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the taub - nut ( newman , unti and tamburino ) space - time @xcite is a stationary , spherically symmetric and non - asymptotically flat solution of the vacuum einstein equation in general theory of relativity . the space - time has topology @xmath0 with lorentzian signature @xcite . the nut space - time is described by two parameters : one is the mass parameter @xmath1 and another one is the nut parameter @xmath2 . there is no modification required in the einstein - hilbert action to accommodate the nut charge @xcite or `` dual mass '' @xcite or `` gravito - magnetic mass '' or `` gravito - magnetic monopole '' @xcite . this dual mass is an intrinsic feature of general theory of relativity . the space - time contains closed time - like curve and null lines . it is a geodetically incomplete space - time @xcite . bonor @xcite has given a new interpretation of the nut spacetime and it describes ` the field of a spherically symmetric mass together with a semi - infinite massless source of angular momentum along the axis of symmetry ' . on the other hand , manko and ruiz @xcite analyzed the mass and angular momentum distributions in case of generalized nut spacetime using komar integral approach . t hooft and polykov @xcite have demonstrated that the magnetic monopole present in certain non - abelian gauge theories . zee @xcite observed that there is an existence of a gravitational analog of dirac s magnetic monopole @xcite . the author is also discussed regarding the mass quantization following the idea of dirac quantization rule . he also claimed that there is certainly no experimental evidence of mass quantization . moreover , he proposed that if mass is quantized there may have profound consequences in physics . for example , if a magnetic monopole moving around a nucleus then the mass quantization rule suggests that the binding energy of every level in the nucleus is also quantized . friedman and sorkin @xcite observed that the gravito - pole may exist in topological solution . dowker @xcite proposed that the nut spacetime as a ` gravitational dyon ' .    the euclidean version of the space - time is closely related to the dynamics of bps ( bogomolnyi - prasad - sommerfield ) monopoles @xcite . the experimental evidence of this dual mass has not been verified till now . there may be a possibilities of experimental evidences in near future and it was first proposed by lynden - bell and nouri - zonoz @xcite in 1998 . letelier and vieira @xcite have observed that the manifestation of chaos for test particles moving in a tn space - time perturbed by dipolar halo using poincare sections . the geodesics structure in euclidean tn space - time has been studied in the ref . @xcite . the gravito - magnetic lensing effect in nut space - time was first studied by nouri - zonoz et al . @xcite in 1997 . they proved that all the geodesics in nut spacetime confined to a cone with the opening angle @xmath3 defined by @xmath4 where @xmath5 is the impact factor . ) for null circular geodesics . ] for small @xmath6 and in the limit @xmath7 , it should be @xmath8 it should also be noted that the opening angle is proportioal to the nut parameter @xmath2 . furthermore , they also examined the lensing of light rays passing through the nut deflector . this properties modified the observed shape , size and orientation of a source . it has been also studied there that there is an extra shear due to the presence of the gravito - magnetic monopole , which changes the shape of the source . the same author also studied the electromagnetic waves in nut space through the solutions of the maxwell equations via newman - penrose null tetrad formalism to further deeper insight of the physical aspects of the dual mass . since the tn space - time has gravito - magnetic monopole that changes the structure of the accretion disk and might offer novel observational prospects @xcite .    the maximal analytic extension or kruskal like extension of the tn space - time shows that it has some unusual properties @xcite . maximal analytic extension is needed in order to understand the global properties of the space - time . misner and taub have shown that tn space is maximally analytic i.e. it has no hausdorff extension @xcite . whereas hajicek @xcite showed that the non - hausdorff property occurs only on the killing horizons and causes no geodesics to bifurcate . chakraborty and majumdar @xcite have derived the exact lense - thirrring precession ( inertial frame dragging effect ) in case of the tn space - time in comparison with the mass - less tn space - time . the _ mass - less dual mass _ ( i.e. tn space - time with @xmath9 ) concept was first introduced by ramaswamy and sen @xcite . they also proved that ` in the absence of gravitational radiation magnetic mass requires either that the metric be singular on a two dimensional world sheet or the space - time contain closed time - like lines , violating causality ' . after that ashtekar and sen @xcite demonstrated that the consequences of magnetic mass in quantum gravity . they also proved that the dual mass implies the existence of ` wire singularities ' in certain potentials for weyl curvature . finally , mueller and perry @xcite showed that the ` mass quantization ' rule regarding the nut space - time .    