l a h l a ) : (6.8)

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83 Because of its denition, the function h l a should b e calculated b y in v estigating the asymptotic expansion of F ret l a for large l On the other hand, b ecause the self-force is kno wn to b e w ell-dened, F ret l a and F S l a are exp ected to ha v e the same largel b eha vior, so h l a can b e determined b y the asymptotic b eha vior of F S l a instead. The singular part F S a of the self-force consists of terms of dieren t order in the limit p p 0 and it has b een sho wn [23] that, in principle, only the rst three of those terms are exp ected to giv e non-zero con tributions, for the eld ( q = ). Ho w ev er, for reasons that will b ecome clear shortly the next order terms are included and F S a = F S ( A ) a + F S( B ) a + F S( C ) a + F S ( D ) a + F S ( E ) a : (6.9) The sup erscripts A; B ; C ; D indicate the dieren t orders, A coming from the most dominan t term, B from the next more dominan t and so on. The sup erscript E refers to all terms of order higher than that of F S( D ) a The mo de-sum regularization pro cedure amoun ts to p erforming the spherical harmonic decomp osition of eac h suc h term, whic h results in an expression of the form F S a = X l ;m ( A l m a + B l m a + C l m a + D l m a + E l m a ) Y l m (6.10) with A l m a corresp onding to F S ( A ) a etc. F or the simple case of a scalar c harge mo ving in a Sc h w arzsc hild bac kground and generating the scalar eld ( q = ), the parameters A l m a ; B l m a ; C l m a and D l m a ha v e b een sho wn [23] to ha v e a v ery simple form so that, when the explicit expression for the spherical harmonics Y l m is substituted in to Equation (6.10), the summation o v er m can b e p erformed and the result is an expression of the form F S a = X l A a ( l + 1 2 ) + B a + C a 1 ( l + 1 2 ) + D a (2 l 1)(2 l + 3) + E a O ( l 4 ) (6.11)

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84 where the regularization parameters A a ; B a ; C a ; D a and E a are l -indep enden t quantities whic h do dep end on the bac kground geometry and the c haracteristics of the orbit. Finally the self-force can b e calculated b y F R a = 1 X l =0 lim p p 0 F ret l a A a ( l + 1 2 ) B a C a 1 ( l + 1 2 ) D a (2 l 1)(2 l + 3) E a O ( l 4 ) : (6.12) One imp ortan t p oin t that should b e made is that the innite sum o v er l m ust b e p erformed, in order for the self-force to b e calculated. Notice, ho w ev er, that the con tributions for large l get less signican t for the terms con taining D a and E a The reason for including these terms is no w clear. Ev en though the sum o v er l of eac h of these t w o terms is exactly equal to 0, including these terms impro v es the con v ergence of the sum. An additional b enet of including these terms is that the appro ximation to F R a b ecomes more dieren tiable, as is explained in [23]. 6.2 Order Calculation of the Scalar Field It should b e ob vious from the analysis of Section (6.1) that, in order to calculate the regularization parameters for the singular eld generated b y a c harge q it is necessary to ha v e an expression of (1 = ) in whic h the order of eac h term is kno wn. The deriv ation of suc h an expression is presen ted in this section for an equatorial circular orbit of radius r o in the Sc h w arzsc hild bac kground. It is noted that the results w ere deriv ed using Maple extensiv ely It is also noted that the deriv ation w as presen ted in [23] where the results of it for the radial deriv ativ e @ r 1 w ere giv en. F or simplicit y the scalar c harge q is set to 1. In order for the calculation of the self-force regularization parameters to b e made easier, the Sc h w arzsc hild co ordinates can b e rotated, as explained in [30]. Sp ecically

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85 new angles and can b e dened in terms of the usual Sc h w arzsc hild angles b y the equations sin cos ( n t s ) = cos (6.13) sin sin ( n t s ) = sin cos (6.14) cos = sin sin (6.15) so that the co ordinate lo cation of the particle is mo v ed from the equatorial plane, where = 2 to a lo cation where sin = 0, for a sp ecic t s Suc h a co ordinate rotation preserv es the index l of an y spherical harmonic Y l m ( ; ). That means that an y Y l m ( ; ) is mapp ed in to a linear com bination of spherical harmonics Y l m 0 ( ; ), where m 0 = l ; : : : ; l Consequen tly eac h l -m ultip ole mo de of the eld or the selfforce that results after summation o v er m is the same, regardless of whic h angles, ( ; ) or ( ; ), are used for calculating it. The b enet of this co ordinate rotation for the calculation of the regularization parameters for the singular part of the self-force can b e understo o d if one remem b ers that in the limit p p 0 the angle is equal to 0. That means that Y l m (0 ; ) has to b e used, for whic h Y l m (0 ; ) = 0 ; for m 6 = 0 (6.16) Y l m (0 ; ) = r 2 l + 1 4 ; for m = 0 : (6.17) So the sum o v er m can, after this co ordinate transformation, b e replaced with just the m = 0 term. Ho w ev er, for the regularization parameters of the singular eld, the limit p p 0 m ust not b e tak en, so the Y l m 's for all m 's m ust b e tak en in to consideration. The m = 0 spherical harmonic m ust b e considered only when the regularization parameters for the self-force are deriv ed from the ones for the scalar

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86 eld and when the regularization parameters for the rst r -deriv ativ e of the self-force are calculated. A commen t on the order of eac h term needs to b e made at this p oin t. The parameter n is used to indicate a term of order x n in the coincidence limit p p 0 In that limit, r r o and 0. That means that the factor ( r r o ) is of order and the factor (1 cos ) is of order 2 A t the end of the calculation, the parameter can b e set equal to 1. The relationships b et w een the Sc h w arzsc hild co ordinates ( t s ; r ; ; ) and the THZ co ordinates ( t; x; y ; z ) for a circular orbit on the equatorial plane of a Sc h w arzsc hild bac kground that w ere giv en in Section (5.1) are used for this calculation. As w as men tioned earlier, 2 = x 2 + y 2 + z 2 in terms of the spatial THZ co ordinates x y and z Ho w ev er, it is clear from Equation (5.26) that the relationship x 2 + y 2 = ~ x 2 + ~ y 2 (6.18) holds b et w een f x; y g and f ~ x; ~ y g so the sum ( ~ x 2 + ~ y 2 ) is used to calculate 2 The Equations (5.22), (5.23) and (5.24) are substituted in to the expression for 2 with the Sc h w arzsc hild angles and replaced b y the new angles and That substitution giv es that the lo w est order term for 2 denoted b y ~ 2 is ~ 2 r o 2 r o 2 M + 2 r 2 o r o 2 M r o 3 M (1 cos ) ; (6.19) where r r o (6.20) and 1 M r o 2 M sin 2 : (6.21)

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87 Clearly ~ 2 is of order 2 in the coincidence limit, as should ha v e b een exp ected. Next the v ariables and r are eliminated from the order expression of 2 in fa v or of the v ariables ~ and b y using Equations (6.19), (6.20) and (6.21). Finally the result is in v erted and the square ro ot is tak en, in order to obtain the order expression of (1 = ). The result of this calculation is a v ery long expression. Here, I only giv e the terms of this expression that are necessary to calculate the regularization parameters of the singular eld, the singular self-force along the radial direction and the r -deriv ativ e of the singular self-force along the radial direction. Exactly ho w that is determined will b ecome clear shortly when the general term ~ p q where p and q stand for in tegers, will b e discussed. The terms of in terest are 1 = 1 1 ~ + + 0 r o 3 M 2 r o ( r o 2 M ) 1 1 r o ~ + 2 r o 3 M 2 r o ( r o 2 M ) r o 3 M 2 r o ( r o 2 M ) 1 3 ~ 3 + 1 r o 3 M 8 r 2 o ( r o 2 M ) 1 r o + M r o 1 2 ~ + 2 r o 3 M 2 r 2 o ( r o 2 M ) 5 r 2 o 22 r o M + 21 M 2 4( r o 2 M ) 2 r 2 o 1 + 5 r 2 o 22 r o M + 21 M 2 8 r 2 o ( r o 2 M ) 2 1 2 2 ~ + O 4 ~ 3 ; 6 ~ 5 + 2 M ( r o 2 M ) 2 r 4 o ( r o 3 M ) ( r o M )( r o 4 M ) 8( r o 2 M ) r 4 o 1 + ( r o 3 M )(5 r 2 o 7 r o M 14 M 2 ) 16 r 4 o ( r o 2 M ) 2 1 2 3( r o + M )( r o 3 M ) 2 16 r 4 o ( r o 2 M ) 2 1 3 ~ + O 3 ~ ; 5 ~ 3 ; 7 ~ 5 ; 9 ~ 7 + O ( 3 ) : (6.22)

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88 6.3 Scalar Monop ole Field As is clear from the previous calculation of (1 = ), the angular dep endence of ev ery term sho ws up in factors of the form ~ p q where p is an o dd in teger and q = 0 ; 1 ; 2 ; : : : The spherical harmonic decomp osition of that factor, whic h is necessary when calculating the regularization parameters for the scalar eld, is giv en in detail in App endix B. The result is ~ p q = 2 r 2 o ( r o 2 M ) r o 3 M p= 2 1 X l =0 l X m = l E p;q l ;m ( r 2 ; M r o 2 M ) Y l m ( ; ) (6.23) where r 2 = 2 ( r o 3 M ) 2 r o ( r o 2 M ) 2 (6.24) and the co ecien ts E p;q l ;m ( r 2 ; M r o 2 M ) are giv en in Equation (B.41). It is stressed that all the r -dep endence of ~ p q resides in the sum P 1s =0 r 2 s and the term ( r 2 ) p 2 + n +1 in Equation (B.41) for E p;q l ;m and is alw a ys prop ortional to p o w ers of = ( r r o ). A t this p oin t, a note on the term ( r 2 ) 2 for an o dd in teger is in order. F or that term, the square ro ot of r 2 m ust b e considered. One migh t think that ( r 2 ) 1 2 = r o 3 M 2 r o ( r o 2 M ) 2 2 1 2 = r o 3 M 2 r o ( r o 2 M ) 2 1 2 : (6.25) But that implies that b eing on the equatorial plane with = 0 and approac hing the particle b y taking the limit r r o w ould giv e for the leading term ~ in the expansion of : ~ = 2 r 2 o r o 2 M r o 3 M 1 2 1 2 r 2 + 1 cos 0 1 2 = 2 r 2 o r o 2 M r o 3 M 1 2 ( r 2 ) 1 2 = r o r o 2 M 1 2 : (6.26)

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89 According to this, the sign of the leading order term of could b e either p ositiv e or negativ e, dep ending on whether r > r o or r < r o That is clearly not correct, since b y denition = p x 2 + y 2 + z 2 whic h is alw a ys p ositiv e. F or that reason, taking the square ro ot of r 2 alw a ys implies ( r 2 ) 1 2 = r o 3 M 2 r o ( r o 2 M ) 2 1 2 j j : (6.27) F or the self-force along the radial direction to b e calculated, the r -deriv ativ e of (1 = ) and the limit r r o ha v e to b e tak en. That means that an y term of order ( r r o ) 2 or higher giv es, after the limit is tak en, no con tribution to the self-force. Ho w ev er, as has already b een men tioned, it is desired to calculate the rst deriv ativ e with resp ect to r of the self-force, whic h is the second deriv ativ e with resp ect to r of the eld, in the limit r r o Consequen tly terms of order ( r r o ) 2 ha v e to b e retained, b ecause they do giv e a con tribution at that limit, while an y terms of order ( r r o ) 5 2 or higher can b e disregarded. As is clear from Equation (B.41) of App endix B, the general term ~ p q in Equation (6.22) has t w o pieces that con tain the r -dep endence. The rst piece comes from the sum o v er s in E p;q l ;m and giv es terms prop ortional to r +2 s Suc h a term should b e k ept only for + 2 s < 5 2 or s < 5 4 2 : (6.28) Since s 0, the r +2 s con tribution can b e immediately disregarded for terms with 5 2 while for terms with < 5 2 a limited n um b er of v alues of s ha v e to b e retained in the sum. The second piece that con tains r -dep endence comes from the term ( r 2 ) p 2 + n +1 and is prop ortional to r p + +2 n +2 Suc h a term should b e k ept only for p + + 2 n + 2 < 5 2 or n < 1 4 p + 2 : (6.29)

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90 Since n 0, all terms of Equation (6.22) for whic h p + 1 2 do not con tribute through the r p + +2 n +2 piece. Com binations of p and for whic h p + < 1 2 should b e examined individually and a sp ecic n um b er of n 's m ust b e retained. The terms of the expansion (6.22) for (1 = ) for whic h just the order (and not the explicit expression) is giv en are terms that fall in b oth categories that according to this analysis can b e ignored. I presen t no w the calculation of the regularization parameters coming from the terms of dieren t order, for the singular eld (1 = ). The abbreviation used for the h yp ergeometric function is F ;m a 2 F 1 ( a; + 1 2 ; j m j 2 + 1; M r o 2 M ) ; (6.30) and it is also noted that the h yp ergeometric function F 0 ; 0 a = 2 F 1 ( a; 1 2 ; 1; M r o 2 M ) (6.31) is denoted b y F a as is done in [23]. Also, for the three sums o v er k and the abbreviation X k ;; = r 2 l + 1 ( l j m j + 1) [( l m + 1)( l + m + 1)] 1 2 [ l 2 ] X k =0 ( 1) k 2 l 2 k ( l k + 1 2 ) ( k + 1)( l 2 k j m j + 1) j m j 2 X =0 ( j m j 2 + 1) ( + 1)( j m j 2 + 1) j m j 2 X =0 ( 1) ( j m j +1 2 ) ( + 1)( j m j 2 + 1) (6.32)

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91 that is used in app endix B is also used here. In addition, the upp er limit of the n -summation is denoted b y N = l j m j 2 k + 2 : (6.33) F or the lo w est order term of Equation (6.22) T ( 1) 1 1 ~ (6.34) the exp onen ts of ~ p q are = 0 ; p = 1 ; q = 0. Equations (6.28) and (6.29) giv e that only the s = 0 and s = 1 terms of the rst piece and only the n = 0 term of the second piece of E 1 ;q l ;m need to b e k ept. Consequen tly T ( 1) = 1 X l ;m A l m Y l m ( ; ) (6.35) where, using Equation (6.23) A l m = r o 3 M 2 r 2 o ( r o 2 M ) 1 2 E 1 ; 0 l ;m = r o 3 M 2 r 2 o ( r o 2 M ) 1 2 X k ;; 2( r 2 ) 1 2 F ;m 1 + N X n =0 ( 1) l + n 2 2 n + 1 2 (2 n + 1)!! ( N + 1) ( N n + 1) h 2 F ;m 1 = 2 + ( n 1 2 ) F ;m 3 = 2 r 2 i (6.36) or, when the explicit expression for r 2 and its square ro ot are substituted A l m = r o 3 M 2 r 2 o ( r o 2 M ) 1 2 X k ;; 2 h r o 3 M 2 r o ( r o 2 M ) 2 i 1 2 j j F ;m 1 + N X n =0 ( 1) l + n 2 2 n + 1 2 (2 n + 1)!! ( N + 1) ( N n + 1) h 2 F ;m 1 = 2 + ( n 1 2 ) F ;m 3 = 2 r o 3 M 2 r o ( r o 2 M ) 2 2 i : (6.37)

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92 As is explained in App endix B, only the ev en m 's should b e included in the sum of Equation (6.35). The regularization parameter A r giv en in [23] for the self-force can b e easily obtained from this result. First the m = 0 parameter is considered, for reasons that w ere explained earlier. That mak es the sums o v er and o v er equiv alen t to the terms with = 0 and = 0. F or the h yp ergeometric function F 0 ; 0 1 it can easily b e pro v en that F 0 ; 0 1 = F 1 = r r o 2 M r o 3 M (6.38) using Equations 15.3.3 of [37]. Then the rst deriv ativ e of A l m with resp ect to r is tak en. That mak es the term prop ortional to F ;m 1 = 2 in Equation (6.37) v anish, while the term prop ortional to F ;m 3 = 2 giv es a factor of whic h v anishes when the coincidence limit is tak en. F or the rst term the deriv ativ e of j j giv es sgn(). Finally the spherical harmonic Y l 0 (0 ; ) is substituted with q 2 l +1 4 This pro cedure giv es @ @ r T ( 1) p p 0 = 2 1 X l =0 dA l 0 dr r = r o r 2 l + 1 4 = 2 ( sgn ()) [ r o ( r o 3 M )] 1 2 r 2 o ( r o 2 M ) 1 p 1 X l =0 ( l + 1 2 ) [ l 2 ] X k =0 ( 1) k 2 l 2 k ( l k + 1 2 ) ( k + 1)( l 2 k + 1) : (6.39) The sum o v er k can b e easily calculated for an y v alue of l using Maple A general pro of that it is equal to p for an y l cannot b e giv en. Ho w ev er, for ev ery v alue of l that w as tried, the sum ended up b eing p whic h giv es @ @ r T ( 1) p p 0 = 2 ( sgn ()) [ r o ( r o 3 M )] 1 2 r 2 o ( r o 2 M ) 1 X l =0 ( l + 1 2 ) (6.40) whic h is the result of [23].

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93 No w, the second deriv ativ e of the term T ( 1) with resp ect to r is tak en, whic h is equal to the con tribution of that term to the rst deriv ativ e of the self-force. The m = 0 comp onen t is considered since the coincidence limit m ust b e tak en, so = 0 and = 0 as w ell. Only the term prop ortional to F ;m 3 = 2 surviv es the dieren tiation, since it is prop ortional to 2 The result is @ 2 T ( 1) @ r 2 p p 0 = 3 1 X l =0 d 2 A l 0 dr 2 r = r o r 2 l + 1 4 = 3 ( r o 3 M ) 3 2 r 2 o ( r o 2 M ) 5 2 1 p F 3 = 2 1 X l =0 ( l + 1 2 ) [ l 2 ] X k =0 ( 1) k 2 l 2 k ( l k + 1 2 ) ( k + 1) l 2 k X n =0 ( 1) l + n 2 2 n ( n 1 2 ) ( l 2 k n + 1)(2 n + 1)!! : (6.41) The zeroth-order term con tribution that is considered in Equation (6.22) is T (0) 0 h r o 3 M 2 r o ( r o 2 M ) 1 1 r o ~ + 2 r o 3 M 2 r o ( r o 2 M ) r o 3 M 2 r o ( r o 2 M ) 1 i 3 ~ 3 : (6.42) F or the piece that is prop ortional to ~ 1 the exp onen ts are = 1 ; p = 1 ; q = 1 for the term that con tains and = 1 ; p = 1 ; q = 0 for the term that do es not con tain In b oth cases, Equations (6.28) and (6.29) indicate that the s = 0 term of the rst piece and the n = 0 term of the second piece of E 1 ;q l ;m should b e k ept. F or the piece that is prop ortional to 3 ~ 3 the exp onen ts are = 3 ; p = 3 ; = 0 for the term that do es not con tain and = 3 ; p = 3 ; = 1 for the term that do es con tain In this case, Equations (6.28) and (6.29) giv e that none of the s -terms and only the n = 0 term of the second piece of E 3 ;q l ;m need to b e k ept. That giv es T (0) = 0 X l ;m B l m Y l m ( ; ) (6.43)

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94 where, from Equation (6.23) B l m = r o 3 M 2 r 2 o ( r o 2 M ) 1 2 r o 3 M 2 r o ( r o 2 M ) E 1 ; 1 l ;m 1 r o E 1 ; 0 l ;m + 3 r o 3 M 2 r 2 o ( r o 2 M ) 3 2 2 r o 3 M 2 r o ( r o 2 M ) E 3 ; 0 l ;m r o 3 M 2 r o ( r o 2 M ) E 3 ; 1 l ;m : (6.44) Substituing the explicit expressions of the co ecien ts E p;q l ;m in to Equation (6.44) for B l m results in B l m = s r o 3 M 2 r 2 o ( r o 2 M ) X k ;; ( N X n =0 ( 1) l + n 2 2 n + 3 2 (2 n + 1)!! ( N + 1) ( N n + 1) r o 3 M 2 r o ( r o 2 M ) F ;m 3 = 2 1 r o F ;m 1 = 2 j j p 2( r o 3 M ) 1 2 r 1 2 o ( r o 2 M ) r o 3 M 2 r o ( r o 2 M ) F ;m 2 1 r o F ;m 1 + 3 j j ( r o 3 M ) 3 2 p 2 r 5 2 o ( r o 2 M ) h 2 r o 3 M r o 3 M F ;m 1 F ;m 2 i ) : (6.45) It is easy to recognize that j j = 3 j j (6.46) so the last t w o terms in the last expression for B l m can b e com bined to giv e a signican tly simpler expression, namely B l m = s r o 3 M 2 r 2 o ( r o 2 M ) X k ;; ( N X n =0 ( 1) l + n 2 2 n + 3 2 (2 n + 1)!! ( N + 1) ( N n + 1) r o 3 M 2 r o ( r o 2 M ) F ;m 3 = 2 1 r o F ;m 1 = 2 + j j ( r o 3 M ) 1 2 p 2 r 5 2 o ( r o 2 M ) h 2( r o 3 M )( r o M ) ( r o 2 M ) F ;m 2 + (4 r o 3 M ) F ;m 1 i ) : (6.47)

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95 As is the case for A l m only the co ecien ts for whic h m is ev en con tribute to the sum in Equation (6.43). The regularization parameter B r calculated in [23] can b e easily deriv ed from the general expression giv en in Equation (6.47). First, the m = 0 co ecien t is considered, whic h also mak es = 0 and = 0. The rst deriv ativ e of the factor j j with resp ect to r in Equation (6.47) giv es zero in the coincidence limit r r o So the only term that con tributes is the one that is prop ortional to If the spherical harmonic Y l 0 (0 ; ) is again replaced with q 2 l +1 4 the result is @ @ r T (0) p p 0 = 1 1 X l =0 @ B l 0 @ r r = r o r 2 l + 1 4 = 1 r o 3 M r 4 o ( r o 2 M ) 1 2 h r o 3 M 2( r o 2 M ) F 3 = 2 F 1 = 2 i 1 X l =0 2 l + 1 2 p 2 [ l 2 ] X k =0 ( 1) k 2 l 2 k ( l k + 1 2 ) ( k + 1) l 2 k X n =0 2 2 n + 3 2 (2 n + 1)!! ( 1) l + n ( l 2 k n + 1) : (6.48) The t w o nite sums o v er k and o v er n can b e easily calculated for an y v alue of l b y using Maple F or an y v alue of l that w as tried, the result w as equal to 2 p 2 2 l +1 but, again, there is no general pro of for that. Substituting this result for the double sum in to Equation (6.48) giv es that @ @ r T (0) p p 0 = 1 r o 3 M r 4 o ( r o 2 M ) 1 2 h r o 3 M 2( r o 2 M ) F 3 = 2 F 1 = 2 i 1 X l =0 1 (6.49) whic h is the same as the result giv en in [23]. The con tribution of T (0) to the deriv ativ e of the self-force can no w b e calculated. The second deriv ativ e of Equation (6.47) has to b e tak en, and the m = 0 comp onen t

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96 to b e considered. Th us, the term prop ortional to v anishes but the second deriv ativ e of the term j j giv es d 2 dr 2 ( j j ) = 2sgn() (6.50) whic h do es giv e a con tribution in the coincidence limit. The result is @ 2 @ r 2 T (0) p p 0 = 2 1 X l =0 d 2 B l 0 dr 2 r = r o r 2 l + 1 4 = 2 ( r o 3 M ) r 5 2 o ( r o 2 M ) 3 2 sgn() p 1 X l =0 ( l + 1 2 ) [ l 2 ] X k =0 ( 1) k 2 l 2 k ( l k + 1 2 ) ( k + 1)( l 2 k + 1) 2 ( r o M )( r o 3 M ) r o ( r o 2 M ) F 2 + 4 r o 3 M r o F 1 : (6.51) Using Equation 15.2.13 of [37] and the previously deriv ed result for the h yp ergeometric function F 1 (Equation (6.38)), a simple expression for the h yp ergeometric function F 2 can also b e obtained and it is F 2 = 1 2 r r o 2 M r o 3 M 2 r o 5 M r o 3 M : (6.52) The nal expression for the con tribution of B l m to the second deriv ativ e of the scalar eld is @ 2 @ r 2 T (0) p p 0 = 2 sgn() p 1 X l =0 ( l + 1 2 ) [ l 2 ] X k =0 ( 1) k 2 l 2 k ( l k + 1 2 ) ( k + 1)( l 2 k + 1) ( r o 3 M ) 1 2 r 7 2 o ( r o 2 M ) 2 (2 r 2 o 4 r o M + M 2 ) : (6.53)

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97 The term of order 1 in Equation (6.22) is considered next. The part of it that needs to b e k ept is T (1) 1 r o 3 M 8 r 2 o ( r o 2 M ) 1 r o + M r o 1 2 ~ + 2 r o 3 M 2 r 2 o ( r o 2 M ) 5 r 2 o 22 r o M + 21 M 2 4( r o 2 M ) 2 r 2 o 1 + 5 r 2 o 22 r o M + 21 M 2 8 r 2 o ( r o 2 M ) 2 1 2 2 ~ : (6.54) F or the part that is prop ortional to ~ the exp onen ts are = 0 ; p = 1 ; q = 1 for the term con taining (1 = ) and = 0 ; p = 1 ; q = 2 for the term con taining (1 = 2 ). F or b oth these terms, Equations (6.28) and (6.29) indicate that only the s = 0 and s = 1 terms of the rst piece and none of the v alues of n of the second piece of E 1 ;q l ;m should b e k ept. F or the part that is prop ortional to 2 ~ 1 the exp onen ts are = 2 ; p = 1 ; q = 0 for the term not con taining and = 2 ; p = 1 ; q = 1 and = 2 ; p = 1 ; q = 2 for the other t w o terms. In this case, only the s = 0 term should b e k ept in E 1 ;q l ;m Consequen tly T (1) = 1 X l ;m C l m Y l m ( ; ) (6.55) where only the ev en m 's con tribute to the ab o v e sum and, from Equation (6.23), the co ecien ts C l m are: C l m = 2 r 2 o ( r o 2 M ) r o 3 M 1 2 r o 3 M 8 r 2 o ( r o 2 M ) E 1 ; 1 l ;m r o + M r o E 1 ; 2 l ;m + 2 h r o 3 M 2 r 2 o ( r o 2 M ) i 1 2 2 r o 3 M 2 r 2 o ( r o 2 M ) E 1 ; 0 l ;m 5 r 2 o 22 r o M + 21 M 2 4 r 2 o ( r o 2 M ) 2 E 1 ; 1 l ;m + 5 r 2 o 22 r o M + 21 M 2 8 r 2 o ( r o 2 M ) 2 E 1 ; 2 l ;m : (6.56)

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98 After the explicit expressions for the co ecien ts E p;q l ;m and for r 2 are substituted in to the expression for C l m the result is C l m = 1 8 r o r r o 3 M r o 2 M X k ;; N X n =0 ( 1) l + n 2 2 n + 3 2 (2 n + 1)!! ( N + 1) ( N n + 1) 2 p 2 2 n + 3 F ;m 1 = 2 r o + M r o F ;m 3 = 2 + 2 p 2 r o ( r o 2 M ) 2 h 2 n + 1 2 n + 3 ( r o 3 M ) 2 F ;m 3 = 2 r o + M r o F ;m 5 = 2 + 4( r o 2 M )(2 r o 3 M ) r o F ;m 1 = 2 2(5 r 2 o 22 r o M + 21 M 2 ) r o F ;m 3 = 2 + 5 r 2 o 22 r o M + 21 M 2 r o F ;m 5 = 2 i : (6.57) Obtaining the result of [23] for the regularization parameter C r for the selfforce is trivial in this case. That is b ecause taking the rst deriv ativ e of C l m with resp ect to r immediately mak es the term with no dep endence v anish and the terms prop ortional to 2 end up ha ving a factor of whic h also v anishes in the coincidence limit. That means that the O ( ) term do es not con tibute to the self-force on the scalar c harge. Things are dieren t for the rst deriv ativ e of the self-force. T aking the second deriv ativ e of Equation (6.57) mak es the term with no dep endence v anish, but the term prop ortional to 2 do es giv e a con tribution whic h m ust b e calculated. Since the coincidence limit is tak en, m = 0 and @ 2 @ r 2 T (1) p p 0 = 1 1 X l =0 d 2 C l 0 dr 2 r 2 l + 1 4 (6.58)

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99 whic h, after a short calculation, giv es @ 2 @ r 2 T (1) p p 0 = 1 1 p r r o 3 M r o 2 M 1 2 r 2 o ( r o 2 M ) 2 1 X l =0 ( l + 1 2 ) [ l 2 ] X k =0 ( 1) k 2 l 2 k ( l k + 1 2 ) ( k + 1) l 2 k X n =0 2 2 n ( 1) l + n ( l 2 k n + 1)(2 n + 1)!! r o 3 M 2 2 n + 1 2 n + 3 F 3 = 2 r o + M r o F 5 = 2 + 4( r o 2 M )(2 r o 3 M ) r o F 1 = 2 2(5 r 2 o 22 r o M + 21 M 2 ) r o F 3 = 2 + 5 r 2 o 22 r o M + 21 M 2 r o F 5 = 2 i : (6.59) The last term of Equation (6.22) that is examined is the term of order 2 The part of it that m ust b e considered is T (2) 2 M ( r o 2 M ) 2 r 4 o ( r o 3 M ) ( r o M )( r o 4 M ) 8( r o 2 M ) r 4 o 1 + ( r o 3 M )(5 r 2 o 7 r o M 14 M 2 ) 16 r 4 o ( r o 2 M ) 2 1 2 3( r o + M )( r o 3 M ) 2 16 r 4 o ( r o 2 M ) 2 1 3 ~ : (6.60) In that term, the exp onen ts for and ~ are = 1 ; p = 1, while the exp onen t of tak es the v alues q = 0 ; 1 ; 2 ; 3 for eac h term. Equations (6.28) and (6.29) indicate that only the s = 0 term of the rst piece con taining the r -dep endence of eac h E 1 ;q l ;m needs to b e k ept and the second piece can b e disregarded. Then, the O ( 2 ) term can b e written as T (2) = 2 X l ;m D l m Y l m ( ; ) (6.61)

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100 where, according to Equation (6.23) D l m = h 2 r 2 o ( r o 2 M ) r o 3 M i 1 2 M ( r o 2 M ) 2 r 4 o ( r o 3 M ) E 1 ; 0 l ;m ( r o M )( r o 4 M ) 8( r o 2 M ) r 4 o E 1 ; 1 l ;m ( r o 3 M )(5 r 2 o 7 r o M 14 M 2 ) 16 r 4 o ( r o 2 M ) 2 E 1 ; 2 l ;m 3( r o + M )( r o 3 M ) 2 16 r 4 o ( r o 2 M ) 2 E 1 ; 3 l ;m : (6.62) Substituting of the co ecien ts E p;q l ;m in to Equation (6.62) giv es D l m = h 2 r 2 o ( r o 2 M ) r o 3 M i 1 2 X k ;; N X n =0 ( N + 1) ( N n + 1) ( 1) l + n 2 2 n + 5 2 (2 n + 3)!! M ( r o 2 M ) 2 r 4 o ( r o 3 M ) F ;m 1 = 2 ( r o M )( r o 4 M ) 8 r 4 o ( r o 2 M ) F ;m 1 = 2 + ( r o 3 M )(5 r 2 o 7 r o M 14 M 2 ) 16 r 4 o ( r o 2 M ) 2 F ;m 3 = 2 3( r o + M )( r o 3 M ) 2 16 r 4 o ( r o 2 M ) 2 F ;m 5 = 2 : (6.63) Repro ducing the result of [23] for D r can b e done b y follo wing the same metho d that w as used for the regularization parameters A r and B r First, the m = 0 co ecien t is considered, whic h mak es = 0 and = 0. The rst deriv ativ e with resp ect to r is then tak en and that simply mak es the factor v anish. Finally the expression of Y l 0 for = 0 is used. The result is @ @ r T (2) p p 0 = 1 1 X l =0 @ D l 0 @ r r 2 l + 1 4 = 1 2 r 2 o ( r o 2 M ) r o 3 M 1 2 M ( r o 2 M ) 2 r 4 o ( r o 3 M ) F 1 = 2 ( r o M )( r o 4 M ) 8 r 4 o ( r o 2 M ) F 1 = 2 + ( r o 3 M )(5 r 2 o 7 r o M 14 M 2 ) 16 r 4 o ( r o 2 M ) 2 F 3 = 2 3( r o + M )( r o 3 M ) 2 16 r 4 o ( r o 2 M ) 2 F 5 = 2 1 X l =0 2 p 2 (2 l 1)(2 l + 3) (6.64)

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101 whic h is the same as the result of [23]. T o deriv e this nal expression, Maple w as used to ev aluate the double sum: [ l 2 ] X k =0 ( 1) k 2 l 2 k ( l k + 1 2 ) ( k + 1) l 2 k X n =0 ( 1) l + n 2 2 n + 5 2 (2 n + 3)!!( l 2 k n + 1) = 4 p 2 (2 l 1)(2 l + 1)(2 l + 3) (6.65) for dieren t v alues of l Ev en though a general pro of that this equalit y holds for ev ery l cannot b e pro vided, this equation w as v eried to hold true for an y v alue of l that w as tried. As far as the con tribution of this term to the deriv ativ e of the self-force is concerned, it can b e easily seen to equal zero since there are no terms that can surviv e after the second deriv ativ e of D l m with resp ect to r is tak en. Consequen tly @ 2 @ r 2 T (2) p p 0 = 0 : (6.66) One nal commen t should b e made ab out Equations (6.41), (6.53) and (6.59). Because these equations in v olv e complicated sums of k and n it ma y b e though t that they are not particularly handy for the calculation of the con tributions of the corresp onding terms to the rst deriv ativ e of the self-force. Nonetheless, all the sums in v olv ed are nite and can alw a ys b e calculated using mathematical soft w are or n umerical co de, whic h renders these equations relativ ely easy to use.

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CHAPTER 7 CALCULA TION OF THE RET ARDED FIELD As w as explained in Chapter 4, when calculating the self-force on a scalar c harge q one needs to kno w the retarded and the singular part of the scalar eld. In order to b e able to pro duce results for the self-force, I presen t here the calculation of the retarded eld generated b y a scalar c harge mo ving on a circular orbit in a Sc h w arzsc hild bac kground, as w ell as its con tribution to the self-force. The metho d used is similar to that presen ted b y Burk o in [31] and the calculation w as outlined in app endix E of [23]. 7.1 Analytical W ork The scalar eld ret at a p oin t p generated b y a scalar c harge q mo ving on a circular orbit of radius r o on the equatorial plane of a Sc h w arzsc hild bac kground of mass M ob eys P oisson's equation r 2 ret ( p ( x a )) = 4 % (7.1) where x a are the Sc h w arzsc hild co ordinates of p oin t p and the source term is giv en b y the in tegral % = q Z ( g ) 1 2 4 ( p p 0 ) d = q Z ( g ) 1 2 ( r r o ) r 2 o ( 2 ) ( n t s ) ( t s t s ( )) d = q ( r r o ) r 2 o ( 2 ) ( n t s ) dt s d 1 : (7.2) 102

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103 In this expression p 0 is the lo cation of the c harge q is the prop er time along the circular orbit of the c harge q and the relationship b et w een the prop er time and the Sc h w arzsc hild time t s giv es dt s d = 1 q 1 3 M r o : (7.3) Then, b y writing P oisson's equation in Scw arzsc hild co ordinates one gets r 2 ret = h 1 2 M r 1 @ 2 @ t 2s + 2 r 2 M r 2 @ @ r + 1 2 M r @ 2 @ r 2 + cos r 2 sin @ @ + 1 r 2 @ 2 @ 2 + 1 r 2 sin 2 @ 2 @ 2 i ret = 4 q ( r r o ) r 2 o ( 2 ) ( n t s ) r 1 3 M r o : (7.4) As w as men tioned in Chapter 6, b oth the source and the eld can b e decomp osed in to spherical harmonics. Sp ecically the spherical harmonic decomp osition of the source can b e done using the spherical harmonic decomp osition of the 3-dimensional function whic h is giv en in [38] % = q ( r r o ) r 2 o 1 X l =0 l X m = l 1 sin 2 Y l m ( 2 ; n t s ) Y l m ( ; ) r 1 3 M r o = q ( r r o ) r 2 o X l ;m e im n t s Y l m ( 2 ; 0) Y l m ( ; ) r 1 3 M r o = X l ;m ( r r o ) 4 r o q l m e i! m t s Y l m ( ; ) (7.5) where m = m n (7.6) and q l m = 4 q r o Y l m ( 2 ; 0) r 1 3 M r o : (7.7)

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104 The retarded eld can b e written as ret ( t s ; r ; ; ) = X l ;m l m ( r )e i! m t s Y l m ( ; ) (7.8) where the time dep endence is the one implied b y the spherical harmonic decomp osition of the function. Substituting the expressions for q l m and ret in to Equation (7.4) results in a dieren tial equation for the radial part l m ( r ) only: d 2 l m dr 2 + 2( r M ) r ( r 2 M ) d l m dr + h 2 m r 2 ( r 2 M ) 2 l ( l + 1) r ( r 2 M ) i l m = q l m r o 2 M ( r r o ) : (7.9) F or p oin ts with r 6 = r o the righ t hand side of Equation (7.9) is equal to zero and the dieren tial equation can b e solv ed n umerically A v ariable that is used in the follo wing is r whic h is dened as r = r + 2 M ln ( r 2 M 1) ; (7.10) and for whic h dr dr = 1 2 M r 1 : (7.11) The b eha vior of l m at the t w o b oundaries needs to b e considered rst. F or large distances a w a y from the orbit, i.e. in the limit r 1 the l m ( r ) is exp ected to b eha v e as an outgoing w a v e, so it is assumed that INF l m ( r ) = e i! m r r 1 X n =0 a n r n : (7.12) This expression m ust satisfy Equation (7.9). The co ecien ts a n can b e found b y asserting exactly that. First, the rst and second deriv ativ es of INF l m ( r ) with resp ect

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105 to r are calculated. These deriv ativ es are d INF l m dr = e i! m r i! m r 2 M 1 X n =0 a n r n + 1 X n =0 ( n + 1) a n r n +2 (7.13) and d 2 INF l m dr 2 = e i! m r 2 m ( r 2 M ) 2 1 X n =0 a n r n 1 + 2 i! m ( r 2 M ) 1 X n =0 h M r 2 M + ( n + 1) i a n r n +1 + 1 X n =0 ( n + 1)( n + 2) a n r n +3 : (7.14) Substituting Equation (7.12) and its t w o deriv ativ es in to Equation (7.9) results to an equation for the co ecien ts a n That equation needs to b e m ultiplied b y ( r 2 M ) 2 in order to get rid of the ( r 2 M ) and ( r 2 M ) 2 factors that app ear in some denominators. An y factors of ( r 2 M ) that sho w up in the n umerators can b e absorb ed in to the sums of r n so that the nal expression con tains sums with p o w ers of r only By setting the factor of eac h r n term equal to zero, a recursion relation for the a n 's is obtained and it is a n = l ( l + 1) n ( n 1) + 4 i! M 2 i! n a n 1 iM ( n 1)(2 n 3) l ( l + 1) n a n 2 + i 2 M 2 ( n 2) 2 n a n 3 : (7.15) F or this recursion relation it is assumed that a 0 = 1 and that a n = 0 for n < 0. The b eha vior of l m ( r ) at the ev en t horizon r = 2 M is no w considered. There, l m ( r ) should b eha v e as an ingoing w a v e, so it can b e assumed to b e HOR l m ( r ) = e i! m r r 1 X n =0 b n ( r 2 M ) n : (7.16)

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106 This expression m ust also b e a solution of Equation (7.9). As in the previous case, the deriv ativ es are calculated rst d HOR l m dr = e i! m r 1 X n =0 i! m + n r b n ( r 2 M ) n 1 1 r 2 1 X n =0 b n ( r 2 M ) n (7.17) and d 2 HOR l m dr 2 =e i! m r 1 X n =0 h 2 i! m M r 2 m r + 2 i! m n + n ( n + 1) r i b n ( r 2 M ) n 2 2 r 1 X n =0 h i! m + n r i b n ( r 2 M ) n 1 + 2 r 3 1 X n =0 b n ( r 2 M ) n : (7.18) Substituting Equation (7.16) and these deriv ativ es in to Equation (7.9) giv es an equation for the co ecien ts b n This equation m ust rst b e m ultiplied b y r 3 so that the factors r 3 r 2 and r that app ear in some denominators v anish. After that, an y factors of r 2 and r that sho w up in the n umerators can b e dealt with b y the substitutions r 2 = ( r 2 M ) 2 + 4 M ( r 2 M ) + 4 M 2 (7.19) r = ( r 2 M ) + 2 M (7.20) so that the nal equation con tains only sums of p o w ers of ( r 2 M ). By setting the co ecien ts of eac h p o w er of ( r 2 M ) equal to zero, the recursion relation for the co ecien ts b n is obtained b n = 12 i! m M ( n 1) + (2 n 3)( n 1) ( l 2 + l + 1) 2 M (4 i! m M n + n 2 ) b n 1 12 i! m M ( n 2) + ( n 2)( n 3) l ( l + 1) 4 M 2 (4 i! m n + n 2 ) b n 2 i! m ( n 3) 2 M 2 (4 i! m M n + n 2 ) b n 3 (7.21)

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107 where b 0 = 1 and b n = 0 for n < 0. The expression (7.12) with the co ecien ts (7.15) is used as the starting solution for the n umerical in tegration of Equation (7.9) from innit y to the radius of the orbit r o This expression is an asymptotic expansion so it is imp ortan t that the in tegration starts at a v alue of r whic h is large enough for the terms of the sum to reac h mac hine accuracy b efore the series b egins to div erge. The result of this rst in tegration is a homogeneous solution of Equation (7.9) for the region r > r o denoted here b y 1 l m ( r ). Then, the expression (7.16) with the co ecien ts (7.21) is used as the starting solution for the n umerical in tegration of Equation (7.9) from the ev en t horizon r = 2 M to the radius of the orbit r o In this case, the in tegration starts at a radius r that is close enough to 2 M for the terms of the sum to reac h mac hine accuracy b efore the series starts to div erge. The result of this in tegration is a homogeneous solution of Equation (7.9) for the region r < r o denoted here as H l m ( r ). All these solutions are sho wn in Figure (7.1). r = 2 M r oH O RI N F( r )H ( r ) Figure 7.1: Solutions obtained with the n umerical co de

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108 Once the t w o solutions are found, they ha v e to b e matc hed at the lo cation of the particle r = r o If l m ( r ) stands for the nal normalized solution, whic h results after m ultiplying H l m ( r ) and 1 l m ( r ) b y the normalization factors ^ A l m and ^ B l m resp ectiv ely then l m ( r ) = 8><>: ^ A l m H l m ( r ) ; for r < r o ^ B l m 1 l m ( r ) ; for r > r o (7.22) the factors ^ A l m and ^ B l m can b e calculated. The ^ is used to a v oid confusion with the regularization parameters A l m and B l m calculated in Chapter 6. The solution itself should b e con tin uous at r = r o : ^ A l m H l m ( r o ) = ^ B l m 1 l m ( r o ) (7.23) but its rst deriv ativ e with resp ect to r also pro vided b y the n umerical in tegration, should b e discon tin uous. The discon tin uit y is determined b y the dieren tial Equation (7.9) when the righ t-hand side term that con tains the function is not set equal to zero. In tegrating Equation (7.9) from r o to r + o giv es Z r + o r o d dr d l m dr dr + Z r + o r o 2( r M ) r ( r 2 M ) d l m dr dr + Z r + o r o 2 m r 2 ( r 2 M ) 2 l m dr Z r + o r o l ( l + 1) r ( r 2 M ) l m dr = q l m r o 2 M Z r + o r o ( r r o ) dr = ) d l m dr r + o r o + 2( r M ) r ( r 2 M ) l m r + o r o Z r + o r o 2 r 2 2 M r ( r M )(2 r 2 M ) r 2 ( r 2 M ) 2 l m dr Z r + o r o 2 m r 2 ( r 2 M ) 2 l m dr Z r + o r o l ( l + 1) r ( r 2 M ) l m dr = q l m r o 2 M ; or, taking in to accoun t the fact that for all the terms of the left-hand side other than the rst one the in tegrand is con tin uous at r = r o and the in tegrals giv e no con tribution ^ B l m d 1 l m dr r o ^ A l m d H l m dr r o = q l m r o 2 M : (7.24)

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109 Equations (7.23) and (7.24) can b e easily solv ed for the factors ^ A l m and ^ B l m and the result is ^ A l m = q l m r o 2 M 1 l m ( r o ) H l m ( r o ) 1 0 l m ( r o ) 1 l m ( r o ) H 0 l m ( r o ) (7.25) ^ B l m = q l m r o 2 M H l m ( r o ) H l m ( r o ) 1 0 l m ( r o ) 1 l m ( r o ) H 0 l m ( r o ) (7.26) where the primes denote deriv ativ es with resp ect to r After the retarded eld is calculated, its con tribution to the self-force along the radial direction can b e obtained. In fact, ev ery l m -comp onen t as w ell as ev ery l -comp onen t can b e calculated b y F ret l m r = d ret l m dr r o (7.27) F ret l r = l X m = l d ret l m dr r o : (7.28) Finally the total con tribution from the retarded eld is F ret r = 1 X l =0 F ret l r : (7.29) A note on the angles and is necessary at this p oin t. The usual Sc h w arzsc hild angles and w ere used for the calculation of the l -mo des of the retarded eld in this c hapter, while the rotated angles and w ere used for the calculation of the singular eld and its regularization parameters in Chapter 6. Questions ma y arise from the fact that the t w o results m ust b e subtracted for the calculation of the self-force. But, as w as men tioned in Chapter 6, the rotation ( ; ) ( ; ) preserv es the spherical harmonic index l so, as long as the summation o v er m precedes the subtraction of the l -mo des, there is no problem with the metho d.

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110 7.2 Numerical Co de The n umerical co de that w as written for the calculation of the radial part of the retarded eld and its con tribution to the self-force is giv en in App endix C. The language used is C. F or the in tegration of Equation (7.9) the adaptiv e stepsize RungeKutta metho d is used, whic h is describ ed in [39]. The functions of [39] that w ere used for the in tegration are: odeint(), rkqs(), rkck() Also, the function plgndr() for the calculation of the Asso ciated Legendre p olynomials, also a v ailable in [39], w as used. The details for all these functions are giv en in [39] and for that reason they are not presen ted here. An eort w as made to k eep the sym b ols used in the co de for the dieren t v ariables iden tical to the ones used in the analysis of this c hapter. In the cases where that w as not p ossible, the commen ts on the co de should mak e the notation clear enough for the reader to follo w.

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CHAPTER 8 APPLICA TIONS AND CONCLUSIONS Some applications of the self-force caclulation are presen ted in this c hapter and some imp ortan t conclusions are dra wn. The eects of the scalar self-force are presen ted in detail. The case of the gra vitational self-force is also discussed. 8.1 Equations of Motion In Chapter 4 it w as explained ho w the self-force on dieren t particles mo ving in curv ed spacetime can b e calculated. Since the ultimate goal is the determination of the motion of those particles, it is imp ortan t to kno w ho w the self-force aects the equations of motion as w ell. Assume that a p oin t particle that carries a nite c harge q (whic h can b e a scalar c harge, an electric c harge or the mass of the particle) is mo ving in a kno wn bac kground spacetime whic h is c haracterized b y the metric g ab F or simplicit y also assume that no other (external to the particle) scalar, electromagnetic or gra vitational elds exist. If the eects of the c harge q are ignored, the p oin t particle mo v es on a bac kground geo desic. If the geo desic is describ ed b y z a ( s ), where s is the prop er time and u a is the tangen t to the geo desic, the geo desic equation is u a r a u b = 0 (8.1) where r a is the co v arian t deriv ativ e with resp ect to the bac kground metric g ab The geo desic is assumed to b e parametrized so that u a u a = 1 : (8.2) 111

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112 If the in teraction of the particle with its o wn elds is to b e tak en in to accoun t, the self-force F a self should b e included in to the equations of motion. Sp ecically if m is the mass of the particle, the equation that holds is mu a r a u b = F b self (8.3) where F a self is giv en b y Equations (4.16), (4.25) and (4.34) for the scalar, electromagnetic and gra vitational eld resp ectiv ely and is of O ( q 2 ). 8.2 Eects of the Scalar Self-F orce In general, Equation (8.3) giv es the eect of the self-force on eac h comp onen t of the 4-v elo cit y of a scalar particle mo ving in spacetime, when the correct expression for the scalar self-force is substituted. If the spherically symmetric Sc h w arzsc hild bac kground is considered, it is easily deduced that only 3 comp onen ts of the 4-v elo cit y are aected b y the self-force. Indeed, as is explained in [40], the Sc h w arzsc hild metric has parit y rerection symmetry so if the initial p osition x a and the initial tangen t u a b oth lie on the equatorial plane ( = 2 ) then the en tire path m ust lie on the equatorial plane as w ell. In addition, due to the spherical symmetry of the Sc h w arzsc hild bac kground, ev ery path can b e brough t b y a rotation to the equatorial plane. Consequen tly one can only consider equatorial orbits, without imp osing an y real restriction on the path of the particle. The analysis that follo ws will b e presen ted in [41] in more detail. Assume that a tra jectory of the Sc h w arzsc hild bac kground is describ ed b y the co ordinates x a : f t s = ( s ) ; r = R ( s ) ; = 2 ; = ( s ) g : (8.4)

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113 If E is the energy p er unit mass and J the angular momen tum p er unit mass of the particle, it can easily b e sho wn that u a = dx a ds = E R ( R 2 M ) ; R ; 0 ; J R 2 u a = g ab u a = E ; R R ( R 2 M ) ; 0 ; J : (8.5) The normalization condition giv en in Equation (8.2) for the 4-v elo cit y giv es a relationship b et w een E J and R sp ecically E 2 R 2 = 1 + J 2 R 2 1 2 M R : (8.6) If the in teraction of the scalar particle with its o wn eld is to b e tak en in to accoun t, the geo desic equation is mo died to u a r a u b = q r b R : (8.7) Calculating the non-v anishing comp onen ts of Equation (8.7) giv es three separate equations, namely du t ds = dE ds = q @ t R (8.8) du ds = dJ ds = q @ R (8.9) d R ds = M R 2 + R 3 M R 4 J 2 + R 2 M R q @ r R : (8.10) In order to predict the eect of the self-force on circular orbits, one can assume that R = 0 and R = 0 at some instan t. Then Equation (8.10) giv es the simple expression J 2 = R 4 R 3 M M R 2 R 2 M R q @ r R (8.11)

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114 whic h giv es the eect of the self-force on the angular momen tum p er unit mass of the scalar particle. Also, Equation (8.6) giv es E 2 = 1 2 M R 1 + J 2 R 2 : (8.12) Com bining Equations (8.11) and (8.12) results in E 2 = ( R 2 M ) 2 R ( R 3 M ) (1 q R @ r R ) (8.13) whic h sho ws the eect of the self-force on the energy of the scalar particle. All the results men tioned up to this p oin t are exact. Since the eect of the scalar c harge q can b e assumed to b e small, the appro ximation that ( R @ r R ) << 1 can b e made, and Equation (8.13) for the energy E can b e written as E = R 2 M p R ( R 3 M ) (1 1 2 q R @ r R ) : (8.14) The orbital frequency is dened as n 0 d dt s = J ( R 2 M ) E R 3 ; (8.15) from Equation (8.5). Substituting Equations (8.11) and (8.14) in to Equation (8.15), and using again the appro ximation ( R @ r R ) << 1, a simple expression for the eect of the self-force on the orbital frequency is obtained: n 0 = r M R 3 h 1 1 2 R ( R 3 M ) M q @ r R i : (8.16) As exp ected, if the in teraction of the particle with its o wn scalar eld is ignored, the w ell-kno wn result n = p M R 3 is reco v ered.

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115 8.2.1 Calculation of the Self-F orce A sample self-force calculation is presen ted in this section. The scalar c harge is assumed to b e q = 1 and to b e mo ving on a circular orbit of radius R = 10 M in a Sc h w arzsc hild bac kground of M = 1. The l m -comp onen ts l m ( r ) of the retarded eld generated b y the scalar c harge and their con tributions F ret l r to the l -m ultip ole mo des of the retarded part of the selfforce are rst calculated for l = 0 ; 1 ; : : : ; 40, using the n umerical co de describ ed in Chapter 7 and giv en in App endix C. Then, the non-zero regularization parameters A r B r and D r are used 1 to subtract o the con tributions of the singular-eld self-force up to O ( l 2 ), according to Equation (6.12). These results are sho wn in Figure (8.1). In that gure, the curv e mark ed F l is the con tribution F ret l r as a function of l The 10 15 20 25 30 35 40 10 -16 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 lFitted ResidualFlA B D E2 E5E3E4 E1 Figure 8.1: Scalar Self-F orce curv es mark ed A B and D sho w the terms ( F ret l r F S l r ) as functions of l where F S l r 1 Remem b er that it w as pro v en that C r = 0 for circular Sc h w arzsc hild orbits, in Chapter 6.

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116 successiv ely includes the con tributions from the regularization parameters A r B r and D r resp ectiv ely It is clear that the series for the self-force con v erges when the terms con taining the regularization parameter B r are included. Ho w ev er, the con v ergence b ecomes m uc h faster the terms con taining the parameter D r are also included. As is rigorously explained in [23], the con tributions from the E a terms in Equation (6.12) are successiv ely of order l 4 l 6 : : : Sp ecically the O ( l 4 ) term is asso ciated with the parameter E 1 r the O ( l 6 ) term is asso ciated with the parameter E 2 r and, in general, the O ( l 2 k ) term is asso ciated with the parameter E k 1 r The parameters E k r are l -indep enden t [23] and can b e determined as follo ws. The con tribution from eac h term is pro v en [23] to b e E k r (2 l + 1) [(2 l 2 k 1)(2 l 2 k + 1) : : : (3 l + 2 k + 1)(2 k + 2 k + 3)] P k +1 = 2 (8.17) where P k +1 = 2 = ( 1) k +1 2 k + 3 2 [(2 k + 1)!!] 2 : (8.18) Assume that the residual ( F ret l r F S l r ) as a function of l resulting after the subtraction of the con tribution of A r B r and D r is calculated. The co ecien ts E k r can b e determined n umerically b y tting that residual, to a linear com bination of terms of the form (8.17) and successiv ely remo ving their con tributions. The curv es mark ed E 1 ; : : : ; E 5 in Figure (8.1) are the residuals resulting after the n umerical t of the parameters E 1 r ; : : : ; E 5 r resp ectiv ely It w as noticed that tting more than four of the co ecien ts E k r did not impro v e the residual, whic h had already reac hed mac hine accuracy b y the fourth tting. 8.2.2 Change in Orbital F requency It w as sho wn in Section (8.2) that the orbital frequency of the circular orbit is aected b y the self-force. Using Equation (8.16) the dierence in frequency can b e

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117 calculated; it is n 0 n n = n n = 1 2 R ( R 3 M ) M q @ r R : (8.19) The eect of the self-force on the orbital frequency is sho wn in Figure (8.2), where the ratio (n = n) is plotted as a function of ( R = M ). The orbital frequency n b efore the self-force eects are included is also plotted for comparison. 0 20 40 60 80 100R/M 10 -4 10 -3 10 -2 10 -1M x Orbital frequency WM DW/W x mM/q2 Figure 8.2: Change in Orbital F requency 8.3 Eects of the Gra vitational Self-F orce Assume that a particle of mass m is mo ving in a bac kground spacetime describ ed b y the metric g ab with no other extrernal elds presen t. Equation (8.3) giv es the w orldline of the particle that includes the self-force eect if F a self is substituted from Equation (4.34). Clearly giv en the initial p osition and 4-v elo cit y of the particle, that w orldline is determined exclusiv ely b y the bac kground metric g ab and the metric

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118 p erturbation h Rab induced b y the particle. Both of these con tributions are solutions of the homogeneous Einstein equations. An observ er making lo cal measuremen ts of the metric in the vicinit y of the particle measures t w o con tributions to it. The rst con tribution comes from the bac kground metric g ab com bined with the h Rab -part of the actual metric p erturbation generated b y the particle. Since b oth these terms ob ey the homogeneous Einstein equations, the observ er sees no source for either one, so no lo cal measuremen t can allo w him to distinguish h Rab from the bac kground metric g ab The second con tribution is h Sab a metric p erturbation also generated b y the particle whic h, according to the observ er, is sourced b y the particle. The lo cal observ er sees the particle mo v e in a b ackgr ound geo desic of the metric ( g ab + h Rab ). Because he only mak es lo cal measuremen ts, he observ es no radiation generated b y the mo ving particle and consequen tly no eect that he could describ e as radiation reaction [22].

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APPENDIX A GR TENSOR CODE F OR THE SINGULAR FIELDS restart;readlib(grii);grtensor();qload(thz);e := epsilon; ord := [t, x, y, z, Bxx, Byy]; EpsEqn := {e^10=0, e^9=0, e^8=0, e^7=0, e^6=0, e^5=0, e^4=0, e^3=0, e^2=0}; Traces := {Ezz=-Exx-Eyy, Bzz=-Bxx-Byy}; # The Ricci Tensor Should Be Zero # grcalc(detg);grmap(detg, series, 'x', e, 4); grmap(detg, convert, 'x', polynom); grmap(detg, expand, 'x'); grmap(detg, subs, EpsEqn, 'x'); grmap(detg, collect, 'x', ord, distributed, factor); grmap(detg, subs, Traces, 'x'); grmap(detg, collect, 'x', ord, distributed,factor); grdisplay(detg);grcalc(g(up,up));grmap(g(up,up), series, 'x', e, 4); grmap(g(up,up), convert, 'x', polynom); grmap(g(up,up), expand, 'x'); grmap(g(up,up), subs, EpsEqn, 'x'); grmap(g(up,up), collect, 'x', ord, distributed, factor); grmap(g(up,up), subs, Traces, 'x'); 119

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120 grmap(g(up,up), collect, 'x', ord, distributed, factor); grdisplay(g(up,up));grcalc(Chr(dn,dn,dn));grmap(Chr(dn,dn,dn), series, 'x', e, 4); grmap(Chr(dn,dn,dn),conv ert, 'x', polynom); grmap(Chr(dn,dn,dn), expand, 'x'); grmap(Chr(dn,dn,dn), subs, EpsEqn, 'x'); grmap(Chr(dn,dn,dn), collect, 'x', ord, distributed, factor); grmap(Chr(dn,dn,dn), subs, Traces, 'x'); grmap(Chr(dn,dn,dn), collect, 'x', ord, distributed, factor); grdisplay(Chr(dn,dn,dn)) ; grcalc(Chr(dn,dn,up));grmap(Chr(dn,dn,up), series, 'x', e, 4); grmap(Chr(dn,dn,up), convert, 'x', polynom); grmap(Chr(dn,dn,up), expand, 'x'); grmap(Chr(dn,dn,up), subs, EpsEqn, 'x'); grmap(Chr(dn,dn,up), collect, 'x', ord, distributed, factor); grmap(Chr(dn,dn,up), subs, Traces, 'x'); grmap(Chr(dn,dn,up), collect, 'x', ord, distributed, factor); grdisplay(Chr(dn,dn,up)) ; grcalc(Chr(dn,up,up));grmap(Chr(dn,up,up), series, 'x', e, 4); grmap(Chr(dn,up,up), convert, 'x', polynom); grmap(Chr(dn,up,up), expand, 'x'); grmap(Chr(dn,up,up), subs, EpsEqn, 'x'); grmap(Chr(dn,up,up), collect, 'x', ord, distributed, factor); grmap(Chr(dn,up,up), subs, Traces, 'x'); grmap(Chr(dn,up,up), collect, 'x', ord, distributed, factor); grdisplay(Chr(dn,up,up)) ; grcalc(Chr(dn,up,dn));grmap(Chr(dn,up,dn), series, 'x', e, 4); grmap(Chr(dn,up,dn), convert, 'x', polynom); grmap(Chr(dn,up,dn), expand, 'x'); grmap(Chr(dn,up,dn), subs, EpsEqn, 'x'); grmap(Chr(dn,up,dn), collect, 'x', ord, distributed, factor); grmap(Chr(dn,up,dn), subs, Traces, 'x'); grmap(Chr(dn,up,dn), collect, 'x', ord, distributed, factor); grdisplay(Chr(dn,up,dn)) ; grcalc(R(dn,dn,dn,dn));

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121 grmap(R(dn,dn,dn,dn), series, 'x', e, 4); grmap(R(dn,dn,dn,dn), convert, 'x', polynom); grmap(R(dn,dn,dn,dn), expand, 'x'); grmap(R(dn,dn,dn,dn), subs, EpsEqn, 'x'); grmap(R(dn,dn,dn,dn), collect, 'x', ord, distributed, factor); grmap(R(dn,dn,dn,dn), subs, Traces, 'x'); grmap(R(dn,dn,dn,dn), collect, 'x', ord, distributed, factor); grdisplay(R(dn,dn,dn,dn) ); grcalc(R(up,up,dn,dn));grmap(R(up,up,dn,dn), series, 'x', e, 4); grmap(R(up,up,dn,dn), convert, 'x', polynom); grmap(R(up,up,dn,dn), expand, 'x'); grmap(R(up,up,dn,dn), subs, EpsEqn, 'x'); grmap(R(up,up,dn,dn), collect, 'x', ord, distributed, factor); grmap(R(up,up,dn,dn), subs, Traces, 'x'); grmap(R(up,up,dn,dn), collect, 'x', ord, distributed, factor); grdisplay(R(up,up,dn,dn) ); grcalc(R(up,dn,dn,dn));grmap(R(up,dn,dn,dn), series, 'x', e, 4); grmap(R(up,dn,dn,dn), convert, 'x', polynom); grmap(R(up,dn,dn,dn), expand, 'x'); grmap(R(up,dn,dn,dn), subs, EpsEqn, 'x'); grmap(R(up,dn,dn,dn), collect, 'x', ord, distributed, factor); grmap(R(up,dn,dn,dn), subs, Traces, 'x'); grmap(R(up,dn,dn,dn), collect, 'x', ord, distributed, factor); grdisplay(R(up,dn,dn,dn) ); grcalc(R(dn,up,dn,up));grmap(R(dn,up,dn,up), series, 'x', e, 4); grmap(R(dn,up,dn,up), convert, 'x', polynom); grmap(R(dn,up,dn,up), expand, 'x'); grmap(R(dn,up,dn,up), subs, EpsEqn, 'x'); grmap(R(dn,up,dn,up), collect, 'x', ord, distributed, factor); grmap(R(dn,up,dn,up), subs, Traces, 'x'); grmap(R(dn,up,dn,up), collect, 'x', ord, distributed, factor); grdisplay(R(dn,up,dn,up) ); grcalc(R(dn,dn));grmap(R(dn,dn), series, 'x', e, 4); grmap(R(dn,dn), convert, 'x', polynom); grmap(R(dn,dn), expand, 'x'); grmap(R(dn,dn), subs, EpsEqn, 'x'); grmap(R(dn,dn), collect, 'x', ord, distributed, factor);

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122 grmap(R(dn,dn), subs, Traces, 'x'); grmap(R(dn,dn), collect, 'x', ord, distributed, factor); grdisplay(R(dn,dn)); # Useful Quantities # grdef(`etadn{(a b)} := -kdelta{^\$t a}*kdelta{^\$t b} +kdelta{^\$x a}*kdelta{^\$x b} + kdelta{^\$y a}*kdelta{^\$y b} +kdelta{^\$z a}*kdelta{^\$z b}`); grcalc(etadn(dn,dn));grdisplay(etadn(dn,dn));grdef(`etaup{(^a ^b)} := -kdelta{\$t ^a}*kdelta{\$t ^b} +kdelta{\$x ^a}*kdelta{\$x ^b} + kdelta{\$y ^a}*kdelta{\$y ^b} +kdelta{\$z ^a}*kdelta{\$z ^b}`); grcalc(etaup(up,up));grdisplay(etaup(up,up));grdef(`eps{[a b c]} := LevCS{d a b c}*kdelta{\$t ^d}`); grcalc(eps(dn,dn,dn));grdisplay(eps(dn,dn,dn)) ; grdef(`EE{(a b)} := kdelta{^\$x a}*kdelta{^\$x b}*e*Exx + kdelta{^\$y a}*kdelta{^\$y b}*e*Eyy + kdelta{^\$z a}*kdelta{^\$z b}*e*Ezz + kdelta{^\$x a}*kdelta{^\$y b}*e*Exy + kdelta{^\$x a}*kdelta{^\$z b}*e*Exz + kdelta{^\$y a}*kdelta{^\$z b}*e*Eyz`); grcalc(EE(dn,dn));grdisplay(EE(dn,dn));grdef(`BB{(a b)} := kdelta{^\$x a}*kdelta{^\$x b}*e*Bxx + kdelta{^\$y a}*kdelta{^\$y b}*e*Byy + kdelta{^\$z a}*kdelta{^\$z b}*e*Bzz + kdelta{^\$x a}*kdelta{^\$y b}*e*Bxy + kdelta{^\$x a}*kdelta{^\$z b}*e*Bxz + kdelta{^\$y a}*kdelta{^\$z b}*e*Byz`); grcalc(BB(dn,dn));grdisplay(BB(dn,dn));

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123 grdef(`h0{(b c)} := kdelta{^\$t b}*kdelta{^\$x c}*2*A*mu*y/((x^2+y^2+z^2)^ (3/ 2)) + kdelta{^\$t b}*kdelta{^\$y c}*(-2)*A*mu*x/((x^2+y^2+z^ 2)^ (3/2 ))`) ; grcalc(h0(dn,dn));grdisplay(h0(dn,dn));grdef(`epsx{[a b]} := eps{c a b}*kdelta{\$x ^c}`); grcalc(epsx(dn,dn));grdisplay(epsx(dn,dn));grdef(`epsy{[a b]} := eps{c a b}*kdelta{\$y ^c}`); grcalc(epsy(dn,dn));grdisplay(epsy(dn,dn));grdef(`epsz{[a b]} := eps{c a b}*kdelta{\$z ^c}`); grcalc(epsz(dn,dn));grdisplay(epsz(dn,dn)); # Chech What The Equations Look Like: # grdef(`h1test{(a b)} := kdelta{^\$t a}*kdelta{^\$t b}*e*h1tt + kdelta{^\$t a}*kdelta{^\$x b}*e*h1tx + kdelta{^\$t a}*kdelta{^\$y b}*e*h1ty + kdelta{^\$t a}*kdelta{^\$z b}*e*h1tz + kdelta{^\$x a}*kdelta{^\$x b}*e*h1xx + kdelta{^\$x a}*kdelta{^\$y b}*e*h1xy + kdelta{^\$x a}*kdelta{^\$z b}*e*h1xz + kdelta{^\$y a}*kdelta{^\$y b}*e*h1yy + kdelta{^\$y a}*kdelta{^\$z b}*e*h1yz + kdelta{^\$z a}*kdelta{^\$z b}*e*h1zz`); grcalc(h1test(dn,dn));grdisplay(h1test(dn,dn)) ; grdef(`htest{(a b)} := h0{a b} + h1test{a b}` ); grcalc(htest(dn,dn));grdisplay(htest(dn,dn));grdef(`htesttrace := htest{^c c}`); grcalc(htesttrace);grmap(htesttrace, series, 'x', e, 4);

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124 grmap(htesttrace, convert, 'x', polynom); grmap(htesttrace, expand, 'x'); grmap(htesttrace, subs, EpsEqn, 'x'); grmap(htesttrace, collect, 'x', ord, distributed, factor); grmap(htesttrace, subs, Traces, 'x'); grmap(htesttrace, collect, 'x', ord, distributed, factor); grdisplay(htesttrace);grdef(`hbartest{(a b)} := htest{a b} (1/2)*g{a b}*htesttrace`); grcalc(hbartest(dn,dn));grmap(hbartest(dn,dn), series, 'x', e, 4); grmap(hbartest(dn,dn), convert, 'x', polynom); grmap(hbartest(dn,dn), expand, 'x'); grmap(hbartest(dn,dn), subs, EpsEqn, 'x'); grmap(hbartest(dn,dn), collect, 'x', ord, distributed, factor); grmap(hbartest(dn,dn), subs, Traces, 'x'); grmap(hbartest(dn,dn), collect, 'x', ord, distributed, factor); grdisplay(hbartest(dn,dn )); # Left-Hand Side Of The Equation # grcalc(hbartest(dn,dn,cd n)); grmap(hbartest(dn,dn,cdn ), series,'x', e, 4); grmap(hbartest(dn,dn,cdn ), convert, 'x', polynom); grmap(hbartest(dn,dn,cdn ), expand, 'x'); grmap(hbartest(dn,dn,cdn ), subs, EpsEqn, 'x'); grmap(hbartest(dn,dn,cdn ), collect, 'x', ord, distributed, factor); grmap(hbartest(dn,dn,cdn ), subs, Traces, 'x'); grmap(hbartest(dn,dn,cdn ), collect, 'x', ord, distributed, factor); grdisplay(hbartest(dn,dn ,cdn )); grdef(`Dhbartest{a b ^c} := hbartest{a b ;^c}`); grcalc(Dhbartest(dn,dn,u p)); grmap(Dhbartest(dn,dn,up ), series, 'x', e, 4); grmap(Dhbartest(dn,dn,up ), convert, 'x', polynom); grmap(Dhbartest(dn,dn,up ), expand, 'x'); grmap(Dhbartest(dn,dn,up ), subs, EpsEqn, 'x'); grmap(Dhbartest(dn,dn,up ), collect, 'x', ord, distributed, factor); grmap(Dhbartest(dn,dn,up ), subs, Traces, 'x'); grmap(Dhbartest(dn,dn,up ), collect, 'x', ord, distributed, factor); grdisplay(Dhbartest(dn,d n,up )); grdef(`DDhbartest{a b} := Dhbartest{a b ^c ;c}`);

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125 grcalc(DDhbartest(dn,dn) ); grmap(DDhbartest(dn,dn), series, 'x', e, 4); grmap(DDhbartest(dn,dn), convert, 'x', polynom); grmap(DDhbartest(dn,dn), expand, 'x'); grmap(DDhbartest(dn,dn), subs, EpsEqn, 'x'); grmap(DDhbartest(dn,dn), collect, 'x', ord, distributed, factor); grmap(DDhbartest(dn,dn), subs, Traces, 'x'); grmap(DDhbartest(dn,dn), collect, 'x', ord, distributed, factor); grdisplay(DDhbartest(dn, dn)) ; grdef(`Rhbartest{a b} := 2*R{a ^c b ^d}*hbartest{c d}`); grcalc(Rhbartest(dn,dn)) ; grmap(Rhbartest(dn,dn), series, 'x', e, 4); grmap(Rhbartest(dn,dn), convert, 'x', polynom); grmap(Rhbartest(dn,dn), expand, 'x'); grmap(Rhbartest(dn,dn), subs, EpsEqn, 'x'); grmap(Rhbartest(dn,dn), collect, 'x', ord, distributed, factor); grmap(Rhbartest(dn,dn), subs, Traces, 'x'); grmap(Rhbartest(dn,dn), collect, 'x', ord, distributed, factor); grdisplay(Rhbartest(dn,d n)); grdef(`LHStest{(a b)} := DDhbartest{a b} + Rhbartest{a b}`); grcalc(LHStest(dn,dn));grmap(LHStest(dn,dn), series, 'x', e, 4); grmap(LHStest(dn,dn), convert, 'x', polynom); grmap(LHStest(dn,dn), expand, 'x'); grmap(LHStest(dn,dn), subs, EpsEqn, 'x'); grmap(LHStest(dn,dn), collect, 'x', ord, distributed, factor); grmap(LHStest(dn,dn), subs, Traces, 'x'); grmap(LHStest(dn,dn), collect, 'x', ord, distributed, factor); grdisplay(LHStest(dn,dn) ); # Assume A Specific Form For The Solution # grdef(`AngM{^a} := [0, 0, 0, A]`); grcalc(AngM(up)); grdisplay(AngM(up)); grdef(`h1{(a b)} := kdelta{^\$t a}*kdelta{^\$t b}*(mu/((x^2+y^2+z^2)^(3/ 2))) (att*etadn{i l}*AngM{^l}*x{^i}*BB{j k}*x{^j}*x{^k} + btt*AngM{^i}*BB{i j}*x{^j}*(x^2+y^2+z^2))

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128 grdef(`hh{(a b)} := h0{a b} + h1{a b}`); grcalc(hh(dn,dn));grdef(`htrace := hh{^c c}`); grcalc(htrace);grmap(htrace, series, 'x', e, 4); grmap(htrace, convert, 'x', polynom); grmap(htrace, expand, 'x'); grmap(htrace, subs, EpsEqn, 'x'); grmap(htrace, collect, 'x', ord, distributed, factor); grmap(htrace, subs, Traces, 'x'); grmap(htrace, collect, 'x', ord, distributed, factor); grdisplay(htrace);grdef(`hbar{(a b)} := hh{a b} (1/2)*g{a b}*htrace`); grcalc(hbar(dn,dn)) ; grmap(hbar(dn,dn), series, 'x', e, 4); grmap(hbar(dn,dn), convert, 'x', polynom); grmap(hbar(dn,dn), expand,'x'); grmap(hbar(dn,dn), subs, EpsEqn, 'x'); grmap(hbar(dn,dn), collect, 'x', ord, distributed, factor); grmap(hbar(dn,dn), subs, Traces, 'x'); grmap(hbar(dn,dn), collect, 'x', ord, distributed, factor); grdisplay(hbar(dn,dn));grcalc(hbar(dn,dn,cdn));grmap(hbar(dn,dn,cdn), series, 'x', e, 4); grmap(hbar(dn,dn,cdn), convert, 'x', polynom); grmap(hbar(dn,dn,cdn), expand, 'x'); grmap(hbar(dn,dn,cdn), subs, EpsEqn, 'x'); grmap(hbar(dn,dn,cdn), collect, 'x', ord, distributed, factor); grmap(hbar(dn,dn,cdn), subs, Traces, 'x'); grmap(hbar(dn,dn,cdn), collect, 'x', ord, distributed, factor); grdisplay(hbar(dn,dn,cdn )); grdef(`Dhbar{a b ^c} := hbar{a b ;^c}`); grcalc(Dhbar(dn,dn,up));grmap(Dhbar(dn,dn,up), series, 'x', e, 4); grmap(Dhbar(dn,dn,up), convert, 'x', polynom); grmap(Dhbar(dn,dn,up), expand, 'x'); grmap(Dhbar(dn,dn,up), subs, EpsEqn, 'x'); grmap(Dhbar(dn,dn,up), collect, 'x', ord, distributed, factor);

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129 grmap(Dhbar(dn,dn,up), subs, Traces, 'x'); grmap(Dhbar(dn,dn,up), collect, 'x', ord, distributed, factor); grdisplay(Dhbar(dn,dn,up )); grdef(`DDhbar{a b} := Dhbar{a b ^c ;c}`); grcalc(DDhbar(dn,dn));grmap(DDhbar(dn,dn), series, 'x', e, 4); grmap(DDhbar(dn,dn), convert, 'x', polynom); grmap(DDhbar(dn,dn), expand, 'x'); grmap(DDhbar(dn,dn), subs, EpsEqn, 'x'); grmap(DDhbar(dn,dn), collect, 'x', ord, distributed, factor); grmap(DDhbar(dn,dn), subs, Traces, 'x'); grmap(DDhbar(dn,dn), collect, 'x', ord, distributed, factor); grdisplay(DDhbar(dn,dn)) ; grdef(`Rhbar{a b} := 2*R{a ^c b ^d}*hbar{c d}`); grcalc(Rhbar(dn,dn));grmap(Rhbar(dn,dn), series, 'x', e, 4); grmap(Rhbar(dn,dn), convert, 'x', polynom); grmap(Rhbar(dn,dn), expand, 'x'); grmap(Rhbar(dn,dn), subs, EpsEqn, 'x'); grmap(Rhbar(dn,dn), collect, 'x', ord, distributed, factor); grmap(Rhbar(dn,dn), subs, Traces, 'x'); grmap(Rhbar(dn,dn), collect, 'x', ord, distributed, factor); grdisplay(Rhbar(dn,dn));grdef(`LHS{(a b)} := DDhbar{a b} + Rhbar{a b}`); grcalc(LHS(dn,dn));grmap(LHS(dn,dn), series, 'x', e, 4); grmap(LHS(dn,dn), convert, 'x', polynom); grmap(LHS(dn,dn), expand, 'x'); grmap(LHS(dn,dn), subs, EpsEqn, 'x'); grmap(LHS(dn,dn), collect, 'x', ord, distributed, factor); grmap(LHS(dn,dn), subs, Traces, 'x'); grmap(LHS(dn,dn), collect, 'x', ord, distributed, factor); grdisplay(LHS(dn,dn));grmap(LHS(dn,dn), subs, {e=1}, 'x'); grmap(LHS(dn,dn), collect, 'x', ord, distributed, factor); grdisplay(LHS(dn,dn));grmap(LHS(dn,dn), collect, 'x', [t, x, y, z, Exx, Eyy], distributed, factor);

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132 (1/3)*etadn{i k}*AngM{^k}*BB{j \$z}*etadn{\$y l}*x{^l}*x{^i}*x{^j} (5/3)*AngM{^k}*etadn{k \$y}*BB{\$z j}*x{^j}*(x^2+y^2+z^2) (5/3)*AngM{^k}*etadn{k \$z}*BB{\$y j}*x{^j}*(x^2+y^2+z^2) (1/3)*AngM{^k}*etadn{k \$y}*etadn{\$z l}*BB{i j}*x{^l}*x{^i}*x{^j} (1/3)*AngM{^k}*etadn{k \$z}*etadn{\$y l}*BB{i j}*x{^l}*x{^i}*x{^j}) /((x^2+y^2+z^2)^(3/2)) ` ): grcalc(h1(dn,dn)):grdef(`hh{(a b)} := h0{a b} + h1{a b}`): grcalc(hh(dn,dn)):grdef(`htrace := hh{^c c}`): grcalc(htrace); grmap(htrace, series, 'x', e, 4): grmap(htrace, convert, 'x', polynom): grmap(htrace, expand, 'x'): grmap(htrace, subs, EpsEqn, 'x'): grmap(htrace, collect, 'x', ord, distributed, factor): grmap(htrace, subs, Traces, 'x'): grmap(htrace, collect, 'x', ord, distributed, factor): grdisplay(htrace):grdef(`hbar{(a b)} := hh{a b} (1/2)*g{a b}*htrace`): grcalc(hbar(dn,dn)):grmap(hbar(dn,dn), series, 'x', e, 4): grmap(hbar(dn,dn), convert, 'x', polynom): grmap(hbar(dn,dn), expand, 'x'): grmap(hbar(dn,dn), subs, EpsEqn, 'x'): grmap(hbar(dn,dn), collect, 'x', ord, distributed, factor): grmap(hbar(dn,dn), subs, Traces, 'x'): grmap(hbar(dn,dn), collect, 'x', ord, distributed, factor): grdisplay(hbar(dn,dn)):grcalc(hbar(dn,dn,cdn)):grmap(hbar(dn,dn,cdn), series, 'x', e, 4): grmap(hbar(dn,dn,cdn), convert, 'x', polynom): grmap(hbar(dn,dn,cdn), expand, 'x'): grmap(hbar(dn,dn,cdn), subs, EpsEqn, 'x'): grmap(hbar(dn,dn,cdn), collect, 'x', ord, distributed, factor): grmap(hbar(dn,dn,cdn), subs, Traces, 'x'): grmap(hbar(dn,dn,cdn), collect, 'x', ord, distributed, factor): grdisplay(hbar(dn,dn,cdn )): grdef(`Dhbar{(a b) ^c} := hbar{a b ;^c}`): grcalc(Dhbar(dn,dn,up)):

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133 grmap(Dhbar(dn,dn,up), series, 'x', e, 4): grmap(Dhbar(dn,dn,up), convert, 'x', polynom): grmap(Dhbar(dn,dn,up), expand, 'x'): grmap(Dhbar(dn,dn,up), subs, EpsEqn, 'x'): grmap(Dhbar(dn,dn,up), collect, 'x', ord, distributed, factor): grmap(Dhbar(dn,dn,up), subs, Traces, 'x'): grmap(Dhbar(dn,dn,up), collect, 'x', ord, distributed, factor): grdisplay(Dhbar(dn,dn,up )): grdef(`DDhbar{(a b)} := Dhbar{a b ^c ;c}`): grcalc(DDhbar(dn,dn)):grmap(DDhbar(dn,dn), series, 'x', e, 4): grmap(DDhbar(dn,dn), convert, 'x', polynom): grmap(DDhbar(dn,dn), expand, 'x'): grmap(DDhbar(dn,dn), subs, EpsEqn, 'x'): grmap(DDhbar(dn,dn), collect, 'x', ord, distributed, factor): grmap(DDhbar(dn,dn), subs, Traces, 'x'): grmap(DDhbar(dn,dn), collect, 'x', ord, distributed, factor): grdisplay(DDhbar(dn,dn)) : grdef(`Rhbar{a b} := 2*R{a ^c b ^d}*hbar{c d}`): grcalc(Rhbar(dn,dn)):grmap(Rhbar(dn,dn), series, 'x', e, 4): grmap(Rhbar(dn,dn), convert, 'x', polynom): grmap(Rhbar(dn,dn), expand, 'x'): grmap(Rhbar(dn,dn), subs, EpsEqn, 'x'): grmap(Rhbar(dn,dn), collect, 'x', ord, distributed, factor): grmap(Rhbar(dn,dn), subs, Traces, 'x'): grmap(Rhbar(dn,dn), collect, 'x', ord, distributed, factor): grdisplay(Rhbar(dn,dn));grdef(`LHS{(a b)} := DDhbar{a b} + Rhbar{a b}`): grcalc(LHS(dn,dn)):grmap(LHS(dn,dn), series, 'x', e, 4): grmap(LHS(dn,dn), convert, 'x', polynom): grmap(LHS(dn,dn), expand, 'x'): grmap(LHS(dn,dn), subs, EpsEqn, 'x'): grmap(LHS(dn,dn), collect, 'x', ord, distributed, factor): grmap(LHS(dn,dn), subs, Traces, 'x'): grmap(LHS(dn,dn), collect, 'x', ord, distributed, factor): grdisplay(LHS(dn,dn));grmap(LHS(dn,dn), subs, {e=1}, 'x'): grmap(LHS(dn,dn), collect, 'x', ordtxyz, distributed, factor): grdisplay(LHS(dn,dn)):

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APPENDIX B DECOMPOSITION OF ~ P Q A detailed description of the spherical harmonic decomp osition of the pro duct ~ p q is presen ted in this app endix, where p is an y in teger and q can b e an y real n um b er, ev en though in ev ery case in whic h this decomp osition will b e used q will b e either an in teger or a half-in teger. The notation used is ~ 2 = r o 2 r o 2 M + 2 r 2 o r o 2 M r o 3 M (1 cos ) (B.1) = 1 M r o 2 M sin 2 (B.2) = r r o : (B.3) In order to simplify the equations, the follo wing sym b ols are used: A = r o 2 r o 2 M (B.4) B = 2 r 2 o r o 2 M r o 3 M (B.5) C = M r o 2 M (B.6) r 2 = 2 ( r o 3 M ) 2 r o ( r o 2 M ) 2 (B.7) 2 = 2 ( r o 3 M ) 2 r o ( r o 2 M ) 2 = r 2 : (B.8) Using Equations (B.5)-(B.8) one gets ~ = B 1 2 1 2 r 2 + 1 cos 1 2 = B 1 2 1 2 2 + 1 cos 1 2 (B.9) ~ p q = B p 2 p 2 q r 2 + 1 cos p 2 = B p 2 p 2 q 2 + 1 cos p 2 : (B.10) 134

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135 F or the decomp osition then it is assumed that ~ p q = B p 2 p 2 q r 2 + 1 cos p 2 = B p 2 1 X l =0 l X m = l E p;q l ;m ( r 2 ; C ) Y l m ( ; ) (B.11) and the co ecien ts E p;q l ;m are calculated b y m ultiplying b oth sides of the ab o v e equation with the complex-conjugate spherical harmonics Y l m ( ; ) and in tegrating o v er = [0 ; ] and = [0 ; 2 ]. Using the results giv en in the discussion of spherical harmonics in [38], the follo wing expression for the co ecien ts is obtained E p;q l ;m ( r 2 ; C ) = ( 1) m s 2 l + 1 4 ( l + m )! ( l m )! Z 2 0 d e im p 2 q Z 1 1 du ( 2 + 1 u ) p 2 P m l ( u ) (B.12) where u = cos and P m l ( u ) are the Asso ciated Legendre p olynomials. In the follo wing it is also assumed that P m =0 l ( u ) = P l ( u ), whic h are the Legendre p olynomials. It should b e noted that, b ecause only ev en p o w ers of sin sho w up in the expression for the co ecien ts E p;q l ;m only the ev en v alues of m con tribute to the sum (B.11) and for all o dd m the co ecien ts E p;q l ;m are equal to 0. In all the equations that follo w, m is assumed to b e ev en. First, the in tegral I p l m = Z 1 1 du ( 2 + 1 u ) p 2 P m l ( u ) (B.13) is calculated. Assuming that m is p ositiv e, Equation 8.911 of [42] giv es the m deriv ativ e of the Legendre p olynomials d m P l ( u ) du m = 1 2 l [ l 2 ] X k =0 ( 1) k (2 l 2 k + 1) ( k + 1)( l k + 1)( l 2 k m + 1) u l 2 k m (B.14)

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136 where ( x ) is the usual -function and [ l 2 ] stands for the largest in teger smaller or equal to l 2 Then, Equation 8.810 of [42] giv es for the asso ciated Legendre p olynomials, for p ositiv e m P m l ( u ) = (1 u 2 ) m 2 1 2 l [ l 2 ] X k =0 ( 1) k (2 l 2 k + 1) ( k + 1)( l k + 1)( l 2 k m + 1) u l 2 k m (B.15) where the righ t-hand side is a p olynomial, since m is ev en. The case of negativ e m can b e dealt with if one tak es in to accoun t the fact that for an y m : P m l ( u ) = ( l j m j + 1) ( l + m + 1) P j m j l ( u ) (B.16) whic h can b e easily deriv ed from Equation 8.752 of [42]. Finally the expression of the asso ciated Legendre p olynomials that is used in the calculation of the in tegral I p l m is found to b e P m l ( u ) = ( l j m j + 1) ( l + m + 1) (1 u 2 ) j m j 2 1 2 l [ l 2 ] X k =0 ( 1) k (2 l 2 k + 1) ( k + 1)( l k + 1)( l 2 k j m j + 1) u l 2 k j m j (B.17) whic h is v alid for all ev en m p ositiv e or negativ e. Substituting Equation (B.17) in to Equation (B.13) giv es I p l m = ( l j m j + 1) ( l + m + 1) 1 2 l [ l 2 ] X k =0 ( 1) k (2 l 2 k + 1) ( k + 1)( l k + 1)( l 2 k j m j + 1) Z 1 1 du ( 2 + 1 u ) p 2 (1 u 2 ) j m j 2 u l j m j 2 k : (B.18)

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137 The in tegral of Equation (B.18) can b e calculated if the term (1 u 2 ) j m j 2 is expanded in p o w ers of u sp ecically I = Z 1 1 du ( 2 + 1 u ) p 2 (1 u 2 ) j m j 2 u l j m j 2 k = j m j 2 X =0 ( j m j 2 + 1) ( + 1)( j m j 2 + 1) Z 1 1 du ( 2 + 1 u ) p 2 u l j m j 2 k +2 : (B.19) The task, then, shifts to the calculation of the in tegral I ( a;b ) = Z 1 1 du u a ( 2 + 1 u ) b 2 (B.20) where a and b are in tegers, with a 0. P erforming in tegration b y parts once results in I ( a;b ) = Z 1 1 u a 1 b 2 + 1 d ( 2 + 1 u ) b 2 +1 = 1 b 2 + 1 ( 2 ) b 2 +1 + ( 1) a 1 b 2 + 1 (2 + 2 ) b 2 +1 + 2 a b + 2 Z 1 1 u a 1 ( 2 + 1 u ) b +2 2 = 2 b + 2 h ( 2 ) b 2 +1 + ( 1) a (2 + 2 ) b 2 +1 i + 2 a b + 2 I ( a 1 ;b +2) (B.21) whic h is a recursion relation for I ( a;b ) Using the fact that I (0 ;b ) = 2 b + 2 h ( 2 ) b 2 +1 + (2 + 2 ) b 2 +1 i (B.22) it can b e pro v en b y induction that the general in tegral I ( a;b ) is equal to I ( a;b ) = a X n =0 2 n +1 ( b + 2)( b + 4) : : : [ b + 2( n + 1)] ( a + 1) ( a n + 1) h ( 1) a n (2 + 2 ) b 2 + n +1 ( 2 ) b 2 + n +1 i : (B.23)

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138 Finally using Equations (B.23) and (B.19) in to Equation (B.18), the in tegral I p l m ends up b eing I p l m = ( l j m j + 1) ( l + m + 1) 1 2 l [ l 2 ] X k =0 ( 1) k (2 l 2 k + 1) ( k + 1)( l k + 1)( l 2 k j m j + 1) j m j 2 X =0 ( j m j 2 + 1) ( + 1)( j m j 2 + 1) l j m j 2 k +2 X n =0 2 n +1 ( p + 2)( p + 4) : : : [ p + 2( n + 1)] ( l j m j 2 k + 2 + 1) ( l j m j 2 k + 2 n + 1) h ( 1) l n (2 + 2 ) p 2 + n +1 ( 2 ) p 2 + n +1 i : (B.24) A short commen t ab out the term ( 2 ) p 2 + n +1 should b e made at this p oin t. If p is an o dd in teger, that term implies that the square ro ot of 2 or of r 2 m ust b e tak en. One thing that ma y b e am biguous is the sign of the square ro ot, + or that m ust b e considered. That p oin t is discussed in Chapter 6 of this dissertation, where the ph ysical in terpretation of the quan tities b ecomes clear and the correct sign to use can b e easily recognized. F or the calculations in this app endix it is enough to in terpret the term ( 2 ) 1 2 as the appr opriate square ro ot of 2 Con tin uing the calculation of the co ecien ts E p;q l ;m substituting Equation (B.24) in to Equation (B.12) sho ws that the next t w o in tegrals to b e done are I 1 = ( 1) l n Z 2 0 d e im p 2 q (2 + 2 ) p 2 + n +1 = ( 1) l n Z 2 0 d e im p 2 q (2 + r 2 ) p 2 + n +1 (B.25) I 2 = Z 2 0 d e im p 2 q ( 2 ) p 2 + n +1 = Z 2 0 d e im p 2 q ( r 2 ) p 2 + n +1 (B.26) and their dierence m ust b e m ultiplied b y the appropriate factor giv en b y Equation (B.24). F or b oth these in tegrals, the exp onen tial e im m ust b e written in terms of

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139 the v ariable In general e im = cos ( m ) i sin( m ) (B.27) but, since only ev en m 's are considered and the remaining of the in tegrand is a function of sin 2 the sin ( m ) part of the exp onen tial giv es 0 once the in tegration is p erformed. So, as far as the in tegrals are concerned, the exp onen tial can b e replaced b y the cos ( m ). Then Equation 1.331 of [42] giv es that for an y ev en m p ositiv e or negativ e, cos ( m ) = j m j 2 X =0 ( j m j + 1)( 1) ( j m j 2 + 1)(2 + 1) cos j m j 2 sin 2 : (B.28) The dieren tial d is written in terms of the dieren tial d d = d 2 C sin cos : (B.29) Also sin 2 = 1 C (B.30) cos 2 = 1 1 C (B.31) and caution should b e used when the square ro ot of these expressions is tak en. Sp ecifically for in [0 ; 2 ) or in [ ; 3 2 ), sin cos 0, so the equation sin cos = r 1 C 1 1 C (B.32) should b e used. On the other hand, for in [ 2 ; ) or in [ 3 2 ; 2 ), sin cos 0, so the equation sin cos = r 1 C 1 1 C (B.33)

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140 should b e used. In I 1 the factor (2 + r 2 ) p 2 + n +1 is rst written as an innite sum (2 + r 2 ) p 2 + n +1 = 1 X s =0 ( p 2 + n + 1) ( s + 1)( p 2 + n s + 1) 2 p 2 + n s +1 r 2 s s : (B.34) The in tegral b ecomes I 1 = ( 1) l n 2 j m j 2 X =0 ( j m j + 1)( 1) ( j m j 2 + 1)(2 + 1) C j m j 2 1 X s =0 ( p 2 + n + 1) ( s + 1)( p 2 + n s + 1) 2 p 2 + n s +1 r 2 s Z 1 1 C d p 2 q s (1 ) 1 2 ( C 1 + ) j m j 2 1 2 : (B.35) Changing the v ariable of in tegration from to y = 1 C and using the denition of the h yp ergeometric functions F ( ; ; r ; z ) = 2 F 1 ( ; ; r ; z ) = ( r ) ( )( r ) Z 1 0 t 1 (1 t ) r 1 (1 tz ) dt (B.36) where: < ( r ) > < ( ) > 0, result in the nal expression for the in tegral I 1 = p ( 1) l n j m j 2 X =0 ( 1) ( j m j +1 2 ) ( + 1)( j m j 2 + 1) 1 X s =0 r 2 s 2 p 2 + n s +2 ( p 2 + n + 1) ( s + 1)( p 2 + n s + 1) F ( p 2 + q + s; + 1 2 ; j m j 2 + 1; C ) : (B.37) T o calculate the second in tegral I 2 one follo ws the same pro cedure that w as used for I 1 the only dierence b eing that the step of expanding the factor (2 + r 2 ) p 2 + n +1

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141 is no w not necessary The result is I 2 = j m j 2 X =0 2( 1) p ( j m j +1 2 ) ( + 1)( j m j 2 + 1) ( r 2 ) p 2 + n +1 F ( q + n + 1 ; + 1 2 ; j m j 2 + 1; C ) : (B.38) The nal expression for the co ecien ts E p;q l ;m ( r 2 ; C ) can b e obtained b y com bining Equations (B.24), (B.37), (B.38) and (B.12). It is E p;q l ;m ( r 2 ;C ) = r 2 l + 1 ( l j m j + 1) [( l m + 1)( l + m + 1)] 1 2 j m j 2 X =0 ( 1) ( j m j +1 2 ) ( + 1)( j m j 2 + 1) [ l 2 ] X k =0 ( 1) k 2 l 2 k ( l k + 1 2 ) ( k + 1)( l 2 k j m j + 1) j m j 2 X =0 ( j m j 2 + 1) ( + 1)( j m j 2 + 1) l j m j 2 k +2 X n =0 2 n ( p + 2)( p + 4) : : : [ p + 2( n + 1)] ( l j m j 2 k + 2 + 1) ( l j m j 2 k + 2 n + 1) ( 2( r 2 ) p 2 + n +1 F ( q + n + 1 ; + 1 2 ; j m j 2 + 1; C ) + ( 1) l + n 1 X s =0 r 2 s ( p 2 + n + 1) ( s + 1)( p 2 + n s + 1) 2 p 2 + n s +2 F ( p 2 + q + s; + 1 2 ; j m j 2 + 1; C ) ) : (B.39) It is reminded that this expression is v alid for all ev en m 's, and that for o dd m 's these co ecien ts are equal to zero. If this expression for the co een ts is used, the equations b ecome v ery long and their dep endence on r 2 whic h is of particular in terest for the calculation of the regularization parameters, b ecomes unclear. In order for the expressions that app ear

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142 in this dissertation to lo ok simpler, the follo wing notation is used for the three nite sums o v er k and whic h app ear in the previous equation and whic h ha v e no r 2 -dep endence X k ;; = r 2 l + 1 ( l j m j + 1) [( l m + 1)( l + m + 1)] 1 2 [ l 2 ] X k =0 ( 1) k 2 l 2 k ( l k + 1 2 ) ( k + 1)( l 2 k j m j + 1) j m j 2 X =0 ( j m j 2 + 1) ( + 1)( j m j 2 + 1) j m j 2 X =0 ( 1) ( j m j +1 2 ) ( + 1)( j m j 2 + 1) : (B.40) It is implied that ( P ;; ) is a m ultiplicativ e factor and an y quan tities app earing to the righ t of it can dep end on k or Also, the notation F ;m ( a ) is used for the h yp ergeometric function F ( a; + 1 2 ; j m j 2 + 1; C ). Finally E p;q l ;m ( r 2 ; C ) = X k ;; l j m j 2 k +2 X n =0 2 n ( p + 2)( p + 4) : : : [ p + 2( n + 1)] ( l j m j 2 k + 2 + 1) ( l j m j 2 k + 2 n + 1) ( 2( r 2 ) p 2 + n +1 F ;m ( q + n +1) + 1 X s =0 r 2 s ( 1) l + n ( p 2 + n + 1) ( s + 1)( p 2 + n s + 1) 2 p 2 + n s +2 F ;m ( p 2 + q + s ) ) : (B.41)

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APPENDIX C NUMERICAL CODE F OR THE RET ARDED FIELD #include #include #include #include #include #include #include #include #include // Convenient macros for debugging #define SHOW(a) "<<#a<<" = "< cmplx ; int kmax,kount; double *xp,dxsav; cmplx **yp; cmplx mu(int l, int m); void fkseh(cmplx *f1, cmplx *f2, cmplx *f3, int k); 143

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144 void fksinf(cmplx *f1, cmplx *f2, cmplx *f3, int k); cmplx coefkeh(int k); cmplx coefkinf(int k); void rkqs(cmplx y[],cmplx dydr[], int n, double *r, double htry, double eps, double yscal[], double *hdid, double *hnext, void (*derivs)(double, cmplx [], cmplx [])) ; void rkck(cmplx y[],cmplx dydr[], int n, double r, double h, cmplx yout[], cmplx yerr[], void (*derivs)(double, cmplx [], cmplx [])) ; void derivs(double r, cmplx y[], cmplx dydr[]) ; void factor(cmplx& coefr, cmplx& coefq, double r); void strline(double x1,double x2,cmplx y1,cmplx y2, double R,cmplx& yR); void odeint(cmplx ystart[],int nvar,double x1,double x2,double eps, double h1, double hmin, int *nok, int *nbad, cmplx& ylast1, cmplx& ylast2, void (*derivs)(double,cmplx [],cmplx []), void (*rkqs)(cmplx [],cmplx [], int,double *, double,double,double [], double *, double *, void (*)(double,cmplx [],cmplx []))) ; const cmplx I(0,1); namespace ScalarWave{ double M = 1; // Mass of the Black Hole int l; // Spherical Harmonic Index int m; // Spherical Harmonic Index double omega; // Frequency of Radiation double w; // Angular Frequency of Orbit double R = 20*M; // Radius of the Orbit double dtdtau = 1/sqrt(1-(3*M/R)); // t-Component of 4-velocity }

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145 #include "rkqs.c" #include "rkck.c" #include "nrutil.c" #include "odeint1.c" #include "sum.c" #include "nrutil.h" #include "sphharm.c" #include "zero.cpp" int main(){ using namespace ScalarWave; M = 1.0 ; w = sqrt(M/(R*R*R)) ; double pi = 4.0*atan(1.0); char* buff; buff = new char[513]; char bh; int nvar = 2; // Number of First Order Differential Equations. int nok = 0; int nbad = 0; int lmax = 31; cmplx ystart[3] ; double mulm; cmplx yLR1, yLR2, yRR1, yRR2; cmplx W1, W2, W3, W4 ; cout << endl; cout << "Radius of the orbit:" << SHOW(R) << endl; ofstream out("DPsi20.dat"); out << "l m mulm yLN2 yRN2" << endl;

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146 // Initialization of the Self-Force Contributions. double Flmt =0; double Flt =0; double Ft =0; double FlmrL =0; double FlrL =0; double FrL =0; double FlmrR =0; double FlrR =0; double FrR =0; for (l=0; l-1 ; m=m-2) { if (m!=0) { omega = m*w ; cout << endl; cout << SHOW(l) << SHOW(m) << endl; cmplx b1 = (l*l+l+1)/(2*M+8*I*omega* M*M) ; cmplx a1 = -I*l*(l+1)/(2*omega) ; double Reh = 1/(100*abs(b1)) + 2*M ; // Starting Radius Close to Event Horizon double Rinf = 100*abs(a1) ; // Starting Radius Close to Infinity // Integrate in the Region (Rs, R] to Get yL(r). double h = 1.0e-5 ; double hmin = 1.0e-10 ;

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147 double eps = 1.0e-12 ; kmax = 0; dxsav = 0.0; // Boundary Conditions at the Event Horizon : cmplx expeh = 1; cmplx Dexpeh = 0; coefkeh(0);double Xn = 1; double last = 1.e+33; for(int k=1; k<100; k++) { Xn *= (Reh-2*M); cmplx bk = coefkeh(k); cmplx next = bk*Xn; if (expeh + next == expeh) { cout << "Machine precision in event-horizon loop" << endl; cout << setprecision(13) << SHOW(k) << SHOW(expeh) << SHOW(next) << endl; break; }if (abs(next) > last) { cerr << "No convergence in event-horizon loop" << endl; cerr << setprecision(13) << SHOW(k) << SHOW(expeh) << SHOW(next) << endl; break; }last = abs(next);

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148 expeh += next; Dexpeh += k*next/(Reh-2*M); if (k < 5) { cout << setprecision(13) << SHOW(k) << SHOW(bk) << SHOW(expeh) << endl; } } expeh =expeh* exp(cmplx(0.0, omega*Reh + omega*2.0*M*log(Reh/(2* M)-1 ) )); Dexpeh = exp(cmplx(0.0, omega*Reh + omega*2.0*M*log(Reh/(2*M )-1 ) ))*Dexpeh + expeh*cmplx(0,omega)*Reh/ (Re h-2* M); ystart[1]=expeh/Reh;ystart[2]=Dexpeh/Reh expeh/(Reh*Reh); cout << setprecision(13) << SHOW(ystart[1]) << SHOW(ystart[2]) << endl; // Wronskian at the event horizon. W1 = Reh*(Reh-2*M)*( ystart[1]*conj(ystart[2] ) ystart[2]*conj(ystart[1] ) ); cout << SHOW(W1) << SHOW(-2*I*omega) << endl ; // Integration of the differential equation for Psi. odeint(ystart,nvar,Reh,R,e ps,h ,hmi n,& nok, &nba d,yL R1, yLR2 (*derivs),(*rkqs));

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149 // Wronskian at the orbit. W2 = R*(R-2*M)* (yLR1*conj(yLR2) yLR2*conj(yLR1)) ; //Solution before the matching. cout << setprecision(13) << SHOW(Reh) << SHOW(yLR1) << SHOW(yLR2) << endl; cout << setprecision(13) << SHOW(W1) << SHOW(W2) << SHOW(-2*I*omega) << endl; //Integrate in the region [R,Rinf) to Get yR(r). h = -1.0e-5; hmin = 1.e-10; eps = 1.e-12; kmax = 0; dxsav = 0.0; //Boundary Conditions at Infinity: cmplx expinf = 1; cmplx Dexpinf = 0; coefkinf(0);Xn = 1; last = 1.e+33 ; for(int k=1; k<100; k++) { Xn *= Rinf; cmplx ak = coefkinf(k) ; cmplx next = ak/Xn; if (expinf + next ==expinf) { cout << "Machine precision in infinity loop" << endl; cout << setprecision(13) << SHOW(k) << SHOW(expinf) << SHOW(next) << endl; break;

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150 }if (abs(next) > last) { cout << "No convergence in infinity loop" << endl; cout << setprecision(13) << SHOW(k) << SHOW(expinf) << SHOW(next) << endl; break; }last = abs(next); expinf += next; Dexpinf += -k*next/Rinf; if (k<5) { cout << setprecision(13) << SHOW(k) << SHOW(ak) << SHOW(expinf) << endl; }} double rstar = Rinf + 2*M*log(Rinf/(2*M)-1) ; expinf = exp(I*omega*rstar)*expin f; Dexpinf = I*omega*Rinf*expinf/(Rinf2*M) + exp(I*omega*rstar)*Dexpi nf; ystart[1] = expinf/Rinf; ystart[2] = Dexpinf/Rinf expinf/(Rinf*Rinf); cout << setprecision(13) << SHOW(ystart[1]) << SHOW(ystart[2]) << endl;

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151 //Wronskian at infinity W3 = Rinf*(Rinf-2*M)*(ystart[ 1]*c onj (yst art[ 2]) -ystart[2]*conj(ystart[1]) ); //Integration of the differential equation for Psi. odeint(ystart,nvar,Rinf,R ,eps ,h,h min ,&no k,&n bad, yRR 1,yR R2, (*derivs),(*rkqs)); //Wronskian at the orbit. W4 = R*(R-2*M)*(yRR1*conj(yRR 2)-y RR2 *con j(yR R1)) ; //Solution before the matching. cout << setprecision(13) << SHOW(Rinf) << SHOW(yRR1) << SHOW(yRR2) << endl; cout << setprecision(13) << SHOW(W3) << SHOW(W4) << SHOW(-2*I*omega)<< endl ; //Matching of the solutions at the orbit. cmplx A = -mu(l,m) yRR1 /(R-2*M)/ (yLR1*yRR2-yRR1*yLR2); cmplx B = -mu(l,m) yLR1 /(R-2*M)/ (yLR1*yRR2-yRR1*yLR2); //Normalized solution. cmplx yLN1 = yLR1*A ; cmplx yLN2 = yLR2*A ; cmplx yRN1 = yRR1*B ; cmplx yRN2 = yRR2*B ; cout << SHOW(yLR1) << SHOW(yLR2) << endl; cout << SHOW(yRR1) << SHOW(yRR2) << endl; cout << SHOW(A) << SHOW(B) << endl; cout << SHOW(yLN1) << SHOW(yLN2) << endl;

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152 cout << SHOW(yRN1) << SHOW(yRN2) << endl << endl; // Contribution to the self-force Flmt = 2*real(I*omega*yLN1*sharm( l,m ,pi/ 2.0, 0.0) ); FlmrL = 2*real(yLN2*sharm(l,m,pi/ 2.0 ,0.0 )); FlmrR = 2*real(yRN2*sharm(l,m,pi/ 2.0 ,0.0 )); out << l <<" "<< m <<" "<< mu(l,m) <<" "<< yLN2 <<" "<< yRN2 << endl; }if (m==0) { cmplx PsiH, dPsiH, PsiI, dPsiI; cout << endl; cout << SHOW(l) << SHOW(m) << endl; getZeroPsiH(&PsiH,&dPsiH) ; getZeroPsiI(&PsiI,&dPsiI) ; cmplx wronsk = R*(R-2.0*M)*(PsiH*dPsiI-P siI* dPsi H); cmplx PsiHN = -mu(l,m) PsiH*R*PsiI/wronsk; cmplx PsiIN = -mu(l,m) PsiI*R*PsiH/wronsk; cmplx DPsiH = -mu(l,m)*dPsiH*R*PsiI/wron sk; cmplx DPsiI = -mu(l,m)*dPsiI*R*PsiH/wron sk; Flmt = real(I*omega*PsiHN*sharm(l ,m, pi/2 .0,0 .0)) ; FlmrL = real(DPsiH*sharm(l,m,pi/2 .0, 0.0) ); FlmrR = real(DPsiI*sharm(l,m,pi/2 .0, 0.0) );

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153 cout << SHOW(DPsiH) << SHOW(DPsiI) << endl << endl; out <
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154 *sqrt(1-(3*M/R))/R; return mulm ; }// Derivatives, For Use in rkqs.c void derivs(double r, cmplx y[], cmplx dydr[]) { cmplx coefr,coefq ; factor(coefr, coefq, r); dydr[1] = y[2] ; // dy/dr dydr[2] = coefr*y[1] coefq y[2] ; // d^2y/dr^2 return ; }// coefr(r) and coefq(r) void factor(cmplx& coefr, cmplx& coefq, double r) { using namespace ScalarWave; int l2 = l*(l+1); double r2 = r*r; double rm = r-2*M; coefr = -( omega*omega*r2/rm/rm l2/r/(r-2.0*M)); coefq = 2.0*(r-M)/r/rm ; }

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155 // Spherical Harmonics. cmplx sharm(int l, int m, double theta, double phi){ cmplx I = cmplx(0,1.0); double c = cos(theta) ; cmplx sh = exp(I*abs(m)*phi) plgndr(l,abs(m),c); float fact = sqrt((2*l+1)*factorial(labs (m)) /(4.0*(4.0*atan(1.0)))/fa cto rial (l+a bs(m ))) ; sh = sh*fact; if(m<0) sh = pow(-1,abs(m))*conj(sh) ; return sh; }// Terms in the expansion of the intial guess for the solution // at the event horizon and at infinity void fksinf(cmplx *f1, cmplx *f2, cmplx *f3, int k){ using namespace ScalarWave; *f1 = cmplx(-k*(k-1)+l*(l+1) 4*omega*M*(k-1) ) / cmplx(0.0, 2*omega*k) ; *f2 = cmplx( 0.0, -M*( (k-1)*(2*k-3) l*(l+1) ) ) /omega/k ; *f3 = cmplx( 0.0, 2*M*M*(k-2)*(k-2)/omega/k ); }void fkseh(cmplx *f1, cmplx *f2, cmplx *f3, int k){ using namespace ScalarWave; *f1 = cmplx( -(2*k-3)*(k-1)+(l*l+l+1) -12*omega*M*(k-1) )

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156 /2.0/M/ cmplx(k*k, 4*omega*M*k) ; *f2 = cmplx( -(k-2)*(k-3)+l*(l+1), -12*omega*M*(k-2) ) /4.0/M/M/ cmplx(k*k, 4*omega*M*k) ; *f3 = cmplx( 0.0, -omega*(k-3) ) / 2.0/M/M/ cmplx(k*k, 4*omega*M*k) ; }cmplx coefkeh(int KK) { using namespace ScalarWave; static cmplx bk_1 = 1 ; static cmplx bk_2 = 0; static cmplx bk_3 = 0; static int k = 0; if (KK == 0) { bk_1 = 1; bk_2 = 0; bk_3 = 0; k = 0; }if (KK != k) { cerr << "sum::coefkeh called out of order, k = << KK << expected << k << endl; assert(KK == k); }cmplx f1 = 0; cmplx f2 = 0; cmplx f3 = 0; cmplx ret; if (k == 0) { k++;return 1; } else { fkseh(&f1, &f2, &f3, k); ret = f1*bk_1 + f2*bk_2 + f3*bk_3;

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157 }bk_3 = bk_2; bk_2 = bk_1; bk_1 = ret; k++;return ret; }cmplx coefkinf(int KK) { using namespace ScalarWave; static cmplx ak_1 = 1; static cmplx ak_2 = 0; static cmplx ak_3 = 0; static int k = 0; if (KK == 0) { ak_1 = 1; ak_2 = 0; ak_3 = 0; k = 0; }if (KK != k) { cerr << "sum::coefkinf called out of order, k = << KK << expected << k << endl; assert(KK == k); }cmplx f1, f2, f3, ret; if (k == 0) { k++;return 1; } else { fksinf(&f1, &f2, &f3, k);

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158 ret = f1*ak_1 + f2*ak_2 + f3*ak_3; }ak_3 = ak_2; ak_2 = ak_1; ak_1 = ret; k++;return ret; }// Scalar wave, zero frequency define D(a) double(a) typedef complex cmplx ; cmplx Ylm(int l, int m, double theta, double phi); void test_zero(double radius); void getZeroPsiH(cmplx* PsiH, cmplx* dPsiH); void getZeroPsiI(cmplx* PsiI, cmplx* dPsiI); const double PI(M_PI); using namespace ScalarWave ; double Bang2LDL2(int L) { assert(L >= 0); double ret = 1; for (int i = L; i > 0; i--) {

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159 ret *= (2.*i)*(2.*i-1)/(i*i); }return ret; }double AnHorizon(double* Anm1, int n, int l) { return *Anm1 = -(*Anm1/(n*n))*(l+n)*(l-n+ 1); }double AnInfinity(double* Anm1, int n, int l) { *Anm1 = (*Anm1/(n*(2*l+1+n)))*(1. *(l+ n)*( l+n )); return *Anm1; }// Wronskian/M for zero frequency scalar modes double WzeroDM(int l) { return (l%2==0?-1:+1)*Bang2LDL2(l )*2* (2*l +1) ; }void getZeroPsiH(cmplx* PsiH, cmplx* dPsiH) { *PsiH = 0; *dPsiH = 0; double b = 1; for (int n = 0; n <= l; n++) { cerr << SHOW(b) << endl; double Rn = pow(R/(2*M),n); *PsiH += b*Rn;

PAGE 166

160 *dPsiH += n*b*Rn/R; cout << SHOW(*PsiH) << SHOW(*dPsiH) << endl; AnHorizon(&b,n+1, l); }assert(b == 0); cout << SHOW(*PsiH) << SHOW(*dPsiH) << endl; }void getZeroPsiI(cmplx* PsiI, cmplx* dPsiI) { *PsiI = 0; *dPsiI = 0; double a = 1; for (int n = 0; n <= 100; n++) { cerr << SHOW(a) << endl; double Rn = pow(2*M/R,n); *PsiI += a*Rn; *dPsiI += -n*a*Rn/R; cout << SHOW(*PsiI) << SHOW(*dPsiI) << endl; if (n>1) { double corr = abs(a*Rn/(*PsiI)) + abs(-n*a*Rn*R/(2*M)/(*dPs iI)) ; if (corr < 1.e-17) { cerr << SHOW(n) << SHOW(corr) << endl; break; } }AnInfinity(&a,n+1,l); }*PsiI *= pow(R/(2*M),-(l+1)); *dPsiI = pow(R/(2*M),-(l+1))*(*dP siI) -(l+1)*(*PsiI)/R;

PAGE 167

161 }void test_zero() { cout << "Entering test_zero" << endl; time_t tenter = time(0); // Set ScalarWave variables: M = 1; // mass of the hole m = 0; // spherical harmonic index R = 20*M; // radius of the orbit w = M/(R*R*R); // angular frequency of orbit omega = m*w; // frequency of the radiation for (int l=0; l < 41; l++) { cmplx PsiH; cmplx dPsiH; getZeroPsiH(&PsiH, &dPsiH); cmplx PsiI = 0; cmplx dPsiI = 0; getZeroPsiI(&PsiI, &dPsiI); cout << "Here is a Wronskian test" << endl; cmplx W = R*R*(PsiH*dPsiI PsiI*dPsiH)*(1-2*M/R); cout << setprecision(12)<< SHOW(l) << SHOW(W) << SHOW(M*WzeroDM(l)) << endl; cout << SHOW(l) << SHOW(abs((W M*WzeroDM(l))/W)) << SHOW(W M*WzeroDM(l)) << endl << endl; cout << SHOW(PsiI) << SHOW(dPsiI) << endl; cout << SHOW(PsiH) << SHOW(dPsiH) << endl; }time_t texit = time(0);

PAGE 168

162 cout<< "test_zero execution time is << difftime(texit, tenter) << seconds" << endl; cout << "Leaving test_zero" << endl; }

PAGE 169

REFERENCES [1] B. F. Sc h utz. Gra vitational w a v e astronom y Class. Quantum Gr av. 16:A131, 1999. [2] M. Da vis. Can sync hrotron gra vitational radiation exist? Phys. R ev. L ett. 28:1352, 1972. [3] S. A. T euk olsky and W. H. Press. P erturbations of a rotating blac k hole. I I I. In teraction of the hole with gra vitational and electromagnetic radiation. Ap. J. 193:443, 1974. [4] E. P oisson. Gra vitational radiation from a particle in circular orbit around a blac k hole. I. analytical results for the non-rotating case. Phys. R ev. D 47:1497, 1993. [5] M. Shibata. Gra vitational w a v es induced b y a particle orbiting around a rotating blac k hole. Eect of orbital precession. Pr o g. The or. Phys. 90:595, 1993. [6] B. R. Iy er and C. M. Will. P ost-Newtonian gra vitational radiation reaction for t w o-b o dy systems. Phys. R ev. L ett. 70:113, 1993. [7] C. Cutler, D. Kennec k, and Eric P oisson. Gra vitational radiation reaction for b ound motion around a Sc h w arzsc hild blac k hole. Phys. R ev. D 50:3816, 1994. [8] E. P oisson and M. Sasaki. Gra vitational radiation from a particle in circular orbit around a blac k hole. V. blac k hole absorption and tail corrections. Phys. R ev. D 51:5753, 1995. [9] S. W. Leonard and E. P oisson. Radiativ e m ultip ole momen ts of in teger-spin elds in curv ed spacetime. Phys. R ev. D 56:4789, 1997. [10] S. A. Hughes. Ev olution of circular non-equatorial orbits of Kerr blac k holes due to gra vitational w a v e emission. Phys. R ev. D 61:084004, 2000. [11] J. M. Bardeen, W. H. Press, and S. A. T euk olsky Rotating blac k holes: Lo cally nonrotating frames, energy extraction and scalar sync hrotron radiation. Astr ophys. J. 178:347, 1972. [12] A. G. Wiseman. Self-force on a static scalar c harge outside a Sc h w arzsc hild blac k hole. Phys. R ev. D 61:084014, 2000. [13] P A. M. Dirac. Classical theory of radiating electrons. Pr o c. R oy. So c. A167:148, 1938. 163

PAGE 170

164 [14] B. S. DeWitt and R. W. Brehme. Radiation damping in a gra vitational eld. A nnals of Physics 9:220, 1960. [15] Y. Mino, M. Sasaki, and T. T anak a. Gra vitational radiation reaction to a particle motion. Phys. R ev. D 55:3457, 1997. [16] Synge. Relativit y: The General Theory (North Holland, Amsterdam, 1960). [17] J. Hadamard. Lectures on Cauc h y's problem in linear dieren tial equations. (Y ale Univ ersit y Press, New Ha v en, 1923). [18] C. W. Misner, K. S. Thorne, and J. A. Wheeler. Gra vitation. (F reeman, San F ransisco, 1973). [19] E. P oisson. The motion of p oin t particles in curv ed spacetime. gr-qc/0306052, 2003. [20] T. C. Quinn. Axiomatic approac h to radiation reaction of scalar p oin t particles in curv ed spacetime. Phys. R ev. D 62:064029, 2000. [21] T. C. Quinn and R. M. W ald. Axiomatic approac h to electromagnetic and gra vitational radiation reaction of particles in curv ed spacetime. Phys. R ev. D 56:3381, 1997. [22] S. Det w eiler and B. F. Whiting. Self-force via a Green's function fecomp osition. Phys. R ev. D 67:024025, 2003. [23] S. Det w eiler, E. Messaritaki, and B. F. Whiting. Self-force of a scalar eld for circular orbits ab out a Sc h w arzsc hild blac k hole. Phys. R ev. D 67:104016, 2003. [24] S. Det w eiler. Radiation reaction and the self-force for a p oin t mass in general relativit y Phys. R ev. L ett. 86:1931, 2001. [25] K. S. Thorne and J. B. Hartle. La ws of motion and precession for blac k holes and other b o dies. Phys. R ev. D 31:1815, 1985. [26] X. H. Zhang. Multip ole expansions of the general relativistic gra vitational eld of the external univ erse. Phys. R ev. D 34:991, 1986. [27] K. S. Thorne and S. J. Ko v acs. The generation of gra vitational w a v es. I. Weakeld sources. Astr ophys. J. 200:245, 1975. [28] L. Barac k and A. Ori. Mo de sum regilarization approac h for the self-force in blac k hole spacetime. Phys. R ev. D 61:061502, 2000. [29] Y. Mino, H. Nak ano, and M. Sasaki. Co v arian t self-force regularization of a particle orbiting a Sc h w arzsc hild blac k hole. Pr o g. The or. Phys. 108:1039, 2003. [30] L. Barac k and A. Ori. Regularization parameters for the self-force in Sc h w arzsc hild spacetime: I.Scalar case. Phys. R ev. D 66:084022, 2002.

PAGE 171

165 [31] L. M. Burk o. Self-force on a particle in orbit around a blac k hole. Phys. R ev. L ett. 84:4529, 2000. [32] L. Barac k and L. M. Burk o. Radiation-reaction force on a particle plunging in to a blac k hole. Phys. R ev. D 62:084040, 2000. [33] L. Barac k. Self-force on a scalar particle in spherically symmetric spacetime via mo de-sum regularization: radial tra jectories. Phys. R ev. D 62:084027, 2000. [34] L. Barac k and A. Ori. Regularization parameters for the self-force in Sc h w arzsc hild spacetime: I I.Gra vitational and electromagnetic cases. Phys. R ev. D 61:061502, 2000. [35] L. Barac k. Gra vitational self-force b y mo de sum regularization. Phys. R ev. D 64:084021, 2001. [36] L. Barac k, Y. Mino, H. Nak ano, A. Ori, and M. Sasaki. Caclulating the gra vitational self-force in Sc h w arzsc hild spacetime. Phys. R ev. L ett. 88:091101, 2002. [37] M. Abramo witz and I. Stegun. Handb o ok of Mathematical Functions. (Do v er Publications, INC., New Y ork, 1972). [38] J. D. Jac kson. Classical Electro dynamics. (John Wiley and Sons, 1975). [39] W. H. Press, S. A. T euk olsky W. T. V etterling, and B. P Flannery Numerical Recip es in C: The Art of Scien tic Computing. (Cam bridge Univ ersit y Press, Cam bridge, 1992), 2nd edition. [40] R. M. W ald. General relativit y (The Univ ersit y of Chicago Press, 1984). [41] S. Det w eiler, B. Whiting, L. Diaz, and E. Messaritaki. The scalar eld self-force eects on circular orbits around a static blac k hole. in preparation. [42] I. S. Gradsh teyn and I. M. Ryzhik. T able of In tegrals, Series and Pro ducts. (Academic Press, 1994).

PAGE 172

BIOGRAPHICAL SKETCH Eirini Messaritaki w as b orn in August 1974, in Iraklio, Greece, where she also grew up. After graduating from high sc ho ol, she w as accepted b y the Ph ysics Departmen t of the Univ ersit y of Crete for undergraduate studies. She receiv ed a bac helor's degree in ph ysics with sp ecialization in computational ph ysics on June 1997. The title of her undergraduate thesis w as `Mon te-Carlo Sim ulations of Comptonization in Spherically Accreting Neutron Stars and Blac k Holes'. In 1997, she started her graduate studies at the Ph ysics Departmen t of the Univ ersit y of Florida. She w ork ed with professor Stev e Det w eiler on the problem of radiation reaction on particles mo ving in the vicinit y of Sc h w arzsc hild blac k holes. 166

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RADIATION REACTION ON MOVING PARTICLES IN GENERAL
RELATIVITY

By

EIRINI MESSARITAKI

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2003

ACKNOWLEDGMENTS

First and foremost I would like to thank Dr. Steve Detweiler, Professor of Physics

at the University of Florida, who was my advisor during my graduate studies. He has

always been very supportive of my work, very willing to help and extremely patient

with me. I feel very lucky to have worked with him during the past 5 years.

I would also like to thank Dr. Bernard Whiting for his help on writing this

dissertation and for his contributions to the research.

Many thanks go to the other members of my committee: Dr. Jim Fry, Dr. An-

drew Rinzler and Dr. Vicki Sarajedini, as well as to Dr. Richard Woodard and Dr. Luz

Diaz, for their useful comments on the dissertation.

I would also like to thank my brothers, my family and my close friends. Their

help and support has always been very important and I could not have come this far

without them.

My research was partly supported by the Institute of Fundamental Theory at

the University of Florida and I am grateful for that.

ACKNOWLEDGMENTS . ...

ABSTRACT . . . . .

ii

. . v

CHAPTER

1 INTRODUCTION . ...........

2 RADIATION REACTION IN FLAT SPACETIME .
2.1 Definitions of the Fields .............
2.2 Derivation of the Equations of Motion ......
2.3 Radiation Reaction .................

. . .

3 RADIATION REACTION IN CURVED SPACETIME . . 14
3.1 Nonlocal Quantities ................... ....... 14
3.2 Green's Functions ................... ....... .. 18
3.2.1 Green's Functions for a Scalar Field ....... ....... 18
3.2.2 Green's Functions for a Vector Field ...... ........ 23
3.2.3 Green's Functions for a Gravitational Field . . .... 27
3.3 Motion of a Charged Particle in Curved Spacetime . .... 32
3.3.1 Equations of Motion .................. .. .. 32
3.3.2 Calculation of the Fields .................. . 34
3.3.3 Conservation of Energy and Momentum . . 37
3.3.4 Radiation Reaction .................. ..... 40

4 SELF-FORCE .. .........
4.1 Scalar Self-Force .. ..........
4.1.1 Direct and Tail Fields ......
4.1.2 The S-Field and the R-Field .
4.2 Electromagnetic Self-Force ........
4.3 Gravitational Self-Force .. .......

5 SINGULAR FIELD FOR SCHWARZSCHILD GEODESICS
5.1 Thorne-Hartle-Zhang Coordinates .. ..........
5.2 Scalar Field of a Charged Particle .. ..........
5.3 Scalar Field of a Dipole .. ...............
5.4 Electromagnetic Potential of a Charged Particle .
5.5 Electromagnetic Potential of an Electric Dipole .
5.6 Electromagnetic Potential of a Magnetic Dipole .
5.7 Gravitational Field of a Spinning Particle ........

5.8 GRTENSOR Code ............... . . ... 77

6 REGULARIZATION PARAMETERS FOR THE SCALAR FIELD 80
6.1 Regularization Procedure ................... ...... 81
6.2 Order Calculation of the Scalar Field ....... .......... 84
6.3 Scalar Monopole Field ................... ...... 88

7 CALCULATION OF THE RETARDED FIELD . . 102
7.1 Analytical Work .......... ................... .. 102
7.2 Numerical Code .......... ................... .. 110

8 APPLICATIONS AND CONCLUSIONS . . . 111
8.1 Equations of Motion ................... ....... 111
8.2 Effects of the Scalar Self-Force ....................... 112
8.2.1 Calculation of the Self-Force ....... ............ 115
8.2.2 Change in Orbital Frequency ... . . 116
8.3 Effects of the Gravitational Self-Force .... . . 117

A GRTENSOR CODE FOR THE SINGULAR FIELDS . . 119

B DECOMPOSITION OF fX-Q .. ....... . . 134

C NUMERICAL CODE FOR THE RETARDED FIELD . . 143

REFERENCES . . . . . . . 163

BIOGRAPHICAL SKETCH . . . . . 166

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

RADIATION REACTION ON MOVING PARTICLES IN GENERAL
RELATIVITY

By

Eirini Messaritaki

August 2003

Chairman: Steven L. Detweiler
M.,. iP r Department: Physics

A particle in the vicinity of a Schwarzschild black hole is known to trace a

geodesic of the Schwarzschild background, to a first approximation. If the interaction

of the particle with its own field (scalar, electromagnetic or gravitational) is taken

into account, the path is no longer a background geodesic and the self-force that the

particle experiences needs to be taken into account.

In this dissertation, a recently proposed method for the calculation of the self-

force is implemented. According to this method the self-force comes from the inter-

action of the particle with the field 9R .= t ~s for a scalar particle; with the

electromagnetic potential AR = A' As for a particle creating an electromagnetic

field; or with the metric perturbation hb = hrb hb for a particle creating a gravita-

tional field. First, the singular fields ys, As and hSb are calculated for different sources

moving in a Schwarzschild background. For that, the Thorne-Hartle-Zhang coordi-

nates in the vicinity of the moving source are used. Then a mode-sum regularization

method initially proposed for the direct scalar field is followed, and the regularization

parameters for the singular part of the scalar field and for the first radial derivative

of the singular part of the self-force are calculated. Also, the numerical calculation of

the retarded scalar field for a particle moving on a circular geodesic in a Schwarzschild

spacetime is presented. Finally, the self-force for a scalar particle moving on a circular

Schwarzschild orbit is calculated and some results about the effects of the self-force

on the orbital frequency of the circular orbit are presented.

CHAPTER 1
INTRODUCTION

The General Theory of Relativity, developed by Albert Einstein at the beginning

of the twentieth century, has been successfully used by scientists for many years to

explain 1.1,.-i .,1 phenomena and to make predictions about ].li ,-i, .1l systems. One of

the most exciting predictions of General Relativity is the existence of gravitational ra-

diation, which can be thought of as a wave-like distortion of spacetime. It is expected

that gravitational radiation from remote .I-r I1. .li,-i .i1 systems will yield very impor-

tant information about those systems, thus answering many questions that scientists

currently have. Even though the detection of gravitational waves has proved to be

a challenging task, recent developments in technology have made scientists confident

that gravitational waves will indeed be detected in the near future. Both earth-based

detectors (such as LIGO and VIRGO) and space-based detectors (such as LISA) are

expected to start operating in the next decade or two. For the data collected by

these detectors to be useful, accurate information about the sources of gravitational

radiation is required. Only if scientists know what the gravitational waves emitted by

specific systems should look like, can they compare them with the patterns observed

and draw conclusions about the pi, -i. .,1 characteristics of those systems. That is why

there has been a lot of interest lately in predicting the gravitational radiation emitted

by different 1.-i 1 1%-- -i. ., systems.

One .i-I1 ,,,li,--i1 .I1 system for which gravitational waves are expected to be de-

tected, specifically by space-based detectors, is the binary system of a small neutron

star or black hole (of mass equal to a few times the mass of the Sun) and a super-

massive black hole (of mass equal to a few million times the mass of the Sun) [1].

Supermassive black holes are believed to exist at the centers of many galaxies, in-

2

eluding our own. It is also believed that their strong gravitational field can capture

smaller stars and black holes, which then move toward the supermassive black hole,

until they are absorbed by it. The exact evolution of the system with time can give

the pattern of the gravitational radiation that the system is expected to emit. Know-

ing that pattern is crucial in distinguishing whether the gravitational waves measured

by the detector come from such a system. It can also help extract information about

many characteristics of the system, such as the masses and the angular moment of

the two components of the binary system.

To determine how the system evolves, the path of the smaller star or black

hole needs to be calculated. That path is largely determined by the influence of the

gravitational field of the supermassive black hole on the small star or black hole; it is

also affected by the interaction of the small star or black hole with its own gravitational

field. This latter interaction, commonly referred to as the self-force, is responsible for

the radiation reaction effects, which cause the decay of the small star's or black hole's

orbit toward the supermassive black hole.

Even though my motivation for studying the self-force effects stems from the

need to know the evolution of this system, radiation reaction effects are present in

other systems that are easier to deal with mathematically. One can think, for example,

of a particle of a certain scalar charge, which moves in a Schwarzschild background

spacetime and creates its own scalar field. There is also the case of a particle that

carries an electric charge and creates an electromagnetic field as it moves in spacetime.

The scalar field of the particle in the first case and the electromagnetic field of the

electric charge in the second case will affect the motion of each particle, causing its

worldline to differ from what it would be if radiation reaction effects were not present.

It is useful to predict the evolution of such systems, mainly because they can give an

idea of how the more difficult system can be handled successfully and not because

they are realistic systems, since they are not expected to be observed in nature.

In the past, various efforts have been made to develop a concrete scheme for

including the radiation reaction effects for scalar, electromagnetic and gravitational

fields in the equations of motion. One approach that has been used extensively in-

volves calculating the flux, both at infinity and at the event horizon of the central

supermassive black hole, of quantities (such as the energy and momentum of the par-

ticle) that are constants of the motion when radiation reaction is not present [2-9].

That flux is then associated with the rate of change of those quantities at the location

of the particle and the evolution of the particle's motion can be predicted. However,

this approach is generally not applicable in the case in which the central black hole

is a rotating black hole, except under some special circumstances [10]. One reason is

that the evolution of the Carter constant [11], which is a constant of the motion that

is not derived by a Killing vector, cannot be calculated using the flux at infinity and

at the event horizon. In addition, this approach does not take into account the very

significant non-dissipative effects of the self-force that the particle's field exerts on the

particle [12]. That is why it is important for the self-force, and not just its radiation

reaction effects, to be calculated. The calculation of the self-force is the subject of

this dissertation.

In Chapters 2 and 3 of this dissertation, I present the innovative work of Dirac

[13] for the radiation reaction effects on an electron moving in flat spacetime and

the theoretical generalization of it to the case of curved spacetime by DeWitt and

Brehme [14]. In Chapter 4, I present different methods for practical calculations of

the self-force that have been proposed in the past, with emphasis on one in particular.

In Chapters 5-8 of this dissertation, I present an implementation of that particular

method for calculating the radiation reaction effects for a scalar, an electromagnetic

and a gravitational field.

CHAPTER 2
RADIATION REACTION IN FLAT SPACETIME

The first successful attempt to provide an expression for the radiation reaction

effects on a particle in special relativity was made by Dirac in 1938 [13]. In his famous

paper, he studied the motion of an electron moving in flat spacetime by using the

concept of conservation of energy and momentum along the electron's worldline. In

this chapter I present the decomposition of the electromagnetic field and the derivation

of the equations of motion, which include the radiation damping effects on the electron.

The scheme for calculating the radiation reaction effects in curved spacetime presented

in this dissertation extends Dirac's work.

For the equations to look as simple as possible, the speed of light is set equal to

1 in this chapter.

It is noted that the metric signature used by Dirac is (+- --) and that is what

is used in this chapter, in order to keep the results identical to those derived by Dirac.

In all subsequent chapters, however, the signature is changed to (- + ++), in order

to adhere to the currently widely used convention. Thus, some of the equations given

in this chapter change in subsequent chapters. Since none of the results described in

this chapter are used in the chapters that follow, that should not cause any confusion

at all.

2.1 Definitions of the Fields

Dirac's analysis starts by assuming that an electron of electric charge q is moving

in flat spacetime, so that the background metric is the Minkowski metric r',. The

worldline of the electron is denoted by

2P, = z,, () (2.1)

where s is the proper time along the worldline. The electromagnetic 4-vector potential

that the electron creates is assumed to satisfy the Lorentz gauge condition

V'A = 0 (2.2)

and obeys Maxwell's equations

V2A 4'jr. (2.3)

It is well-known that the Lorentz gauge condition leaves some arbitrariness for the

electromagnetic potential. In Equation (2.3), j, is the charge-current density vector

of the electron, namely

Sqj (xo zo) 6(x Z) 6(X2 J 2) 6(x3 Z) ds. (2.4)

The electromagnetic field associated with a 4-vector potential A" is, in general, given

by the equation

F^ = V"A" V^A. (2.5)

It is clear that, unless the boundary conditions of the problem are specified,

Equations (2.2) and (2.3) do not have a unique solution. In fact, adding any solution

of the homogeneous equation

V2A, 0

(2.6)

(which represents a source-free radiation field and obeys the Lorentz gauge condition)

to a solution of Equation (2.3), gives a different solution of Equation (2.3) that also

obeys Equation (2.2).

One solution of interest for the problem at hand is the retarded electromagnetic

field F'"" created by the electron, which is derived by the well-known Lienard-Wiechert

potentials Aet that obey Equations (2.2) and (2.3). Assuming that there is also a

radiation field Fi incident on the electron, the actual field in the neighborhood of

the electron is the sum of those two fields

FCt = Fret F (2.7)

Another interesting solution is the one given by the advanced potentials Adv. It is rea-

sonable to expect that the advanced potentials play a role symmetrical to the retarded

potentials in this problem. If the outgoing radiation field leaving the neighborhood

of the electron is Fo(, then
FV = Fv + Fo.V (2.8)

The radiation emitted by the electron is given by the difference of the outgoing and

F r wd -- Fut F. (2.9)

Using Equations (2.7) and (2.8) for the actual field, an alternative expression for the

radiation field can be obtained

F = F F (2.10)

The important implication of Equation (2.10) is that, because the retarded and the

advanced fields are uniquely determined by the worldline of the electron, so is the

radiation emitted by the electron. It becomes clear from the analysis of Section (2.2)

that the radiation field is responsible for the deviation of the electron's worldline from

a background geodesic, because that field shows up in the equations of motion, in the

It is significant to realize that Equation (2.10) for the radiation field is consistent

with what one would expect for the radiation of an accelerating electron. To under-

stand that, one should recall that the radiation produced by an accelerating electron

can be calculated using the retarded field F/, that the electron creates, at very large

distances away from the electron and at much later times than the time when the

acceleration takes place. At those distances and times the advanced field vanishes

and Equation (2.10) gives the correct result. The advantage of Equation (2.10) is

that it gives the radiation produced by the electron at any point in spacetime, so it

can be used for the radiation in the neighborhood of the electron.

A field that is needed for the analysis of Section (2.2) is the difference of the

average of the retarded and advanced fields from the actual field, denoted by f""

f "" Fjj (fe~ + F ). (2.11)
2 ac r ret advi)

Using Equations (2.7) and (2.8) with Equation (2.11)

f" = -(F~ l + F."). (2.12)
2

Because the incoming and outgoing radiation fields are derived from potentials that

satisfy the homogeneous Equation (2.6), the field fl" is sourceless. It is also free from

singularities at the worldline of the electron.

2.2 Derivation of the Equations of Motion

The main concept used by Dirac in deriving the equations of motion for the

electron moving in flat spacetime is the conservation of energy and momentum. The

first step is to surround the worldline of the electron by a thin cylindrical tube of

constant radius. This radius is assumed to be very small, specifically smaller than

any length of ] li, -i .1l significance in the problem. The objective is to calculate the

flow of energy and momentum across the three-dimensional surface of the tube, using

the stress-energy tensor Tl, calculated from the actual electromagnetic field. The

stress-energy tensor is given by

4 7 T9p = Fact ,V Fact + 1 pp Fact ac3 F`a (2.13)
P 4 act.

The flow of energy and momentum out of the surface of the tube is equal to the

difference in the energy and momentum at the two ends of the tube.

In the following, dots over quantities denote differentiation with respect to the

proper time s. To simplify the notation, (dz,/ds) is set equal to v,,. One then obtains

the equations

vv = 1, (2.14)

v., -= 0, (2.15)

v, *: + .,,'' = 0. (2.16)

To calculate the stress-energy tensor, the electromagnetic 4-vector potential and

the electromagnetic field need to be calculated first. The retarded potential at a point

x, generated by an electron moving on the worldline z,(s) is given by

Aretp q (2.17)
cl t (xe z v)

calculated at the retarded proper time, which is the value of s that solves the equation

(x, z,) (x" z") 0.

(2.18)

An equivalent expression for the retarded potential, obtained from Equation (2.17)
by using elementary properties of the 6 function, is

Aet. =- 2 q f 6[(x, z) (xv z)] ds (2.19)

where Tint is a proper time between the retarded and advanced proper times. By
differentiating Equation (2.19) and using Equation (2.5), the retarded electromagnetic
field is obtained

Fret, -2 q q i d -ip) 6[(x z) ( )] ds. (2.20)
Fret / (y T y[ \ )

By using again some of the elementary properties of the 6 function, an equivalent
expression for the retarded field is derived

Fret p, d z (x-z-j z( z )] (2.21)
A (x ^ zA ^ ds 1i (x" z) (2.21)

Again, all quantities are calculated at the retarded proper time.
Since the goal is to calculate the stress-energy tensor at the two ends of the
world-tube surrounding the worldline of the electron, it can be assumed that the
point x, is very close to the worldline. Specifically

x = z/(so) + 7/' (2.22)

where the 7t's are very small. Then, the fields can be Taylor-expanded in the 7,'s.
In the expansions that follow, all the coefficients are taken at the proper time so. It
can be assumed that the retarded proper time is so a, where it is reasonable that a
is a small positive (Ii.ml, il' of the same order as 7,.

The detailed calculations needed to obtain the Taylor expansions of the electro-

magnetic fields are tedious but straightforward. They can be found in Dirac's paper;

here I present only the results of that calculation.

Since 7- is a space-like vector, it can be assumed that 7y,7 = -2, where E is

a positive number. Taylor-expanding the retarded field and keeping only the terms

that do not vanish in the limit E -4 0 gives

Fretpv q (1 j)V [ v) >- 1 (v- __ v) (1 -- 7-VA)
+ 1
-68 15%v (vs'> Vv7p) 2 6 1/
2 2
+ 2 KQV '4v)].

(2.23)

The advanced field can be obtained by changing E to -E and changing the sign of the

whole expression. By using the retarded and advanced fields in Equation (2.10) the

radiation field on the worldline is obtained

4q
Frad v 3 ( v V &,vj). (2.24)

The actual field then becomes

Fact p = fiv + q [1 7Av^]- ( E 1 7EU)

1 1
+ -(1 +2 V-)( v, (2.25)
1 ( J_ vv)j/
+ E-1

To calculate the energy and momentum flow out of the tube, the stress-energy

tensor needs to be calculated. In fact, only its component along the direction of y, is

necessary. Substituting Equation (2.25) into Equation (2.13):

4 i T ,,p F.atpu F 't vp F. ,3 F13a
Sa t p act t
q2 (1 7 )-1[ 1 -4 +
= q2 -1 (2 E 2 \VA) 'Y

-I E(1 +t 3 ]Av) t
2 3

+ qE- lvf4.

(2.26)

The flow of energy and momentum out of the surface of the tube is given by

integrating this component of the stress-energy tensor over the surface of the tube.

The result is

( 12 E-1
2

q v" f, ) ds

where the integration is over the length of the tube.

As mentioned earlier, this flow of energy-momentum depends only on the con-

ditions at the two ends of the tube. That means that the integrand must be a perfect

differential, namely

1 2 -1
c

q Vf" f = B,

(2.28)

for some B,. Equation (2.15) and the fact that f"" is antisymmetric in its indices put

a restriction on B., specifically

12
,PB q2 E-1 ,
v/* / 5Qev

q v. f" 0.

(2.29)

Thus, the simplest acceptable expression for B. is

BP, kvg

(2.27)

(2.30)

for some k independent of s. From Equation (2.28) one obtains

k = 2 2-1 (2.31)

where m must be a constant independent of E in order for Equations (2.28) and (2.29)

to have a well-defined behavior in the limit E -> 0. Finally, the equations of motion

for the electron are

m i,.= q v, (2.32)

and m plays the part of the rest mass of the electron. This result is used in Section

(2.3) to derive an expression for the radiation damping effects on the moving electron.

The field f" defined in Section (2.1) allows the equations of motion (2.32) to

be expressed in a simple form. However, in practical applications one would prefer

to have the incident radiation field Fi~" in the equations of motion, as Fi~ is usually

given. By substituting Equations (2.7) and (2.10) into Equation (2.11), an expression

for f^' is obtained, that involves the incident field and the radiation field:

f'"= F' + Ftad" (2.33)

Then, the equations of motion become

1
m t,, =q2 vvFad, (2.34)

The first term of the right-hand side of Equation (2.34) involves the incident

radiation field and gives the work done by that field on the electron. The second

term of the right-hand side involves the radiation field emitted by the electron and

is the term of particular interest when one considers radiation reaction. This term

gives the effect of the electron radiation on itself and is present even if there is no

external radiation field, namely even if Fi = 0. The fact that the radiation reaction

effects on the electron can be described by using the electromagnetic field F` is

one of the important results of Dirac's work and is the idea used when motion of

particles in curved spacetime is considered, to calculate radiation reaction effects on

those particles.

It is worth writing the equations of motion in terms of the characteristics of the

worldline of the electron and the external radiation field, which can be achieved by

using Equation (2.24) for the radiation field of the electron

m ', = q v, Fi,,;, + 2 q 2 + 2 q2 tV" v. (2.35)

It is an important and a very interesting feature of Equation (2.35) that, in addition

to the first derivative of vI, its second derivative shows up as well. A discussion of

that fact and of some of its implications is presented in [13], but that is beyond the

scope of this dissertation.

CHAPTER 3
RADIATION REACTION IN CURVED SPACETIME

Dirac's study of the radiation reaction effects on an electron gave a relatively

simple result, because the electron was assumed to be moving in flat spacetime. The

analysis becomes significantly more complicated for particles moving in curved space-

time. DeWitt and Brehme [14] were the first to study the motion of an electrically

charged particle in a curved spacetime using Green's functions. Later, Mino, Sasaki

and Tanaka [15] generalized that analysis for the case of a particle creating a grav-

itational field while moving in curved spacetime. Since Green's functions are used

extensively in subsequent chapters to determine the self-force effects on different par-

ticles, it is appropriate to present the analyses of DeWitt and Brehme and of Mino,

Sasaki and Tanaka before proceeding.

In this and all subsequent chapters, geometrized units are used, meaning that

Newton's gravitational constant and the speed of light are both set equal to 1. Note

also that the signature of the metric is changed to (- + ++) from now on, so minor

changes in some of the equations previously mentioned should not be surprising.

3.1 Nonlocal Quantities

One of the main characteristics of any Green's function is that it connects two

points in spacetime. It does that by propagating the effect of the source, from the point

where that source is located (source point) to the point where the field needs to be

calculated (field point). Since Green's functions are inherently nonlocal quantities, the

discussion about them can be facilitated if some more elementary nonlocal quantities

are introduced first.

The most general class of nonlocal quantities is the class of bitensors, which

are simply tensors whose indices refer to two points in spacetime. In the following,

x denotes the field point and z denotes the source point. In order for the indices for

each point to be easily identifiable, all unprimed indices refer to the field point x and

all primed indices refer to the source point z.

A very important (1ll..1l il' for the study of nonlocal properties of spacetime is

the biscalar of geodesic interval s(x, z). It is the magnitude of the invariant distance

between the points x and z as measured along a geodesic that joins them and is a

non-negative quantity. It is defined by the equations

V VVs = V's V,,s 1, (3.1)

lim s 0, (3.2)

where V" denotes covariant differentiation with respect to the background metric

g"". It is obvious that s must be symmetric under interchange of its two arguments,

namely

s(x, z) = s(z, x). (3.3)

With the signature of the metric being (- + ++), the interval s is spacelike when the

+ sign holds in Equation (3.1) and timelike when the sign holds. The points x for

which s = 0 define the null cone of z.

It is more convenient to use a different qi(111.m il to measure the invariant distance

between the source point and the field point, a (qil.,li i l' that is, however, related to

s(x, z). That quantity is Synge's [16] world function and is defined by

(x, z) -s 2(x,z). (3.4)
2

By using the defining equations for s one can deduce that a has the properties

1 1
V'u VcUa VO'a V/,a
2 2

(3.5)

(3.6)

limr a 0.
XZ

Also, a is positive for spacelike intervals and negative for timelike intervals.

Another very significant nonlocal q(1.,,l il'v is the bivector of geodesic parallel

displacement, denoted by g,0,(x, z). The defining equations for it are

V3g",' V3T 0, V3'g0' Va (T 0

lim g9'
X--Z

(3.7)

(3.8)

S"'
Oa

Equation (3.7) signifies that the covariant derivatives of gaa' are equal to zero in the

directions tangent to the geodesic joining x and z. Equation (3.8) expresses the fact

that g,,, is equal to the Kronecker-6 when x = z. The bivector of geodesic parallel

displacement also has the property that

9aa' (x, z) ga'a(z,x).

(3.9)

In the following, the determinant of ga', is denoted by

g I ga, I,

6 1 9 .

(3.10)

(3.11)

The effect of applying the bivector of geodesic parallel displacement to a local vector

A"' at z is a parallel transport of that vector from point z to point z, along the

and

geodesic that connects the two points. The result is a local vector A" at point x. In

general, g,,, can be used to transform any bitensor that has indices referring to the

two points x and z to a tensor whose indices refer to one point only, for example

9, 9, T3' 1T36 T T (3.12)

How useful the bivector of geodesic parallel displacement is in studying nonlocal pror-

erties of spacetime can be understood if expansions of a bitensor about one point are

considered. In order for a bitensor to be expanded about a certain point in spacetime,

all its indices must refer to that specific point. For bitensors for which that is not the

case, their indices must first be homogenized by applying g'', and then the expansion

about the specific point can be taken.

A bivector that is very useful for the Hadamard [17] expansion of the Green's

functions is defined by

Daa'(x, z) = -VaV, T(x, z) (3.13)

and a biscalar relating to it is its determinant

D=- D, | (3.14)

DeWitt and Brehme proved that

lim D ',, (x, z) = gaa (z) (3.15)

which shows that the biscalar D is i'ii, v.ii-lii-. at least when x and z are close to

each other. In fact, D is the Jacobian of the transformation from the set of variables

{zW', X"}, which specify the geodesic between x and z in terms of its two end points,

to the set of variables {z", Vc'a}, which specify the geodesic in terms of one of its

end points and the tangent to the geodesic at that end point. DeWitt and Brehme

also showed that D obeys the differential equation

D-1 V0(D Va) =4. (3.16)

Instead of the biscalar D, a different biscalar is used in the Hadamard expansions

of the Green's functions. That biscalar is denoted by A and is defined as

A(x, z) =[g(x, z)]-D(x, z). (3.17)

DeWitt and Brehme proved that A can be expanded in terms of derivatives of a and

that the expansion is

A 1+ R Voa Va + 0(s3), for x -_ z (3.18)

where R" is the background Ricci tensor.

3.2 Green's Functions

The goal of the work of DeWitt and Brehme was to study the radiation damping

effects on a particle of a given electric charge that moves in curved spacetime. To

do that it is necessary to study the vector field that represents the electromagnetic

potential created by the particle. However, the equations for the scalar field are

less complicated and thus easier to deal with. Also, the lack of indices makes the

results more transparent. For those reasons the scalar Green's functions and their

corresponding fields are discussed first.

3.2.1 Green's Functions for a Scalar Field

A point particle of scalar charge q which is moving on a worldline F : z"'(r),

where T is the proper time along the geodesic, creates a scalar field y which obeys

Poisson's equation:

V2 = -4 7 Q.

(3.19)

In Equation (3.19), Q is the source function for the point particle. Specifically

L(y) =q (-g)- 64(y z(r)) dr

(3.20)

where y is some point in spacetime. It is desired to express the different solutions of

Equation (3.19) as integrals containing Green's functions, namely

(3.21)

so the properties of the various Green's functions are emphasized in this section.

One function of importance is the symmetric Green's function, Gsym(x, z), which

satisfies the inhomogeneous differential equation

(3.22)

V2Gsym(x, z)

The Hadamard form [17] of this function is

1
Gsym(x, z) [U(x, z) 6(a) V(x, z) ((-a)]
87

(3.23)

where 6 is the step function that equals 1 if the argument is greater than zero and

equals 0 otherwise. U(x, z) and V(x, z) are biscalars that are free of singularities and

symmetric under the interchange of x and z, namely

U(x, z) = U(z, x)

(x) 4 I

(-g) G(x, y) Q(y) dy4

J spacetime
4 q/ G[x, z(T)] dr,

(3.24)

(-g)- 64(X- z).

and

V(x, z) V(z, x). (3.25)

They can be determined by expanding the solution of Equation (3.22) in powers of a,

in the vicinity of the geodesic F. The biscalar U(x, z) satisfies the differential equation

1
U-1(x, z) VoU(x, z) = A-1(x, z) VoA(x, z) (3.26)
2

and the boundary condition

lim U(x,z) 1 (3.27)
X-Z

and is given by

U(x,z) =[A(x,z))
1 (3.28)
=1 + -R, Var Va + O(S3), x -- F
12

where s is the proper distance from the point x to F measured along the spatial

geodesic which is orthogonal to F. The biscalar V(x, z) satisfies the homogeneous

differential equation

V2V(x, z) 0 (3.29)

and is given by

12
V(x, z) Rz) + 0(s), x -> P. (3.30)

It is noteworthy that the symmetric Green's function vanishes for a > 0, that is for

spacelike separation of the points x and z. Also, because both U(x, z) and V(x, z)

are symmetric under the interchange of x and z, so is Gsym:

Gsym(x, z) =Gsym(z, x).

(3.31)

If & = d((x, z(T))/dT, the scalar field ,''r(x) associated with the symmetric

Green's function is

W -x) = [U(x, z) 6(a) V(x, z) G(-1)] dr
2 j(3.32)
[qU(x, z) [qU(x,z) q( et +j ( 3.
L 2 Jret+ 2 ]- ,v 2 _v

and consists of two parts. The first part is the one that contains the biscalar U(x, z)

and the 6-function 6(a) and is referred to as the direct part. This is the term that

corresponds to a = 0, which is the part of the field that comes from the retarded

and advanced proper times (Tret and Tadv respectively), namely the proper times that

correspond to the intersection of the geodesic F with the past and future null cone of

the point x. In other words, the direct part has the same singularity on the null cone

that the symmetric scalar field has in flat spacetime. The second part is the one that

contains the biscalar V(x, z) and the step-function O(-a) and is referred to as the

tail part. This term gives the part of the field that corresponds to a < 0, which is the

part that comes from the interior of the past and future null cone of x. It is the part

of the field that is due to the curvature of spacetime and vanishes in flat spacetime.

The symmetric Green's function can be separated into the retarded and ad-

vanced parts, which constitute two very important Green's functions themselves.

Specifically it can be written as

1
Gsym(x, z) = [Gret(, z) + Gadv(, z) (3.33)
2

where the retarded and advanced Green's functions are given by

Gret(x, z) = 2 [f(x), z] Gsym(x, z), (3.34)

Gadv(x, z) = 2 [, Z(x)] Gsym(x, z).

(3.35)

In Equations (3.34) and (3.35), E(x) is any spacelike hypersurface that contains x.
The step-function O[Z(x), z] = 1 O[z, Z(x)] is equal to 1 when z lies to the past
of E(x) and vanishes when z lies to the future of E(x). Both the retarded and the
advanced Green's functions satisfy the inhomogeneous differential equation

V2Gret(x, z) V2Gadv(,) _(g) 64(x, z). (3.36)

They also have the property that

Gret(x, z) Gadv(z, x). (3.37)

The retarded and advanced scalar fields, both solutions of the inhomogeneous
differential Equation (3.19), can also be expressed as integrals of the respective Green's
functions

'(x) = 4 q Gret [x, z(r)] d = 2 z) q V(, z) dr, (3.38)

1 2 J(r 1} r+00
adv (x) = 4 r q Gadv[x, Zr) dr 2 q -z) q V(x, z) d. (3.39)
1 2( I -dv Jdv
The retarded field '/" (x) is the actual scalar field that results from the scalar particle
and is singular at the location of the particle.
The last scalar Green's function that is significant for the analysis that follows
is the radiative Green's function Grad, which is defined in terms of the retarded and
advanced Green's functions in a manner analogous to the radiation field defined by
Dirac in flat spacetime, namely

Grad(x, z) = Get(x, z) Gadv(x, z).

(3.40)

It is obvious by this definition that Grad(x, z) satisfies the homogeneous differential

equation

V2Grad(x, z) =0. (3.41)

By Equation (3.37) one can infer that Grad(x, z) is antisymmetric under the inter-

change of its arguments

The corresponding radiation field equals the difference between the retarded and

."() = ,.r- (x) dv(x) 4 i q Grad[x, z(r)] dr (3.43)

and is a sourceless field because it satisfies the homogeneous differential equation

V2 1,'r- (x) 0. (3.44)

Consequently, ,-' is smooth and differentiable everywhere in space.

The Green's functions and the corresponding scalar fields that were mentioned

in this section are used extensively in Chapter 4, where the self-force on scalar particles

is discussed.

3.2.2 Green's Functions for a Vector Field

Now that the scalar Green's functions and their properties have been discussed,

it is straightforward to define analogous Green's functions for the electromagnetic

vector potential A" generated by an electrically charged particle moving on a geodesic

F: z"'(r). Assuming that A" satisfies the Lorentz gauge condition

V7A0 = 0,

(3.45)

Maxwell's equations for it become

V2Ac R`A3

-4 Jo,

(3.46)

and, again, the goal is to express the vector potential as an integral that contains a

Green's function and the source J", specifically

AX) s4 peacetime

(-g)i GC,(x, y) J(y) d4y.

(3.47)

The symmetric Green's function for a vector field obeys the equation

V2Gasym (, z)

R 3 Gsym' (x, ;)

9 2 g ,,a 64(, z)

and the Hadamard form of it is

1
-[uI ,(x, z) 6(a)
87

V1, (x, z) G(

The biscalar U,,, is given by the differential equation

(2 V3U,, + 00, A-1 V3A) V,3a

with the boundary condition

lim U", (x, z) = g0, (z).
x-z

As shown in [14], the solution is unique

Ua,(x, Z) A /( Z) g00'(x, Z)

S[1+ R3'7' V7a V7y + O(s3)] g,, for x z
12

(3.48)

Gsym (x, z)
aa' ^

(3.49)

(3.50)

(3.51)

(3.52)

where Equation (3.18) was used to derive the final expression. For the bivector V,,,

DeWitt and Brehme prove that

1 1
lim V0 '(x, z) -= g (I R gg R) + 0(s), for x -> F. (3.53)
x-z 2 6

The symmetric electromagnetic vector potential is calculated by

A (x) 47 (-g) Gs(x, y) J(y) d4y

i(-) g [A(x, y) ga(x, y) 6(a) V0(x,y) (-)} J((y) d4y

(3.54)

Just as in the case of the symmetric scalar field, the symmetric electromagnetic poten-

tial also consists of the direct part, that contains the 6-function and comes from the

retarded and advanced proper times, and the tail part, that contains the O-function

and is the contribution from within the past and future null cone. An interesting fea-

ture is the appearance of the bivector of geodesic parallel displacement in the direct

part. It signifies that the electromagnetic radiation is parallel propagated along the

null geodesic that connects the points x and z.

The symmetric Green's function can again be separated into the retarded and

Gsym(x, ) I [G ret(, (advx, z) (3.55)
aa' \*r z! 2 9 [^aa'^i z) aa' \ -0.00

which are given by the equations

ret (x, z) 2 6[E(x), z] Gsy(x, z) (3.56)
~7LLYI~-L -~1 IV ii~~l /i U~~y \LU ,i

(3.57)

Gadv, z) = 2 8[z, (x)] Gsym(, (z).

They are solutions of the inhomogeneous differential equation

V72G e(x, zI ret,(, z) V2C /redv/ (x R adv/, z)
G Ilaail y z) l\a L v Gaa' z) a Ra3Ga' {ZL -

-g 2 ga(, z)J (x, z)

(3.58)

and have the property that

(3.59)

The retarded and advanced electromagnetic potentials are then given by

A t (x)= 4 p
J spacetime

(-gGre (x, y) J (y) d4 y

(3.60)

Av (x) 4 (-g GC (x,y) J3(y) dy (3.61)
/ spacetime

and Art(x) is the actual potential generated by the moving charged particle.

The radiative Green's function is defined in terms of the retarded and advanced

Green's functions by the equation

rtad/ )\ _tret /(X, Z)\ adv \
aa' X, (xz) a' )z) Gaa (, z)

and satisfies the homogeneous differential equation

(3.62)

G,,, (X, Z)

3 rad (x, z) 0.
a 03a'.

Using the symmetry expressed in Equation (3.59) for the retarded and advanced

Green's functions it can be inferred that

aLrI\X, Z- ,,,Z, X.

(3.64)

(3.63)

G ret (X, ) Gadv( x)

The corresponding radiation vector potential is equal to the difference between

the retarded and advanced vector potentials

(3.65)
S-g)l x, y) j(y) d4 (3.65)
Sspacetirme

and is a source-free field, because it satisfies the homogeneous differential equation

V2A d ) R aA;d(x) = 0. (3.66)

The important quantities for the motion of a charged particle are not the vec-

tor potentials but rather the electromagnetic fields. The electromagnetic field that

corresponds to a vector potential A0 is calculated by:

F,3(x) = VA,(x) V7AP(x). (3.67)

For completeness, the definitions of the symmetric and radiation fields are given, in

terms of the retarded and advanced fields:

sym /(Frt +F adv (3.68)

3.2.3 Green's Functions for a Gravitational Field

Even though the work of DeWitt and Brehme did not cover the motion of

a massive particle creating its own gravitational field, the Green's functions for a

gravitational field are presented here, for completeness. A more detailed description

is given by Mino, Sasaki and Tanaka in [15] and it is that analysis that is followed

here.

Assuming that a massive pointlike particle is moving on a geodesic F : z"'(r)

and is inducing a perturbation h,,(x) on the background, the trace-reversed metric

perturbation is

(3.70)

where h(x) and h(x) are the traces of hP"(x) and of hP"(x) respectively. It is assumed

that the trace-reversed metric perturbation obeys the harmonic gauge condition

Vh"(x) = 0.

(3.71)

In this gauge, the linearized Einstein equations [18] become:

V2h + 2Rs3 h76

167rTa

(3.72)

to first order in the metric perturbation. The Green's functions of interest in this

case are bitensors and are used to express the trace-reversed metric perturbation in

integral form

h3(x) = 167 time
/ spacetime

(-g) G':S(x, y) T, (y) d4y.

The symmetric Green's function G03'y' (x, z) satisfies the differential equation

V2GSM (x, z) +2 Rs 0 (x) Gy 6 (x, z)
Psym / V sym Y

-2 g'((x, z) g3'(x,z) 64 Z) (3.74)
(-g)2

and the Hadamard form for it is

yG'('x, ) [Uay's'(x, z)6(a~) Vapy's'(x, z)(--)].8
sym 8x

(3.73)

(3.75)

hy,(x) = hy,(x) g[v,,(x)h(x)

The bitensor U8Y'3'' (x, z) is the solution of the homogeneous differential equation

[2 V7 U`y3' (x, z) + py' (, z)] VAo-
A

(3.76)

with the boundary condition

(3.77)

Mino, Sasaki and Tanaka [15] prove that the solution is

2 g7'((x, z) g)' (x, z) A( z).

(3.78)

The bitensor V3P'6' (x, z) is divergence-free

V3Vo'~''(x, z) 0

and satisfies the homogeneous differential equation

V2Vy'5' (, z) + 2 c Vvy7''(x, z)

(3.79)

(3.80)

with the boundary condition

(3.81)

It is proven in [15] that the solution is

-9g' R,1 ~ ,'a'(z) + 0(r), for x -+ F.

limn U,37'5' (x, z)

lim 2g"(' (x, z) g9)"'(x, z).
X-zz

Ucpy'Gs'(x, z)

lim VP,7's' (, z)
o(->0

Voap,3y' (x, z)

(3.82)

The symmetric gravitational field is obtained by

hpm(x) = 167 g (-g G's' (x, y) T6 ^(y) d4y
Sspacetime

= 2 (-9) [Uo^(x, y)6(a) VoyS'(x, y) (-T)] T7s(y) d4y
J spacetime

(3.83)

Its direct part is the part that contains the 6-function and gives the contribution from

the null cone of point x. Its tail part is the part that contains the O-function and

gives the contribution that comes from within the null cone of point x.

The symmetric Green's function can be separated to the retarded and advanced

Green's functions

a 'S'(,, z) = [G%,t'(x, z) + G '6 (x, z)1
sym G' t *v )

which are given by the equations

(3.84)

G '(x, z) = 2 e[Z(x), z] G`s'y(x, z)

Gad-"''(x, z) = 2 e[z, Z(x)] G'7'(x, z).

(3.85)

(3.86)

They are solutions of the inhomogeneous differential equation

V2G'/y' s(x, z) +2 RC3 (x) G"-' '(x, z)

V2G Cd-' (x, z) + 2 R ,(x) G"d'v (x, z)

-2 g(-(x, Z) g)1'(X, Z) 64 -
(-9g)
(3.87)

and have the property that

(3.88)

G'ret (x, z) G-" 6(z)

The retarded and advanced gravitational fields are given by

hret = 6 (-g G (x, y) Ts(y) y (3.89)
spacetime

16r (- g) C v(x,T y) T7 (y) d4y (3.90)
spacetime

and hr' is the actual trace-reversed gravitational perturbation induced by the moving

particle.

The radiative Green's function is defined in terms of the retarded and advanced

Green's functions by the equation

Ga'(x, z) G'y''(x, z) G`p'- (x, z) (3.91)
"rad '- ret -- adv r ,

and satisfies the homogeneous differential equation

V2G'5'(x, z) + 2 R (x) G^""'(x, z) 0. (3.92)

Using the symmetry expressed in Equation (3.88) for the retarded and advanced

Green's functions, a symmetry property for the radiative Green's function can be

derived, namely

Goy6'(, z) -G Y' (z, x). (3.93)

Finally, the radiative gravitational perturbation is given by the difference be-

tween the retarded and advanced gravitational perturbations

ha13 (X h a,(-X ha,3 (X'i
2 ((3.94)
=167 (--g) 'G ,)T(t)d4
Jspacetime

and is a solution of the homogeneous differential equation

V2r7/ ) 2 R d (x) = 0. (3.95)

3.3 Motion of a Charged Particle in Curved Spacetime

Following DeWitt and Brehme's analysis, I present the equations of motion and

the radiation reaction effects for a particle carrying an electric charge e. Studying the

vector case is useful in comparing the result derived in curved spacetime with Dirac's

result for flat spacetime. Similar analyses can be done for the scalar case and the

gravitational case. Those analyses can be found in a recent review of the subject by

Poisson [19].

3.3.1 Equations of Motion

Let's assume that a particle of bare mass mo and electric charge e is moving in

a curved spacetime of metric g,, on a worldline described by F : z(r), where T is the

proper time.

In the following, the notation u' = (dz"'/dr) = z"' is used. Also, the symbol

(D/dr) is used to denote absolute covariant differentiation with respect to the proper

time r along the worldline.

The particle generates an electromagnetic vector potential A"(x) and an elec-

tromagnetic field Fo3(x) given by Equation (3.67). The action of the system contains

three terms; the first term comes from the particle, the second from the electromag-

netic field and the third from the interaction between the two:

S Sparticle Sfield Sinteraction, (3.96)

where

Sparticle =-Mo j [-g0,a' (z) U' U' d (3.97)

Sfield Fp Fp3 g d4x (3.98)
167

Sinteraction = e A, (z) ua' dT. (3.99)

In determining the equations of motion, the stationary action principle can be

applied to S for variations in the potential A" and for variations in the worldline z"'

separately. Varying A" gives

V3Fc3 4 7J" (3.100)

where J" is the current density defined as

JP(y) e g,(y, z) u'64(y,z) d. (3.101)

The electromagnetic field Fcp is invariant under a gauge transformation of the form

An -- An + VcA(x), where A(x) is any scalar function of x. For convenience, A(x)

can be chosen in such a way that the electromagnetic potential satisfies the Lorentz

gauge condition

VoA" 0 (3.102)

in which case Equation (3.100) for the electromagnetic field becomes

V2AS(x) R]A3(x) = -4Jc(x). (3.103)

Equation (3.103) allows one to determine the electromagnetic field, once the worldline

of the charged particle is known.

On the other hand, varying the worldline zc' gives

Dua'
mo = eF c (z) u '. (3.104)
(IT(ZU

Equation (3.104) allows one to calculate the worldline of the particle, once the elec-

tromagnetic field is specified.

As is well-known, the electromagnetic field given by Equation (3.103) is diver-

gent at the location of the particle and cannot be used in Equation (3.104). An

alternative method for studying the motion of the particle is to take advantage of the

conservation of energy and momentum for the particle and the field, the same method

that was used by Dirac in his analysis of the motion of an electron in flat spacetime.

3.3.2 Calculation of the Fields

It becomes obvious in Section (3.3.3) that in order for the conservation of energy

and momentum to be implemented along the worldline of the charged particle, it

is necessary for the retarded and advanced fields generated by the particle to be

calculated. A brief description of the calculation is given here; the details can be

found in [14].

In the following, Tret and Tadv are the proper times along the worldline of the

particle, where the past and future null cones of x respectively intersect that worldline.

They are referred to as retarded and advanced proper times. Also, re is the proper

time at the point where the hypersurface E(z) intersects the worldline of the particle.

All these points are shown in Figure (3.1).

x Z(x)

Sret

null cone

Z

Figure 3.1: Retarded and advanced proper times

Starting with Equation (3.60) for the retarded potential, and using the Hada-

mard expansion for the Green's function, a simple expression for A"t can be obtained

A" (x) 47 d4y (-g)i G (xy) J3(y)

47re d4 y(- g) 2 e[Y(x), y] [Ua3(x, y) (T) Va (x, y) (-)] x
J ~8r
x dT g ,(y, z) ua' 64(yz)

=e JX Uo (x, z) 6(a) u' dT ef V0&(x, z) 0(- ) un' dT

nil 1' f1 Tret
7 Tr et -e Vl,'(x, z) uC dT.
LV pOU J ret -Joo
(3.105)

By a similar calculation, starting with Equation (3.61), the advanced potential can
be calculated

av UC' r+o00
A (dv e, e Va (x, ) u dr. (3.106)
LVa'o- T v dv

Equations (3.105) and (3.106) give the covariant Lienard-Wiechert potentials.

The retarded and advanced electromagnetic fields can be calculated by Equation

(3.67) if the respective potentials are differentiated. The result is

a f Du8'
FUVet/adv J ,V u' (-Y'u + u7 V Du d ) (V77 uT')-3

[V,(U.,Va UpV,a) uCu + (UviV UU U Va) ]
dT

x (V77,( u-')-2
+ (VUc, VUlc, + V'CV ,a ,Va) n' (V/,a T ')-1
STret / dv
SeI T (V,,V ,V,, ,) u'/ dT.
(3.107)

In Equation (3.107), the lower of the two signs corresponds to the retarded field and

the upper sign corresponds to the advanced field. This is an important expression

because it gives the electromagnetic field at point x in terms of the characteristics of

the worldline of the charged particle. The significance of this becomes clear when the

final equations of motion of the charged particle are written down.

As becomes obvious in Section (3.3.3), when calculating the balance of energy

and momentum between the particle and the field along the worldline, the values of

the fields very close to the worldline need to be known. Expanding the retarded and

advanced fields in powers of a and its derivatives is a tedious but straightforward

calculation, which yields a complicated expression for the two fields, Equation (5.12)

of [14].

Following Dirac's ideas, the total field in the vicinity of the charged particle can

be written as the sum of the retarded field created by the particle plus the external

incoming field

F,= F + F, (3.108)

which, by using Equations (3.68) and (3.69), can be written in terms of the radiation

and symmetric fields

SF + + F sm. (3.109)

This breaking up of the total field is very important because, after implementing the

conservation of energy and momentum along the worldline of the particle, only the

radiation field F rd and the external incoming field Fc, show up in the equations of

motion of the charged particle.

Since the radiation field is the difference between the retarded and advanced

fields, only two terms of Equation (3.107) are of interest here. Specifically, it is

sufficient to write

Fret/adv C (gpag '31) 2 | ( -2 D '

f 1, ( (T), (T')) d (3.110)
2 J,

+ Ftra + (higher order terms)

where, again, the lower sign corresponds to the retarded field and the upper sign

corresponds to the advanced field. In Equation (3.110), the notation

fa0',,(X, z) = Vlrl(Xz, z) V Val ,(, z) (3.111)

is used. The term Fetr is, in fact, a sum of terms and is the same for the retarded

and the advanced fields, so it vanishes when the difference of the two is taken.

3.3.3 Conservation of Energy and Momentum

The first step, when considering the conservation of energy and momentum, is

the construction of a cylindrical surface that surrounds the worldline of the charged

particle. That surface is referred to as the world-tube from now on. As in Dirac's

analysis, the radius c of the world-tube is very small, because the balance of energy

and momentum between the particle and the field needs to be calculated in the vicinity

of the particle's trajectory. DeWitt and Brehme [14] explain the procedure for the

construction of the world-tube in great detail. For the needs of the present calculation,

the world-tube can be thought of as the three-dimensional surface that is generated

by the particle if the particle is surrounded by a sphere of a very small radius and is

then left to move along its worldline, starting at proper time Tr and ending at proper

time T2. In the following, the cylindrical surface is denoted by E, its two end caps are

denoted by Ei (corresponding to the earliest proper time Ti) and E2 (corresponding

to the latest proper time 72). The total volume enclosed by the world-tube is denoted

by Vt.

The stress-energy tensor T"' of the system consists of one part that comes from

the particle and one part that comes from the field

To T + T

where

TP =mof g cg u"'u' 64 dr (3.113)

1 1
T (F4 F, g.'3FF6) (3.114)
47 7 4

where Fp is the total field in the vicinity of the particle. It is the sum of the field

generated by the particle plus any external incoming field that is present.

The conservation of energy and momentum for the system can be expressed as

VTc3 0. (3.115)

When integrating Equation (3.115) over the world-tube, special attention should be

given to the fact that the integral (f VpTc3d4x) is not a vector in the curved spacetime

that is being considered here. That means that Gauss's theorem cannot be used as

usual, to convert the integral over the volume Vt to an integral over the surface of the

world-tube. The natural way to overcome this difficulty is to consider the integral

f g.'VpTP, d4x instead. In this integral, the bivector of geodesic parallel displacement
is used, so that any contributions to the integral by the point x are referred back to

the fixed point z, which is assumed to correspond to proper time T. That integral is,

then, a contravariant vector at point z, so Gauss's theorem can be used. That gives

(f 12 Tt d f (v +if) T3 d4x = 0 (3.116)

where dYE is the surface element along the surface of the world-tube.

The limit of the radius of the world-tube going to zero is taken next. Also the

integral over the surface Y of the world-tube is expressed as a double integral, over

the proper time and over the solid angle. In addition, the proper times Tr and T2 are

let to approach r. If their infinitesimal difference is denoted by dr, Equation (3.116)

becomes

mo dr + lim go'T"3dE = 0. (3.117)

It is in the calculation of the integral of the second term that the symmetric field and

the radiation field mentioned in Equations (3.68), (3.69) and (3.109) are very useful.

As was mentioned earlier, the result contains only the incoming external field,

the radiation field and the tail field and is

Duo' 1 1 '
mD = eFin a +-eFrad, a' [12 z(T)) '(T')dT' (3.118)
dT- 2 2--o

where m is the renormalized observed mass of the charged particle

m mo+ lim Ie -1 (3.119)
e-o 2

An equivalent expression is derived by using Equation (3.110) and the definition

of the radiation field given in Equation (3.69), and it is

Duo' 2 D2 U' Du Dup,
m = eFin ,' + -e2 U
d1r 3 dr2 dr dr (3.120)
+2 fu f ,(Z-(T), Z(T')) ')d'.
~I J--0o

Equation (3.120) contains only the ].li',-i .,1 characteristics of the worldline of the

charged particle and the incoming external field.

A short discussion about Equations (3.118) and (3.120) is in order. First it is

significant to notice that, except for the last term that contains the (Iil.,,I il' fa0a',

these equations are identical to Equations (2.34) and (2.35) derived by Dirac for flat

spacetime. Since that term contains the biscalar V,,,, it comes from the tail part of

the retarded electromagnetic field.

Also, it is interesting that the integral that shows up in these two equations

needs to be calculated from -oo to the proper time T, meaning that knowledge of the

entire past history of the particle is required. That is something that renders the use

of Equation (3.120) impractical and is a point that is also discussed in the following

chapters.

CHAPTER 4
SELF-FORCE

In general, the self-force can be due to a scalar, an electromagnetic or a gravita-

tional field. Two different methods for the self-force calculation are presented in this

chapter. The analysis is given in detail for the scalar self-force, due to the simplicity

of the notation. The results are also given for the electromagnetic and gravitational

self-force.

The first method, in which the direct and tail parts of the retarded fields are

used, was described by DeWitt and Brehme in [14] for the scalar and electromagnetic

fields and by Mino, Sasaki and Tanaka in [15] for the gravitational field. An axiomatic

approach to this method was also presented by Quinn [20] for the scalar field and by

Quinn and Wald [21] for the electromagnetic and gravitational fields.

The second method, in which the singular-source part and the regular remainder

of the retarded fields are used, was proposed by Detweiler and Whiting in [22], where

they described it for the scalar, electromagnetic and gravitational fields.

As in Chapter 3, z denotes a point along the worldline of the moving particle

and x any point in spacetime. The limit of x -> F is the coincidence limit. The

retarded and advanced proper times, Tret and Tadv, are the proper times at the points

where the null cone of x intersects the worldline F (see Figure (3.1)). Primed indices

refer to z and unprimed ones refer to x.

4.1 Scalar Self-Force

A particle of scalar charge q is assumed to be moving in a background spacetime,

described by the background metric gab. For simplicity it can be assumed that there is

no external scalar field and consequently, if the scalar charge q is small, the lowest order

approximation for the worldline F : z''(T) of the particle is a background geodesic,

where T is the proper time. But the scalar field y generated by the particle interacts

with the particle, inducing a self-force on it. The self-force gives F an acceleration,

so, to higher order, the particle does not trace a background geodesic.

4.1.1 Direct and Tail Fields

The discussion of the scalar field self-force in [14] and [20] shows that the self-

force on the particle can be calculated by

I 7 particle (4.1)

where .,~" is the scalar field that interacts with the particle and is given by

SqU(x- z) q V(x (T)) d. (4.2)
L 2 \7ret 0_

It can be inferred from the discussion of the scalar Green's functions of Section

(3.2.1) that the first term of the right-hand side of Equation (4.2) comes from the

direct part of the retarded field generated by the moving particle. This first term

is finite and differentiable at the location of the particle and its contribution to the

self-force is the curved-spacetime generalization of the Abraham-Lorentz-Dirac force

of flat spacetime. This contribution results from the moving particle's acceleration, if

the worldline F is not a geodesic, and can be expressed in terms of the acceleration of

F and components of the Riemann tensor [14, 15, 20, 21].

The integral term of the right-hand side of Equation (4.2) comes from the tail

part of the retarded field of the moving particle. Its contribution to the self-force

represents the result of the scattering of the retarded field of the particle, due to the

curvature of spacetime. Taking the derivative of the tail part of ", gives one term

that comes from the implicit dependence of the retarded proper time on x and one

term that comes from the dependence of the integrand on x, as was calculated by

Quinn in [20]:

STrrtet
= -q V(x, z(Tret)) [Varret(x)] q (V, V(x, z(r))} dr
7t (4.3)
=-q V 'rVa o q rV, V(x, z(7)) } dr
( I--ret J_ O
c q R(x\ ret
S(x z) +O(r) q {Va V(x, z(r)) } dr.

In Equation (4.3), r is the proper distance from x to F measured along the spatial

geodesic that is orthogonal to F.

The spatial part of the first term of the derivative calculated in Equation (4.3)

is not well defined when x is on F, unless R(x) = 0 there. That makes this approach

problematic since, for the self-force to be calculated, the derivative must be calculated

at the location of the particle. To overcome this difficulty, V.' -" can be averaged

over a small spatial 2-sphere surrounding the particle, thus removing the spatial part.

Then the limit of the radius of the 2-sphere going to zero can be taken and a finite

contribution to the self-force is obtained.

Another complication comes from the fact that, in order for the contribution of

the tail term to the self-force to be calculated, knowledge of the entire past history of

the moving particle is necessary, as is clear from the integral term in Equation (4.3).

It is also important to note that the field just like the field ',-' is not a

llv-i, illyy well-understood field. That is because it is not a solution to any particular

differential equation.

4.1.2 The S-Field and the R-Field

In order to provide an alternative scalar field that gives the self-force on the

moving particle, Detweiler and Whiting [22] took advantage of the fact that adding a

homogeneous solution of a differential equation to an inhomogeneous solution of the

same differential equation, gives a new inhomogeneous solution. Specifically, since

the biscalar V(x, z) is a symmetric homogeneous solution of the scalar differential

Equation (3.29), it can be added to the symmetric Green's function GsYm(x, z) to give

a new symmetric inhomogeneous solution GS(x, z) of Equation (3.22)

Gs(x, z) = Gsy(, ) + V(, z)
87
[U(x, z)6(a) V(x, z)O(-a) + V(x, z)] (4.4)
87r
1
[U(, z)6(a) + V(x, z)e(a )],
87

which obeys

V2GS(x, z)= -(-g)4(z ). (4.5)

The superscript S is used to indicate that this Green's function obeys the differential

equation that contains the Source term. This new symmetric Green's function has

support on the null cone, coming from the term that contains the 6-function, and

outside of the null cone, at spacelike separated points, coming from the term that

contains O(a). It has no support within the null cone, as shown in Figure (4.1).

The local expansion of the biscalar V(x, z) given in Equation (3.30) is sufficient for

the purposes of this dissertation, since the singular Green's function is only used for

points x close to the worldline of the particle.

The Regular-Remainder Green's function is defined in terms of the retarded and

singular Green's functions as

GR(x, z) = Get(x, z) Gs(x, z) =

I {2e[E(x), z][U(x, z)6(a) V(x, z)O(-a)] [U(x, z)6(a) + V(x, z)0(a)]},

(4.6)

direct

-- and GS

Tret

null cone

Z

Figure 4.1: Support of Gdirect, Gtail and Gs.

and it is obvious that, by construction, it obeys the homogeneous differential equation

V2GR(x, z) =0.

(4.7)

Clearly, GR(x, z) has no support inside the future null cone of x.
Using the new Green's functions, the fields ys and 9R can be defined. First,
the singular field is

s(x) = 4q fGS[x, z(T)] dT
[qU(x,z) [q U(xz) dv q ad
2(a o ,et r 2o e Tqtdv

and obeys Poisson's equation

(4.8)

V(z, z) dr,

V2yS(x) = -4(L.

(4.9)

It is noteworthy that the singular field does not depend on the entire past history

of the moving particle, but only on its motion between the retarded and advanced

proper times.

The regular-remainder is defined as

(Wx)=,~X) S(Wx)

-[qU(x,z) Tdv
2 7-etro

(4.10)

q +- V(x, z) dr,
-0 2 2Tret

and by definition obeys the homogeneous differential equation

V2R(x) = 0.

(4.11)

Since the field yR is a source-free field, it is smooth and differentiable at any point in

spacetime and consequently at any point along the worldline of the particle as well.

That is the most significant property of it, as far as the calculation of the self-force is

concerned.

To compare the scalar fields ,, and R, the difference of the two is calculated

from Equations (4.10) and (4.2)

R ) --`(X)

2 t

V(x, z) dr.

(4.12)

Using Equation (3.30) for the biscalar V(x, z) to expand the above integrand in the

coincidence limit, the difference of the two fields becomes

R(x) + O(r)]dT
2 1ret 12

12

(4.13)

R(.) ,. -- (X)

where the fact that the difference between the advanced and the retarded proper times

is equal to

Tadv Tret 2r + 0(r2), for x -> F (4.14)

was used. Taking the derivative gives

VaR V. .' + q Va[rR(x)] + (terms that vanish as x F). (4.15)
12

The second term of the right-hand side of Equation (4.15) gives an outwardly pointing

spatial unit vector and cancels the first term of the right side of Equation (4.3). Con-

sequently, the fields 9R and -I'" give the same self-force. In addition, the averaging

procedure that was necessary for the calculation of the self-force using ,- '" is not

required when 9R is used.

A self-force calculation that has been demonstrated in [23] showed an additional

benefit of calculating the self-force using the difference of the retarded and singular

fields. The retarded field can be computed numerically and, since the singular field

depends only on the motion of the particle between the retarded and advanced proper

times, the knowledge of the entire past history of the particle is not required. That is

a significant advantage of this method compared to the one that involves the direct

and tail fields.

Additionally, it is important that all the scalar fields used in this analysis are

specific solutions of the homogeneous or inhomogeneous Equation (3.19). That makes

them well-defined pir, -i .,1 fields, a property that the direct and tail scalar fields do

not have.

For those reasons, in this dissertation the self-force is calculated using

. = qV i 'p

(4.16)

4.2 Electromagnetic Self-Force

A detailed analysis for the self-force in terms of the direct and tail electromag-

netic potentials was done in [14] and was presented in Chapter 3. In this section, the

calculation of the self-force using the singular-source and the regular remainder fields

is presented.

It is assumed that a particle of electric charge q is moving on a worldline F :

za'(r) in a background described by the metric gab. For the purposes of this section

it can be assumed that there is no external electromagnetic field and the lowest order

approximation to the particle's motion is a background geodesic. The particle creates

an electromagnetic potential AP and an electromagnetic field Fab, which interact with

the particle, causing a self-force to act on it and making its worldline to deviate from

a background geodesic to order q2. In the Lorentz gauge

VaAa 0, (4.17)

the electromagnetic potential generated by the particle can be calculated by using

Maxwell's equations

V2A R" Ab -4Ja (4.18)

and the different solutions of interest were described by the Green's functions of

Section (3.2.2).

If a potential A"f, analogous to ,", is used to calculate the self-force in this

case, its tail part is

Aself (tail) (x) -q Vab (X, Z()) Ub' dr. (4.19)
--OO

Taking the derivative of it gives

S'Tret
(tail) q LVab Ub' VcT ret x q f (VbUb)dT

-q (b'd' -b'd') b- q
2 6 r -- r o

(4.20)

7c[Vab'Ub'] dr,

where Equation (3.53) for the bivector Vab' was used to derive the last expression.

It is obvious that the same problems that were encountered in the scalar case are

encountered here as well. The first term is not well-defined unless (Rab'- ga'b'/R)u b

is zero at the particle's worldline, the entire past history of the moving particle needs

to be known and the potential used in the self-force calculation is not a solution of

Maxwell's equations.

To overcome those difficulties, the singular Green's function in the neighborhood

of the particle is defined analogously to the scalar case

Ga,(x, z):

GsY(x, z) + VI (x,z)
(aalx, 8) Vaa, (X, Z)
1
--[Uaa(X, Z)((T) + Vaa (X, Z)G()].
87r

(4.21)

It obeys the inhomogeneous differential equation

V2G a,(2, Z) RabGsa, z)

(4.22)

- gaa'
-g 29aa164(2, z)

and gives the singular electromagnetic potential

As 47 (-g)- GS (,y)b(y)d4y, (4.23)

which is a solution of the inhomogeneous Maxwell's Equations (4.18). By definition,

both Gsb, and As have no support inside the past and future null cone of x. The

V7Aself
7c, Aa

regular remainder is defined by

A(x)- A t(x)- Aa(x). (4.24)

It has no support within the future null cone of x and it obeys the homogeneous

Maxwell's equations. Consequently, it is smooth and differentiable everywhere in

space, including every point along the worldline of the particle. The electromagnetic

self-force can be calculated by the equation

= qg" (VA VbA) b (4.25)

and is well-defined at the particle's location. It is stressed, again, that both As and

A4 are well-defined solutions of Maxwell's equations.

4.3 Gravitational Self-Force

It is now assumed that a particle of mass m is moving on a worldline F : za (r),

in a background described by the metric gab. The particle causes a metric perturbation

hab on the background metric. This perturbation obeys the harmonic gauge

Vahab 0 (4.26)

where hab is the trace-reversed version of the metric perturbation

1
hab hab 2gabhc. (4.27)

The linearized Einstein equations for it are

V2hab 2Rcbdhcd= --16Tab.

(4.28)

This metric perturbation interacts with the particle, causing a self-force to act on it

and forcing its worldline to deviate from a background geodesic.
If hs'f, which is not a solution of Einstein's equations, is used to calculate the
self-force, its tail part contains the bitensor Vabc'd' given in Equation (3.82). The
contribution of this tail term to the derivative comes partly from the dependence of

the proper time Tret on the point x. Specifically

T et
V f (tail) LM 'Vabc/d/'d'' Vfr] ie ri Vf [Vabc'd'U'ud] dT
-Ir f (4.29)
M[g7' ag b a (f Z- f)j m t Vabre VII/, ]dT.(4
L J X---r f _O

Just like before, the first term is not well-defined at the particle's worldline, unless

Rc'a'd'b'U'cUd' equals zero along the worldline. Also, knowledge of the entire past
history of the particle is required in order to calculate the integral term.
The singular Green's function Gabc'd'(x, z) in the neighborhood of the particle
can be defined as

S bc'd'(X z) = G(sm z) + Vabc'd'(X, Z)
1 8r (4.30)
[Uabc'd' (, Z)6(T) Vabc'd' (, Z) (o)1.

It gives the singular field h b which is an inhomogeneous solution of Equation (4.28)
and has no support inside the past or future null cone of x

hSb = 16 l(-g)-Gbd(x, y)Td(y)d4y. (4.31)

The regular remainder is defined as

h b(x) h x h (t).
ab ab -ha-'~b Wl

(4.32)

It obeys the homogeneous differential equation

V2 + 2RaCb d 0, (4.33)

which means that it is smooth and differentiable everywhere in spacetime. Also, it

has no support within the future null cone of x.

Then the self-force can be calculated by

S= (gb_ + iab)icd(vR 5Vbhh) (4.34)

which is well-defined at the location of the particle. It is important that both hR, and

hs, are solutions of specific differential equations.

A significant conclusion can be drawn at this point. As befits the problem, the

self-force is causing the particle to move on a geodesic of the metric (gab + hR), to

order m2. This metric (gab + hR) is, itself, a homogeneous solution of the Einstein

equations. An extended discussion of that point is given in [22] and [24] and is also

presented in Chapter 8 of this dissertation.

CHAPTER 5
SINGULAR FIELD FOR SCHWARZSCHILD GEODESICS

It was seen in Chapter 4 that the singular field is important in calculating the

self-force. It is the part that must be subtracted from the retarded field to give the

regular remainder, which is then differentiated to give the self-force. It is also clear

that, since the singular field obeys the inhomogeneous Poisson, Maxwell or Einstein

equations, it depends on the worldline of the moving particle and on the background

spacetime.

The coordinates that are used in this chapter to caclulate the singular field

are the Thorne-Hartle-Zhang coordinates, abbreviated from this point on as THZ

coordinates. Those coordinates were initially introduced by Thorne and Hartle [25]

and later extended by Zhang [26]. A short discussion on them is presented in Section

(5.1) of this chapter.

The detailed calculation of the singular field of a scalar charge q for geodesics in

a Schwarzschild background was presented in [23]. A brief discussion about that field

is given here for completeness, since the regularization parameters associated with it

are calculated in Chapter 6. The singular field is also calculated for geodesics in a

Schwarzschild background, for a dipole generating a scalar field, for an electric charge,

an electric dipole and a magnetic dipole each generating its own electromagnetic field

and for a spinning particle generating a gravitational field. For all these calculations,

MAPLE and GRTENSOR were used extensively. The GRTENSOR code is explained in

Section (5.8) and presented in Appendix A. In Sections (5.3) through (5.7), only the

results that the code gives are presented.

5.1 Thorne-Hartle-Zhang Coordinates

The Poisson, Maxwell and Einstein equations for the singular field assume a

relatively simple form when they are written in a coordinate system in which the

background spacetime looks as flat as possible. In the following, it is assumed that

the particle is moving on a geodesic F in a vacuum background described by the

metric gab. Also, R is a representative length scale of the background geometry, the

smallest of the radius of curvature, the scale of inhomogeneities of the background

and the time scale of curvature changes along F.

A normal coordinate system can always be found so that, on the geodesic F,

the metric and its first derivatives coincide with the Minkowski metric [18]. Such a

normal coordinate system is not unique. The one used here is the THZ coordinate

system and is used only locally, close to the worldline of the particle. Specifically it

is assumed that the background metric close to the worldline of the particle can be

written as

gab = lab + Hab
4 (5.1)
r ab + 2Hab + 3Hab + 0( ),

where ]ab is the flat Minkowski metric in the THZ coordinates (t, x, y, z) and

p2 2 +2 2. (5.2)

Also

2HbdXadXb = xijj(dt2 + dxl) + 4kpqBqZPXidtdXk
2Habd a db__i 2(kpq k
20 2 + 5
21 L zxJxk #2ikX k [Xijpq k P pqI
(5.3)

p3 p4
3HabdXadXb O( )abdX adb + 0(R )idxidx. (5.4)

In this and the following, a, b, c and d denote spacetime indices. The indices

i, j, k, I, n,p and q are spatial indices and, to the order up to which the calculations

are performed, they are raised and lowered by the 3-dimensional flat space metric ij.

The dot denotes differentiation with respect to the time t along the geodesic. Also,

Qijk is the 3-dimensional flat space antisymmetric Levi-Civita tensor.
If Hab consists of the terms given in Equation (5.3), the coordinates are second-

order THZ coordinates and are well defined up to the addition of arbitrary functions

of O(+-3). If Hab also includes the terms given in Equation (5.4), the coordinates

are third-order THZ coordinates and are well defined up to the addition of arbitrary

functions of O( ).

The tensors S and B are spatial, symmetric and trace-free and their components

are related to the Riemann tensor on the geodesic F by

E Rtitj (5.5)

,i ~PiRpqqjt. (5.6)
2

They are of O(L) and their time derivatives are of O(-).

For the calculation of the singular fields that follows, only the first two terms of

Equation (5.3) are included in gab. The remaining two terms are of 0(-) and must

be included in a higher-order calculation which must also take into account terms

coming from the 3Hab part of the metric.

The 'gothic' form of the metric is also defined as

gab -Jgab
g gga(.7

(5.7)

and its difference from the Minkowski metric is

Hb a g ab ab

For the lowest order terms of Hab it can be found that

4
Hab 2Hab 3Hab (+ o( 4

where

2Htt -2i jxi
2 Htk 210 [ X iXk k i 2]

2H5 2- (ij)pq qk k I ep(i 8) X 2
2` Ox )C qkXpX q P.

2Ht o ( P3 0 ( P4
R 3 RO(4) s ^ -O( -) .

(5.8)

(5.9)

(5.10)

(5.11)

(5.12)

(5.13)

Using the simple symmetry properties of the tensors S and B, the following

relationships can be shown for their components and the spatial THZ coordinates

(x, y, z):

k ki k
Cijk l + Ceikl j

ijk + ikl

(5.14)

(5.15)

Cjkl i

and

(--ijkSxl i+ ijkknk

(-eijk~lna + tijkaln k

- eilkXj- + k Xli)
_ikl B Xj + Ckl i)X
iklIknX + Cjkl1k in al

(5.16)

(5.17)

These relationships are used in the following sections when calculating the singular

fields for different sources to simplify the expressions for those fields.

The calculation of the regularization parameters shown in Chapter 6 requires

one to know the exact relationship between the THZ coordinates and the background

Schwarzschild coordinates. A general procedure for finding that relationship for a

given geodesic is to try to satisfy the three basic properties of the second order THZ

coordinates:

On F: the coordinate t measures the proper time along the geodesic and the

spatial coordinates x, y and z are equal to 0. Also, gab = lab and the first

derivatives of the metric vanish there.

At linear, stationary order: Hi = 0(x3).

The coordinates satisfy the harmonic gauge: ,gab = 0(x2).

The details of this procedure are described in Appendix A of [23].

For circular orbits in a Schwarzschild background of mass M, the THZ coor-

dinates have been calculated in Appendix B of [23] using specific properties of the

spherically symmetric background, which is much simpler than following the proce-

dure just described. Their functional relationships with the Schwarzschild coordinates

are given here in order to facilitate the discussion of the calculation of the singular

fields and the regularization parameters. The Schwarzschild coordinates are (ts, r, 0, q)

and the Schwarzschild metric is

ds2 = -( 1- ) dt2 (1 )M-dr12 r2d2 r2 sin2 Od2. (5.18)
r F

The circular orbit given by = Qts has orbital frequency equal to 2 = (Mro 3), at

58

First, the functions X, y, Z, which are Lie derived by the Killing vector =

- + Q2, are chosen

X (5.19)
(1 2M

y r sin(O) sin(O- ts) o (5.20)
ro 3M
Z rocos(0). (5.21)

Then the functions x, y, z, also Lie derived by the Killing vector ", and the function

t are defined in terms of X, y, Z

r sin 0 cos(9 Qts) ro
(1 2M)-
ro

X2 Y2o 3M z2
2 + ro 2Mj

*2- M2 2 + y2( o 3A/
2r(5 2M)(r,- 3M)-
M [ (r2 r + 3M2)
r5(1- 2Mro 3M) 8(ro 2M)
S(2 o 114r 8 123M2) 142
* --- (28r2 114r .- + 123 M2) + (14r2
28 014 0

+ _-(3r3 7
56(ro 2M )2
M2y2Z2(7ro 18M)
4(ro 2M)

y rsin0sin(O-

14 L
S[- 2X

M

)(3ro

8M) + 3Z2 (r

2M)2]

48r .1 + 33M2)

4r2.1 + 337r .1 2 430M3)

MZ4 (3r0o + 22M),
56

ro 3M)
-F024

2n(\

(5.22)

(5.23)

+2 o -- 3M+1 72
(ro 2M/
y 2 2

0M o
2(14r( 69r), 3M2) L 2Z2(r
+y2(14r 69r .fT + 89M2) + 2Z2(ro

15M)

2M)(7ro- 24M)],

M
r2a(1 2M
SF V

MZ
z rcos0+ 2 [-M) X2(2r o 3M) + 2(r 3M) + Z2(r- 2M)]
2r (r o- 3M)
MXZ
+14rM(1 2M)-[MX2(13ro 19M) (5.24)
14r5(1 -2M (ro 3M)

y2(14r 36r .I + 9M2 )+ Z2(o 2M)(14ro 15M)],

3M i rQy
To
Qti F 2 ro0- 3M MZ2l

(ro M) + )/y2 o + M
-r(1 2M (ro 3M) 2 3(ro 2M) (5.25)
QMXy
QMX- X2(r 2 11r _t + 11M2)
14r (r- 2M)(ro 3M) r
y2(13r2 -45r .1t + 31M2) + Z2(13ro- 5M)(ro 2M)].

Finally

x x= cos(Qtts)- sin(Qtts)
(5.26)
y = J sin(Qtts) + ycos(Qtts)

where Qt -Q 3M

There are two coordinate systems of interest. The first system is (t, x, y, z)

which is a non-inertial coordinate system that co-rotates with the particle, meaning

that the x axis always lines up the center of the black hole and the center of the

particle. The y axis is always tangent to the spatially circular orbit and the z axis

is always orthogonal to the orbital plane. As was already mentioned, for the spatial

coordinates of this system it holds that: I x= I-y =~ Iz = 0.

The second system is (t, x, y, z) and is a locally inertial and non-rotating system

in the vicinity of F. However, when viewed far away from F these coordinates appear

to be rotating due to Thomas precession, as is clear from the terms involving the sine

and the cosine of (Qtts) in Equations (5.26). It is also noted that: L p2 IJ- z

L Vt = 0, but Lx, I~y and I~t are not equal to zero. This second system is used

to calculate the singular fields. In order to avoid confusion of the field point x and

the source point z with the THZ coordinates, the notation needs to be changed. The

field point is denoted as p and its coordinates as xa and the source point is denoted

as p' and its coordinates as x\$,.

5.2 Scalar Field of a Charged Particle

The singular field generated by a particle that carries a scalar charge q and

is moving on a geodesic F : p'(r) (where, as usual, r is the proper time) of the

Schwarzschild background must obey Poisson's equation

V2S = -47 (5.27)

where the V2 is written in the THZ coordinates and the source term is

Q= q ( -g) 4 p p'()) (dr. (5.28)

That singular field is derived in [23] and is equal to

is + O( ). (5.29)

Here, instead of following the exact calculation of the singular field, which is performed

in detail in [23] following a similar derivation in [27], I prove that it does indeed satisfy

the scalar field equation to the order specified in Equation (5.29).

The differential operator of the scalar field equation becomes, in THZ coordi-

nates:

gV7aV 'iS a a, ab .'S) aa(Hb .)(5.30)
(5.30)
= rh.. ,.~j: Huiaa9s 2Hta(i at)ys Haitc.,t..

If the field s = (q/p) is substituted into this equation, the first term gives the

expected 6-function singularity and the last two terms vanish since p does not depend

on the time t. An explicit calculation shows that for the second term, 2H3 x Dc rA; is

equal to zero. The remainder 3H" gives a term that scales as O(-)). So

)Vavv() = -4iq6"(x) + O( ), for 0. (5.31)
p R4 R

Consequently, for the remainder to be removed a term of O(1) must be added to

q/p. So s q/p + O(-) is an inhomogeneous solution of the scalar wave equation
and the error in this approximation is C2.

5.3 Scalar Field of a Dipole

The calculation of the singular scalar field generated by a dipole moving on

a geodesic in a Schwarzschild background is presented in this section. The dipole

moment is assumed to have a random orientation and its THZ components are denoted

as K = (0, K, Ky, K).

In this and the following sections, the subscripts (or superscripts) (0), (1) and

(2) are used to indicate the order of significance of each term or component. The

subscript (0) refers to the most dominant contribution, the subscript (1) refers to

the next most significant correction, which is calculated for the singular fields, and

the subscript (2) refers to the next correction, the order of which is predicted for the

singular fields. It is important to realize that multiplying two terms of order (1) does

not necessarily give a term of order (2), because of the fact that there is no O(-)

correction to the metric (see Equation (5.1)) and the fact that the first correction to

the metric is of O(-) while the second is of O( ).

The scalar field of a dipole can be thought of as having the form

s ts o0) + 1) 2)

(5.32)

and it obeys the differential equation

V(0+1+2) QS V+1+2) ( ) + 1) ,2)) -47ro (5.33)

where the source term is given by

K [64v (p- p'())] g dr. (5.34)

The zeroth-order term is the scalar field generated by a dipole that is stationary

at the origin of a Cartesian coordinate system and it obeys the lowest order differential

equation
Vo7o o) T -4 (5.35)

It is equal to
s Kx' Kx + Ky Kz
() 3 (2 +2 + 2)3/2 (536)

The first-order term obeys the differential equation derived from Equation (5.33)

(V2 () + 72 lS o V2 )I/1S 2 )TS (5.37)
V(O) (1) (1IO) =0 = ) V (o)1) 1) 0)(7)

which means that the Vt) of the zeroth-order part of the field is the source term in

the scalar differential equation for the first-order part of the field. In general, that

source term is expected to contain the S's and the B's and to give the first-order

correction coming from the dipole's motion on the Schwarzschild geodesic. In this

case, Equation (5.37) gives that

o2 a2 O2
(a2 2 4 2 (1>)= 0 (5.38)

so the first order correction to the field can be set equal to zero.

The source for the next order correction comes from the part of the V2 that is

of O( i) acting on the zeroth-order scalar field so) The differential equation is of

the form

ata J O) o( x aai o) O( 4 ). (5.39)

That means that the next order term must be of O( ).

Finally, the singular scalar field of a dipole moving on a Schwarzschild geodesic

is equal to
Ts Kx + Kyy + Kzz
3S + O( ). (5.40)
P3 R3

5.4 Electromagnetic Potential of a Charged Particle

In this section, the singular electromagnetic potential generated by a charge q

moving on a Schwarzchild geodesic is calculated. Its general form is

A = A +(o) + A(1)+ AS(2). (5.41)

Since in the vacuum background the Ricci tensor is

Rab = 0, (5.42)

this electromagnetic potential, as well as all those calculated in this chapter, obey the

vacuum Maxwell's equations in curved spacetime

o+1+2) A (0+1+2) -47a. (5.43)

The zeroth-order term obeys the differential equation

(5.44)

V2 a = -47 J
'(O)AS(O) -4d

the source term being

J (q (-g- [p_ -p'()] dr, 0, 0, 0). (5.45)

The solution of this differential equation is the well-known Coulomb electromagnetic

potential

AS(o) =(, 0, 0, 0). (5.46)

The first-order correction to this electromagnetic potential obeys the differential

equation derived from Equation (5.43)

V72 o)A -V A2 I (5.47)

The first-order part of V2 (which contains the S's and the B's) acting on the zeroth-

order electromagnetic potential is the source term for the first-order correction. Sub-

stituting the THZ components of A'I()

A() (A(l), A(1), A(), A()) (5.48)

into the differential equation results in four differential equations, one for each one

of these components. Each equation relates a specific sum of second derivatives of a

component to a sum of terms of the form:

qx- f- or the t-component,
5 (5.49)
qB xx a
for the spatial components,
P3

where the dots denote appropriate indices.

Solving these four equations is straightforward, once one notices that the solu-

tion should have the form

Stcicj' for the t-component,
p (5.50)
qcp.B' xkxi for the p-spatial component,
P

each term multiplied by an appropriate algebraic factor. Substituting these expres-

sions into the differential equations gives a set of simple algebraic equations for these

factors, which can be easily solved to give the final expression for the first-order cor-

rection:

A() q Ceijxij, i jiBkXJk, q k yijXLik XJk, zijLikX X (5.51)

The next order correction comes from the part of the V2 that is of O(-) acting

on the zeroth-order electromagnetic potential. It gives a differential equation for each

component of A~(2) of the form

a a
S,,As(2) =0( ( ) x O .A(o) O( (5.52)

which indicates that the next order correction must be of O(-). It is noteworthy that

the first-order correction Ad(1) does not appear in the equation for the second-order

correction. That is because it only shows up in terms that involve the O(-) part of

the metric, which are of the form

0( ) x2A O(A)( (5.53)

and must be included in a higher-order calculation.

Finally, the singular electromagnetic potential of a charge q that is moving on

a geodesic in a Schwarzschild background is equal to

!([I- IijxixJ1, X (5.5x 4)
A= 2 [- Jk 2 k 2 + k O( ). (5.54)

5.5 Electromagnetic Potential of an Electric Dipole

The calculation of the singular electromagnetic potential of an electric dipole is
presented in this section. The dipole moment is assumed to point at some random

direction and its THZ components are qa = (0, q", qY, q).
The singular electromagnetic potential can be written as

A = As(o) + A(1) + AS(2) (5.55)

and obeys the vacuum Maxwell's equations

V(o+1+2) A(o+1+2) -47 (5.56)

where the source is

ja (- /qi[64(p -p'())] gdr, 0, 0, o). (5.57)

The zeroth-order term is the electromagnetic potential generated by an electric

dipole that is stationary at the origin of the Cartesian coordinate system, so it obeys
the differential equation

(o)As(o) = -47J"

(5.58)

The solution to this equation is well-known and has only a t-component

A- () 0, 0, (5.59)

The first-order correction to the electromagnetic potential obeys the differential

equation

V )(1) 7(1) A(o) (5.60)

If its THZ components are assumed to be

A ) (A(), A ), A A )) (5.61)

and are substituted into Equation (5.60), the differential equation for A~,) becomes a

set of four second-order differential equations for these components. Each differential

equation relates a sum of second derivatives of a component to a sum of terms of the

form

q- xx- for the t- component,
Sp (5.62)
q x-x- x for the spatial components,
P7

where, again, the dots denote appropriate indices.

These differential equations indicate that the solution should be equal to a sum

of the terms
qijkX i xjx k qi i XkX k
-SJk and (5.63)

for the t-component and a sum of the terms

P ei LJ3 k P i k P k
Cifq 1xxkx ciqI3klXXX 3 q ikXXlXl
S(5.64)
eP jqk flk xkxl Eijk qiL3PjXkX1 c ijk qiL3jxp k X 1
P3 P3 3

for the p-spatial component, each term multiplied by an appropriate numerical co-

efficient. In fact, using Equations (5.15) and (5.17), the last two terms that are

expected to show up in the solution for the p-component can be eliminated in favor

of the remaining four. Substituting these expressions into the differential equations

gives simple abgebraic equations for the coefficients. The final expressions for the

components of A(,) are

At 1 qiSJkx i x k
(l) 2 p3
2 *(5 .6 5 )
p i 1ik JP k & j
A % + +
s(1) 2p 2p3 2p3

The order of the next term in the expansion of the electromagnetic potential

can be predicted. It is the solution of the differential equation whose source term

comes from the 0( ) part of the V2 acting on Apo. That differential equation has

the form
ap
aOA (2) = O( ) A O( ). (5.66)
3 JAOPR

Consequently, the second-order correction must be of O(-). The terms that involve

the first-order correction A4(1) do not contribute to the equation for the second-order

correction, because they involve the O(-L) part of the metric and that results in terms

of0( ).
Finally, the singular electromagnetic potential for an electric dipole moving on

a Schwarzschild geodesic is equal to

At qxi iq xI xxk + 0( ),
p L +(5.67)
A4 3 e qP x'k +X k kP 1 k kPli xxkl] + 0( ).

5.6 Electromagnetic Potential of a Magnetic Dipole

In this section, the singular electromagnetic potential of a magnetic dipole mov-

ing on a geodesic of the Schwarzschild background is calculated. The magnetiza-

tion m' is assumed to point at some random direction and its THZ components are

ma = (0, m", mZ).

The singular electromagnetic potential can be written as

AS AS(o) + AS() + A(2) (5.68)

and obeys the vacuum Einstein equations

(+1+2) A(o+1+2) -47Ja (5.69)

where the source term is

Jo =0,
i m 4 (5.70)
J m p p'(T))] g d.

The zeroth-order term is the electromagnetic potential generated by a magnetic

dipole that is stationary at the origin of a Cartesian coordinate system and is the

solution of the differential equation

VO)As(O) -47J. (5.71)

Its THZ components are

A(o) (0, 't P3 j )3 (5.72)

70

The first-order term obeys the differential equation derived from Maxwell's

Equations (5.69)
72 o)A -V2 A) (5.73)

which indicates that the source term for AS() is the VI) of the zeroth-order part of

the potential. The THZ components of A are

A() = (A( A, A( s(1), A()) (5.74)

and when substituted into Equation (5.73) the result is a set of four second-order

differential equations, one for each one of those four components. Each equation

relates a sum of second derivatives of one component to the source term which consists

of terms of the form:

mB x-x-x-
..5 for the t-component,

p (5.75)
...X 7 for the spatial components,
p7

where the dots denote the appropriate indices.

Solving these differential equations is tedious but not difficult, because each

equation involves only one component of AS(1) and only one of the two tensors S and

B. A careful look at the equations indicates that the solution should be a sum of the

terms
miB X k k mijkX i XjX k
3x -and 3JX (5.76)
p3 p3

for the t-component and a sum of the terms

eJmi Sxkxklxl Emifklxk1 1 kP ?kik Xi1X
P P3 P 3
p3 p(5.77)
el m Xk x jk 1 tijkMi Epj k xl1 ijkmi Xk Xlxp
p3 p3 p3
p p p3?

for the p-spatial component, with appropriate numerical factors in front of each term

so that the equations are satisfied. Using Equations (5.15) and (5.17), the first and

last terms expected to show up in the final expression for the p-component can be

eliminated, since they can be expressed as linear combinations of the remaining four

terms. Substituting these expressions into the differential equations gives a system of

four algebraic equations for those factors. Solving these algebraic equations is trivial.

The result is that the first-order components of the electromagnetic potential are:

A[ iBjk XiX k BijiXjxkXk]
[ 3 (5.78)
1 1 1 .
A() l [-P kl J kX Pijjk ilJkX ijkMiS x]
8(1 p3 2 22

The order of the next correction to the singular electromagnetic potential can

be predicted. It is the solution of the differential equation that has the O(&) part

of the V2 acting on the zeroth-order electromagnetic potential as the source term.

Specifically, it looks like

p 1
aOjAS(2) = 0( ) x ijA A(o) 0( (5.79)

meaning that the Ag(2) correction is of O(-). As in the previously studied cases, the

terms that involve the first-order correction AS(1) do not contribute to this equation,

since they involve the O(-) part of the metric which results in terms of O(h).

Finally, the singular electromagnetic potential for a magnetic dipole moving on

a Schwarzschild geodesic is equal to

A t [mBjkX i XX k 2_ iik k] + O( )
S 3 16 3 R3
A mC x rx P mi kl X3xkx 1 mkil X kl 1 miSpjXkXlXl (
S = +0 + 3. 2 3)

(5.80)

5.7 Gravitational Field of a Spinning Particle

The calculation of the singular gravitational field of a spinning particle mov-

ing on a Schwarzschild geodesic is presented in this section. The particle is as-

sumed to have a small angular momentum pointing at some random direction A

(0, A', AY, A-) in THZ coordinates.

The singular gravitational field can be written as

hs ab = h h() h(2) (5.81)
h Sab a Sab S ab

and obeys the linearized Einstein equations

a2 (+1+2) c dh(O+l+2) -16r Tb (5.82)
(0+1+2) Sab + 2~(0+1+2) ab 'Scd -16 7ab (582)

where

hsab =hsab 9Sgabhs (5.83)

is the trace-reversed version of hsab.

The zeroth-order part of the singular field is the gravitational field generated by

a particle with angular momentum A" that is stationary at the origin of a Cartesian

coordinate system.

For the angular momentum pointing along the z-axis, that is the well-known

Kerr solution with the mass set equal to zero. Since the angular momentum is assumed

to be small and the effects of the mass of the particle are not taken into account,

only the terms of the Kerr metric that are linear in the angular momentum need to

be considered. In the Boyer-Lindquist coordinates (tBL, r, 0, q) around the spinning

particle, hjab (where the superscript Z denotes that this is the part of the zeroth-order

gravitational field coming only from the z-component of the angular momentum) is

equal to
0001
S0 0 00

h4b --A sin2 0 0 0 0 (5.84)
0 0 0 0

1 0 0 0)
1000

Using GRTENSOR, this expression can be easily converted into the equivalent expres-

sion in THZ coordinates

0 y -x 0

h ab 2Ay 0 (5.85)
P -x 0 0 0

0 0 0 0
0000

The relationships between the Boyer-Lindquist and the THZ coordinates used for

this conversion are the usual relationships between the spherical and the Cartesian

coordinates, namely

t =tBL

r = V2 + y2 + z2
r-2--2 (5.86)
0 = arctan (5
z
= arctan .

This is sufficient because the corrections to these relationships that involve the angular

momentum would give terms of higher order in the angular momentum and must be

ignored in this analysis, since only first-order terms in the angular momentum are

kept.

The components of the angular momentum along the x and y axes must be

treated separately, because of the axial symmetry of the Kerr metric. The analyses

74

for the angular momentum being along the x and y axes are very similar to that

for the angular momentum being along the z axis and the only change comes from

the different form that the relationships (5.86) have. Specifically, when the angular

momentum points along the x axis, the axial symmetry is around the x axis and the

relationships used are

t =tBL

r = 2 + y2 +

0 = arctan -V2
x

(5.87)

= arctan -.
y

For the angular momentum pointing along the y axis, the axial symmetry is around

the y axis and the relationships are

t = tBL

r 2 + y2 + z2

a a 2 z
0 arctan -
Y

(5.88)

x
= arctan -.
z

Adding all the contributions that result from this analysis, the zeroth-order singular

gravitational field becomes

0

(-Ayz + Azy)

(A'z Azz)

S(-A"y + Ax)

(-A'z + Azy) (A'z Axz) (-A"y + Ayx)

0 0 0

0 0 0

0 0 0
(5.89)

h(0) 2
S ab 3
p

The first-order correction to this gravitational field obeys the differential equa-

tion derived from Equation (5.82):

72 ) h d() c 2 (o) R2 hC d((O)
o) hS ab+ (0) a b S cd (1)S ab (1) a b S cd (5.90)

so the first-order V2 and the first-order Riemann tensor acting on the zeroth-order

solution give the source for the first-order correction. The first-order correction must

be a symmetric tensor so it is assumed to be equal to

J j(t1) Q1) hil) Q1)
(1) 1) h(1)x h(1)xy h(1)xz

S ab ) () h)(5.91)
"ty xy h2y hj"yz
h(l) h,(l) () h(l)

Substituting it into the first-order equation results in 10 differential equations. There

is one set of four differential equations for the four diagonal components, each equation

containing all four diagonal components. There is also one differential equation for

each one of the t i components and one differential equation for each one of the i j

components, for i / j. In each equation, a sum of second derivatives of components

is related to a sum of terms of the form

A.B x.x x-xx
AB -x-x for the t t and p q components,

(5.92)
for the t p components,
p7

where the dots denote the appropriate indices for each term.

Solving the differential equations in this case is slightly more complicated than

in the previous cases, mainly because of the fact that four of them involve all diagonal

components rather than only one of them. Still, the process becomes significantly

easier if one notices that the t t component must be an appropriate sum of terms

of the form

AiLjkX i xk
p3

A'BijXjXkXk
and A 3j
P3

each t p component must be a sum of terms of the form

EpijAiS xlXkXk EpijA' l xlxk EpijAl~zxZxkxk
P3 P3 P
Epi^1 x Il fk 1icflA ` Xk Ck i31 A 'S3k p l k
p3 P3 P3
c AlgSxixlxk iA lAig~8x'XkXk cijAiS"xPxlxk
p3 P 3 P 3

and each p q spatial component must be a sum of terms of the form

ApBqkXkXlXl Aq,3pkXkXlXl Ap3klXqXkXl Aq,3klXpXkXl
p3 P 3 3 3
Ak3pqX k Xx11 Ak klXpXqXl pgqAiBjkXixJxk 1pqA'BL3ijXXkXk
P3 P3 P3 P3
Ak pkXqxlxl AkBqkXpXlXl AkBplXqX kX AkBqlXpXkXl
p3 p3 p3 p3

(5.93)

(5.94)

(5.95)

with appropriate numerical coefficients in front of each term. Equations (5.15) and

(5.17) can again be used to eliminate the first and last terms in favor of the remaining

four, for the expression for the t -p components. Substituting these sums into the

differential equations gives algebraic equations for the coefficients, which are fairly

easy to solve. The result is that the first-order correction to the gravitational field

has components

SAjBjk xixjxk
-2 j
P3
i [pjAjS xxlxk cpjA'Skxlxk+ 3cj,A SpjXlXkXk

3 [2AiX'X + A'BiXpXqXj + 2A'2 ,,,, A p

3 A(pBq)ix'xx A(pXq)Bijx'x + -q, (Aijki A'B3ijXjXk
(5.96)

h(1)
S tt

h(0)
S tp

h(S)
S pq

The order of the next correction can be predicted. That correction is the solution

of the differential equation

772 h (2) d (2) Y2 h(0) R C da (0)
0) hab + 2 (0) b S cd (2) S ab 2(2) a b S cd (5.97)

where V2) and R(2) b come from the 0( ) part of the metric. So the equations
look like

a, i bjh O( ) x aijh(O)b O(- ) (5.98)

and the second correction to the gravitational field is of O(-). Again, h)b does not

show up in the equations for the second correction, since it relates to the 0( _-) part
of the metric and results in terms of O(R).

Finally, the singular gravitational field due to a spinning particle moving on a

Schwarzschild geodesic is equal to

hS A Bjk i k 0(
p3 Rp
hs=~ -2- c- (pij A
hstp -2 + 1 [pA3 XXlXk EpjA'SXXlXk k +pi cjAjX'1Xk k]

+0()
R3
hspq [2AiB pqxXj + A'I3ijXpXqXj + 2A'B,,. rj AiB3j(pXq)X

3 A(pB)X'XX 3A(pXq)ijX'Xa + q,, (AiL3jkeik A'3ijXrjkk)]
+04().

(5.99)

5.8 Grtensor Code

The GRTENSOR code (running under MAPLE) used to derive the differential
equations for the first order correction to the singular field is given in Appendix A.

Since the case of the gravitational field is the most complicated one, the analysis is

presented for the gravitational field generated by a spinning particle with the angular

momentum pointing along the THZ z-axis. The analyses for the scalar fields and the

electromagnetic potentials are very similar, and can be easily deduced from that for

the gravitational field.

An effort was made to keep the symbols in the code in accordance with the ones

used in this chapter for the various quantities. In the situations where that is not the

case, the comments in the code should make the notation clear enough for the reader

to follow.

The parameter e is used to keep track of the order of each term in the com-

ponents of the tensors S and B and is set equal to 1 at the end. Throughout the

calculation, only first order terms are kept. Specifically, the Christoffel symbols and

the components of the Riemann and Ricci tensors are calculated first and all their

terms that are of order higher than 1 are set equal to zero. Doing that makes the sub-

sequent analysis significantly simpler and the running time of the code significantly

shorter.

After the various quantities associated with the problem are calculated, the

test tensor hbartest(a,b), whose exact dependence on the THZ coordinates is not

specified, is used as a trial solution in the linearized Einstein equations and the source

term coming from the zeroth-order solution h0(a,b) is examined. That helps identify

the terms that should be expected to show up in each component of the solution

hbar(a,b). Specifically, it helps determine which tensor's components, S (denoted as

EE in the code) or B (denoted as BB), should show up in each component of hbar(a,b)

and gives an idea of how they should be contracted to the spatial THZ coordinates

x, y, z. The terms that result from this analysis are multiplied by algebraic factors

and the appropriate sum is substituted into the linearized Einstein equations. The

79

result is a simple system of algebraic equations which can be easily solved to give the

values of the algebraic factors. That completes the solution.

The last component of this analysis is a simple confirmation performed for the

gravitational field that was calculated, that the algebraic coefficients obtained do,

indeed, give the required solution. The confirmation is simply done by replacing the

initially unknown algebraic coefficients with their exact values in the expression for

the solution and substituting that expression into the Einstein equations. Despite the

fact that it was not explicitly mentioned in Sections (5.3)-(5.7), that confirmation was

performed for all singular fields and potentials that were calculated.

CHAPTER 6
REGULARIZATION PARAMETERS FOR THE SCALAR FIELD

A mode-sum regularizarion procedure for the scalar singular field is presented

and implemented in this chapter. This regularization procedure was first proposed by

Barack and Ori in [28], where they described the calculation of the regularization pa-

rameters for the direct part of the self-force on a particle carrying a scalar charge. The

procedure was later implemented by different groups for the calculation of the regular-

ization parameters for the direct part of the self-force on a scalar charge on different

geodesics [28-33] and also for non-geodesic motion [31, 33] around a Schwarzschild

black hole. The calculation of the regularization parameters has also been performed

for the direct part of the electromagnetic self-force [34] and for the direct part of the

gravitational self-force [34-36], for arbitrary geodesics around a Schwarzschild black

hole.

Even though the regularization procedure was initially described for the contri-

bution of the direct part of the scalar field to the self-force, it can be used equally

successfully for the contribution of the singular scalar field to the self-force, as was

demonstrated in [23]. In that paper, the regularization parameters for the self-force

on a scalar charge in circular orbit around a Schwarzschild black hole were calculated

and the self-force results ended up being in excellent agreement with the results that

were derived using the direct scalar field [28-31].

This chapter begins with an outline of the regularization procedure for the scalar

self-force. The description closely follows that given by Barack and Ori [28] for the

direct self-force but is presented here for the singular self-force instead. Then, the

regularization parameters are calculated for the singular scalar field (rather than the

scalar self-force) of a charged particle that moves on an equatorial circular orbit in

a Schwarzschild background and the results of [23] for the scalar self-force are repro-

duced. Finally, the regularization parameters for the first derivative of the singular

part of the self-force are also calculated.

6.1 Regularization Procedure

As was shown in Chapter 4, the self-force on a particle that carries a scalar

charge q can be calculated from the equation

S q lim VR q liImV (, ) (6.1)
p-p' pp'

where p' is the point along the worldline of the charged particle on which the self-force

needs to be calculated and p is a point in the vicinity of p'.

It is assumed that the charge q is moving in a Schwarzschild background of

mass M and the Schwarzschild coordinates are (ts, r, 0, y). For the calculation of the

retarded field, the source term in Poisson's Equation (3.19) can be decomposed in

terms of spherical harmonics and the retarded field can be written as

00 1

1=0 m=-1

Then, the Im-components of '"t can be calculated numerically. That calculation

is discussed in great detail in Chapter 7. Here it is just noted that the important

property of the "';.'s and of their first r-derivatives is that they are finite at the

location of the particle, even though t-t is singular there.

If the spherical harmonic decomposition of the singular field is also considered,

Equation (6.1) becomes

00 1
R= q lim V, 1 .m q lim a- (6.3)
1=0 m=-1 l,m

It is helpful to define the multiple 1-modes of the two contributions to the self-force,

which result after performing the m-summation of each term individually in Equation

(6.3). Specifically

_Flret -q77,, 1.n,
In (6.4)
iFl- qVa 7, Y ,,
in

which gives for the self-force

.-T lim, F t ) (6.5)

In Equation (6.5), the difference in the multiple 1-modes must be taken before the

summation over I is performed.

From this point on, the discussion of the regularization procedure becomes spe-

cific to the problem of the scalar field (q/p), since more detailed results are available

for this case. However, a similar analysis can be done for any other scalar field. The

goal is to find a function hl, such that the series

(ZF hl,) (6.6)

converges. When such a function is found, the self-force can be written as

= (EFt hl,) E, (6.7)

where

Ea0 lim (.Fs hi.). (6.8)
p___1p ,

Because of its definition, the function hi, should be calculated by investigating the

.1- Inil.litic expansion of 17t for large 1. On the other hand, because the self-force is

known to be well-defined, FJt and Fs,~ are expected to have the same large-/ behavior,

so ha, can be determined by the ..i-i.mptotic behavior of FS, instead.

The singular part Fs of the self-force consists of terms of different order in the

limit p --- p' and it has been shown [23] that, in principle, only the first three of those

terms are expected to give non-zero contributions, for the field (q/p). However, for

reasons that will become clear shortly, the next order terms are included and

S= S(A) S(B) S(C) (D) + ) (6.9)

The superscripts A, B, C, D indicate the different orders, A coming from the most

dominant term, B from the next more dominant and so on. The superscript E

refers to all terms of order higher than that of F (D). The mode-sum regularization

procedure amounts to performing the spherical harmonic decomposition of each such

term, which results in an expression of the form

.-S \ A n ln + lm i lm lrn\1
__ = (A\$ +B~ + nl + D + Em)Y (6.10)
I,m

with A'" corresponding to F. A), etc. For the simple case of a scalar charge moving

in a Schwarzschild background and generating the scalar field (q/p), the parameters

A B C~m and DI" have been shown [23] to have a very simple form so that, when
the explicit expression for the spherical harmonics Yj, is substituted into Equation

(6.10), the summation over m can be performed and the result is an expression of the

form

1 1 Da E+4)] (
= [A, ( + + B + Ca E,7,((1-27 1)+
2 + (+l- (21- 1) (21+ 3)

where the regularization parameters Aa, Ba, Ca, Da and Ea are 1-independent quan-

tities which do depend on the background geometry and the characteristics of the

orbit.

Finally, the self-force can be calculated by

10
.FR [ lim Tl
= (6.12)
1 1 D_
Aa(1 + ) B Ca Ea
2 (1 + 1) (21- 1)(21 + 3)

One important point that should be made is that the infinite sum over 1 must be

performed, in order for the self-force to be calculated. Notice, however, that the

contributions for large I get less significant for the terms containing Da and Ea. The

reason for including these terms is now clear. Even though the sum over I of each of

these two terms is exactly equal to 0, including these terms improves the convergence

of the sum. An additional benefit of including these terms is that the approximation

to F R becomes more differentiable, as is explained in [23].

6.2 Order Calculation of the Scalar Field

It should be obvious from the analysis of Section (6.1) that, in order to calculate

the regularization parameters for the singular field generated by a charge q, it is

necessary to have an expression of (1/p) in which the order of each term is known.

The derivation of such an expression is presented in this section for an equatorial

circular orbit of radius ro in the Schwarzschild background. It is noted that the

results were derived using MAPLE extensively. It is also noted that the derivation was

presented in [23] where the results of it for the radial derivative ,( () were given. For

simplicity, the scalar charge q is set to 1.

In order for the calculation of the self-force regularization parameters to be made

easier, the Schwarzschild coordinates can be rotated, as explained in [30]. Specifically,

new angles 6 and D can be defined in terms of the usual Schwarzschild angles by the

equations

sin 0 cos( ts) cosO (6.13)

sin 0 sin(q ts) = sin cos D (6.14)

cos0 = sin sin D (6.15)

so that the coordinate location of the particle is moved from the equatorial plane,

where 0 = to a location where sin = 0, for a specific ts. Such a coordinate

rotation preserves the index 1 of any spherical harmonic Ym,(O, q). That means that

any Yim(0, 9) is mapped into a linear combination of spherical harmonics Ym( (b),

where m' = -1,...,1. Consequently, each 1-multipole mode of the field or the self-

force that results after summation over m is the same, regardless of which angles,

(0, q) or (6, D), are used for calculating it.

The benefit of this coordinate rotation for the calculation of the regularization

parameters for the singular part of the self-force can be understood if one remembers

that in the limit p -- p' the angle 6 is equal to 0. That means that Yi(0, D) has to

be used, for which

Ym(0, ) 0, for m / 0 (6.16)

m(0,) = ,for m 0. (6.17)

So the sum over m can, after this coordinate transformation, be replaced with just

the m = 0 term. However, for the regularization parameters of the singular field,

the limit p -- p' must not be taken, so the Yim's for all m's must be taken into

consideration. The m = 0 spherical harmonic must be considered only when the

regularization parameters for the self-force are derived from the ones for the scalar

field and when the regularization parameters for the first r-derivative of the self-force

are calculated.

A comment on the order of each term needs to be made at this point. The

parameter is used to indicate a term of order x" in the coincidence limit p p'.

In that limit, r or and 6 0. That means that the factor (r ro) is of order e

and the factor (1 cos 6) is of order e2. At the end of the calculation, the parameter

e can be set equal to 1.

The relationships between the Schwarzschild coordinates (ts, r, 0, q) and the THZ

coordinates (t, x, y, z) for a circular orbit on the equatorial plane of a Schwarzschild

background that were given in Section (5.1) are used for this calculation. As was

mentioned earlier, p2 = 2 y2 + z2 in terms of the spatial THZ coordinates x, y and

z. However, it is clear from Equation (5.26) that the relationship

x2 2 2 + 2 (6.18)

holds between {x, y} and {x, y}, so the sum (2 + y2) is used to calculate p2. The

Equations (5.22), (5.23) and (5.24) are substituted into the expression for p2, with the

Schwarzschild angles 0 and ) replaced by the new angles 6 and D. That substitution

gives that the lowest order term for p2, denoted by p2, is

2 roA2 + o 2o- 2M
o 2 2r2 3 cos (), (6.19)

where

A r ro (6.20)

and
M
x 1 sin2 I. (6.21)
ro 2M

Clearly, p2 is of order e2 in the coincidence limit, as should have been expected. Next
the variables 0, D and r are eliminated from the order expression of p2 in favor of
the variables A, p and X, by using Equations (6.19), (6.20) and (6.21). Finally, the
result is inverted and the square root is taken, in order to obtain the order expression

of (1/p).
The result of this calculation is a very long expression. Here, I only give the
terms of this expression that are necessary to calculate the regularization parame-
ters of the singular field, the singular self-force along the radial direction and the
r-derivative of the singular self-force along the radial direction. Exactly how that is
determined will become clear shortly, when the general term A"~'-q, where a, p
and q stand for integers, will be discussed.
The terms of interest are

1 _1 +
P P
+o\ r0o 3M 1 1 A [ 2r 3M r, 3M 1] A3
L2ro(ro 2M) ro7 + 2r,(r, 2M) 2r,(r, 2M) M
if r, 3M 1 ro M 1
r (ro- 2M) x ro X2]
2ro -3M 5r -22r +21M2 1 5r 22r .+ 21M2 1 A2
+ 2r(ro 2) 4(ro 2M)2 8ro(r 2M)2 X
A4 A6
+o( ,

+ [_ M(r 2M) (ro M)(ro 4M) 1
2r4 (ro 3M) 8(r 2M)r4 x
(r 3M)(5r 7r .[ 14M2) 1 3(r + M)(r 3M)2 1
16r(ro 2M)2 X 16r2(ro 2M)2 x X3
(A3 A5 A7 A9 }
+0( 7, '

+ O(3).

(6.22)

6.3 Scalar Monopole Field

As is clear from the previous calculation of (1/p), the angular dependence of

every term shows up in factors of the form p"f-q, where p is an odd integer and q =

0,1, 2,.... The spherical harmonic decomposition of that factor, which is necessary

when calculating the regularization parameters for the scalar field, is given in detail

in Appendix B. The result is

2r 2(r 2MA)p/2 ] q M
x-'" ro : EP.2 )Y e,) (6.23)
l 0 n 1 rm=- -

where
2 (r 3(6.24)
2r0(r 2M)2

and the coefficients E'72, ,M) are given in Equation (B.41). It is stressed that

all the r-dependence of pP-9 resides in the sum Z:o m2 and the term (72)i+n+l, in

Equation (B.41) for E'f, and is always proportional to powers of A (r ro).

At this point, a note on the term (72)2 for an odd integer /3 is in order. For

that term, the square root of 72 must be considered. One might think that

1 1
r, 3M 2 r 3M 2
(y2)i o 2 o_. (6.25)
(2r2) oo 2M)2 2roo 2M)2 (.

But that implies that being on the equatorial plane with O 0 and approaching the

particle by taking the limit r ro, would give for the leading term p in the expansion

of p:

2M 1 172

2ro 3M (6.26)

ro 2- A.
ro 2M

According to this, the sign of the leading order term of p could be either positive or

negative, depending on whether r > ro or r < ro. That is clearly not correct, since by

definition p = 2+ y2 z2, which is always positive. For that reason, taking the

square root of 72 always implies

r 3M 2
2 A. (6.27)
[2r0(r0 2M)2 j

For the self-force along the radial direction to be calculated, the r-derivative of

(1/p) and the limit r -- ro have to be taken. That means that any term of order

(r ro)2 or higher gives, after the limit is taken, no contribution to the self-force.

However, as has already been mentioned, it is desired to calculate the first derivative

with respect to r of the self-force, which is the second derivative with respect to r

of the field, in the limit r -- ro. Consequently, terms of order (r ro)2 have to be

retained, because they do give a contribution at that limit, while any terms of order

(r ro) or higher can be disregarded.

As is clear from Equation (B.41) of Appendix B, the general term AOPX in

Equation (6.22) has two pieces that contain the r-dependence. The first piece comes

from the sum over s in Ef', and gives terms proportional to -a+2s. Such a term should

be kept only for a + 2s < or
5 a
< (6.28)
4 2

Since s > 0, the a+2s contribution can be immediately disregarded for terms with

a > while for terms with a < | a limited number of values of s have to be
2' 2
retained in the sum. The second piece that contains r-dependence comes from the

term (_2) +'+1 and is proportional to 1p+a+2n+2. Such a term should be kept only

forp + a n 2 + < or

n< a (6.29)
4 2

Since n > 0, all terms of Equation (6.22) for which p + a > 1 do not contribute

through the 7p+a+2n+2 piece. Combinations of p and a for which p + a < should be

examined individually and a specific number of n's must be retained. The terms of the

expansion (6.22) for (1/p) for which just the order (and not the explicit expression)

is given are terms that fall in both categories that according to this analysis can be

ignored.

I present now the calculation of the regularization parameters coming from the

terms of different order, for the singular field (1/p). The abbreviation used for the

hypergeometric function is

FA 2F1(,A++1; ), (6.30)
2' 2 2M

and it is also noted that the hypergeometric function

1 M
F,' 2F1 (, 2 1; ) (6.31)
2' r, 2M

is denoted by Fa, as is done in [23]. Also, for the three sums over k, v and A the

abbreviation

( ) 21+1 F(l ( 1 l -|m|+1)
( 2 [( -m+1)F(l +m+ 1)]+

F r(1-k + 2 r(I + 1)
S(- 2 -F(k +i)F(-2k- Iml + ) F(+ 1)F( + 1)
Iml
I(1 )A2
F(A +1)F( -A+1)
(6.32)

that is used in appendix B is also used here. In addition, the upper limit of the

n-summation is denoted by

N = I m 2k+ 2v.

(6.33)

For the lowest order term of Equation (6.22)

-11
T(-1) C -

(6.34)

the exponents of A" X- q are a 0, p

1, q 0. Equations (6.28) and (6.29) give

that only the s = 0 and s = 1 terms of the first piece and only the n

0 term of the

second piece of E ,'q need to be kept. Consequently

(6.35)

where, using Equation (6.23)

Am 2\ ro 2 1M) ,
A2, (r ,m
1 2r0 o 2M) I o

2ro 2M)

(6.36)

S( 1)1+22n+i F(N ) [ ?
n (2n +)!! F(N n + 1/2
n-=0

(n FA,m 2]
(T )3/2 J

or, when the explicit expression for 72 and its square root are substituted

r -f 4 r- 13Ma3r
m(ro 2 AM) kvA 2ro( 2M)2

S(-1) +n22n+i F(N ) [,
O (2n + 1)!! F(N -n 1) L (
n-=0

IA Tr03o 3M Y
2 3/2 2ro(ro 2 M)2 j

(6.37)

T(1_) --1 Y AYl (', ()

2(q)2 F1A

(Y){
kfeA '

As is explained in Appendix B, only the even m's should be included in the sum of

Equation (6.35).
The regularization parameter A, given in [23] for the self-force can be easily

obtained from this result. First the m = 0 parameter is considered, for reasons that

were explained earlier. That makes the sums over A and over v equivalent to the

terms with A = 0 and v = 0. For the hypergeometric function F1' it can easily be

proven that

F '0- F (6.38)

using Equations 15.3.3 of [37]. Then the first derivative of Am with respect to r is

taken. That makes the term proportional to F1 2 in Equation (6.37) vanish, while the

term proportional to F, gives a factor of A which vanishes when the coincidence

limit is taken. For the first term the derivative of |A| gives sgn(A). Finally, the

spherical harmonic Yo(0, 1) is substituted with 21. This procedure gives

2 [o( 3d)]d 1 1
-2(-sgn(A)) T -r 2M1) (+ 2) (6.39)

-12 -2k F( k +)
k-0 F(k + 1)F( 2k + 1)

The sum over k can be easily calculated for any value of 1, using MAPLE. A general

proof that it is equal to rT for any 1 cannot be given. However, for every value of 1

that was tried, the sum ended up being /7, which gives

S[ro(r 3M)]i 1
T_) pp' -2( sgn(A)) o(o ) (6.40)
Sr(ro- 2M) ( + 2)

which is the result of [23].

Now, the second derivative of the term T(_1) with respect to r is taken, which

is equal to the contribution of that term to the first derivative of the self-force. The

m = 0 component is considered since the coincidence limit must be taken, so A = 0

and v 0 as well. Only the term proportional to F, 2 survives the differentiation,

since it is proportional to A2. The result is

02T(-1)
0r2
p->p'

-3 d2Alo 21+ 1
dr2 47

0 0
(ro 3M) 2 1 1
e F3 / 2

>2 ( k F(1 k + 1) 1-2k 22(n )
0 F(k+ 1) F(I 2k -n 1)(2n +1)!
(6.41)

The zeroth-order term contribution that is considered in Equation (6.22) is

SC[ ro 3M 1
T(o) L2ro(ro 2M) X

S2rr3M ro 3M ] A3 (6.42)
ro p 2ro(ro 2M) 2ro(ro 2M) 3- (.2

For the piece that is proportional to Ap-1 the exponents are a =1, p -1, q = 1 for

the term that contains X and a =1, p = -1, q = 0 for the term that does not contain

X. In both cases, Equations (6.28) and (6.29) indicate that the s = 0 term of the first

piece and the n = 0 term of the second piece of ET1,q should be kept. For the piece

that is proportional to A3"-3 the exponents are a = 3,p = -3, = 0 for the term

that does not contain X and a = 3, p = -3, = 1 for the term that does contain X.

In this case, Equations (6.28) and (6.29) give that none of the s-terms and only the

n =0 term of the second piece of E-3,q need to be kept. That gives

(6.43)

T(o) coy BimYm(,, m)
imT

where, from Equation (6.23)

B r-, 1F rM 3 ,
m A 2r2(r, 2M) 2r,(r, 2M) ^i

[2r (ro.- 2M)]

[ 2ro 3,0
2r(ro 2M) EIz

SE,
7,0 ~

ro 3M E3,11
2r(r 2M) 'm 1

Substituing the explicit expressions of the coefficients Ef"' into Equation (6.44) for

Bim results in

R ro 3M
2r(ro k,v,A

A = (-1)+n22n+ 2 (N +1) roo -3M
S0 (2n + 1)!! (N n 1) (r 3/2

vA (ro- 3M) ro- 3M F,,.
r (ro 2M) 2ro(ro 2M)

A3 (ro ,-3M) r2ro 3M fA
IA /2r (ro 2M)L ro -3M

Fo 1

(6.45)

F0

It is easy to recognize that

A3
^1 -

(6.46)

so the last two terms in the last expression for BRi can be combined to give a signifi-

cantly simpler expression, namely

Sro- 3M
2rB (r 2M) )

A : (-_1)i+ 22n+ F(N ) 3M Am
-=0 (2n +l)!! F(N- n +1) 2ro(ro-2M) 3/2

1
AA (ro 3M)
2r (ro 2M)

2(r 3M)(ro M) (4r
(ro 2M)

AI rAm+
1ia
F0
(o 1/2 )

o 3M)F1

(6.47)

(6.44)