in @xcite , the author has been studied @xmath10 time - dependent tensorial perturbations of lorentzian tn space - time and proved that lorentzian tn space - time is unstable . geodesics of accelerated circular orbits on the equatorial plane has been studied in detail of the nut space using frenet - serret procedure @xcite . however , in the present work we wish to investigate the complete geodesic structure of the tn space - time in the equatorial plane . we compare the circular geodesics in the tn space - time with mass - less tn space - time and zero nut parameter by _ analyzing the effective potential graphically for both null cases and time - like cases_. the presence of the dual mass can changes the geodesic structure in comparison with the mass - less dual mass and zero dual mass . this is clearly manifested in the effective potential diagram . we also differentiate _ graphically _ the isco , mbco and cpo of the said space - time in comparison with the mass - less cases . moreover , we examine the circular geodesics in the @xmath11 plane for different values of energy i.e. @xmath12 , @xmath13 and @xmath14 in case of tn and mass - less tn spacetime , and plotted graphically . furthermore , we have studied more exotic cases i.e. extreme cases , where we find surprising results that the radii of three important class of orbits namely , the radius of isco ( @xmath15 ) , the radius of mbco ( @xmath16 ) and the the radius of cpo ( @xmath17 ) are coincident with the horizon . from the best of my knowledge , this is the first we have observed in this work for any spherically symmetric , stationary and _ non - asymptotically _ flat spacetime showing such feature . finally , we compute the cm energy for these spacetimes and we have found that for non - extreme tn bh , the cm energy is finite whereas for _ extreme _ tn bh the cm energy is diverging in nature . the circular geodesics studied earlier by several authors @xcite but they have not been studied in more graphically . we here show the differences between the two spacetime with the mass parameter and the mass - less parameter in visually . it may be noted that circular orbits of arbitrary radii are not possible , there exists a minimum radius below which no circular orbits are possible . the geodesic structure has been studied earlier for schwarzschild bh @xcite , reissner nordstrm ( rn ) bh @xcite , kerr - taub - nut ( ktn ) bh @xcite and kerr - newman - taub - nut ( kntn ) more recently @xcite . in this work , we have specialized on the cases when the parameter @xmath18 . by studying the geodesic structure , we can extract more information about the back ground space - time . different observables like lense - thirrring effect , gravitational time delay , gravitational bending of light etc . all are the phenomenon related to the geodesic structure of the space - time . this is one of the major motivation behind to study them . the structure of the manuscript is as follows . in section [ nut ] , we discuss the basics of tn bh . in section [ egtn ] , we study the equatorial geodesic properties of the said bh . section [ ex ] devoted to study the extreme tn bh . the conclusions are given in the section [ dis ] . the metric is given by @xcite @xmath19 where , @xmath1 denotes the gravito - electric mass or adm mass and @xmath2 denotes the gravito - magnetic mass or dual mass or magnetic mass of the space - time . it is clearly evident that there are two types of singularities are present in the metric ( [ tn1 ] ) . one is at @xmath20 which give us the killing horizons or bh horizons : @xmath21 @xmath22 is called event horizon and @xmath23 is called cauchy horizon .    from fig . [ eq1 ] , we can see the horizon structure of tn and mass - less tn bh . there is a qualitative difference between two horizon structure with nut parameter and without nut parameter . , title="fig:",scaledwidth=45.0% ] , title="fig:",scaledwidth=45.0% ]    in the limit @xmath24 , one obtains the schwarzschild bh . interestingly , when @xmath9 , we get mass - less tn space - time and the horizons at @xmath25 the other type of singularity occurs at @xmath26 and @xmath27 , where the determinant of the metric component vanishes . misner@xcite first demonstrated that in order to remove the apparent singularities at @xmath26 and @xmath27 , @xmath28 must be identified modulo @xmath29 . provided that @xmath30 . it should be noticed here that the nut parameter actually measures deviation from the asymptotic flatness at infinity which may be manifested in the off - diagonal components of the metric and this is happening due to presence of the dirac - misner type of singularity . when , @xmath31 the tn metric describes a bh , otherwise it has a naked singularity . when @xmath32 , we find extreme tn bh . the characteristics of the variation of @xmath33 with radial coordinate is shown in fig . [ eq2 ] . with @xmath34 for tn and massless tn bh . [ eq2],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn and massless tn bh . [ eq2],title="fig:",scaledwidth=45.0% ]    the `` red - shift factor '' ( @xmath35 ) @xcite for tn bh is given by @xmath36 and for mass - less tn bh , it is @xmath37 the variation of redshift factor with radial coordinate is shown in fig . [ eqrs ] .     for tn and massless tn bh . [ eqrs],title="fig:",scaledwidth=45.0% ]   for tn and massless tn bh . [ eqrs],title="fig:",scaledwidth=45.0% ]    the `` red - shift''(@xmath38 ) @xcite is given by @xmath39 for mass - less tn bh , it is @xmath40 the variation of redshift with radial coordinate is depicted in fig . [ eqz ] .     for tn and massless tn bh . [ eqz],title="fig:",scaledwidth=45.0% ]   for tn and massless tn bh . [ eqz],title="fig:",scaledwidth=45.0% ]    the thermodynamic properties of tn bh could be found in detail in @xcite and the bh temperature of @xmath41 denotes event horizon of the bh . ] was calculated there @xmath42 in the limit @xmath43 , @xmath44 , this indicates that there must exists _ extreme _ tn bh . the geodesic properties have been studied in sec . since the equatorial plane is the good location where we can see the causal characteristics of the geodesics so in this section we would like to study them . it should be noted that @xmath45 , a constant is called circular geodesics . also , the study of geodesics in the tn space - time is governed by the laws of conservation of energy and angular momentum because the space - time is independent of the coordinates @xmath28 and @xmath46 . the tn space - time possesses time - like isometry generated by the time - like killing vector @xmath47 whose projection along the four velocity @xmath48 of geodesics : @xmath49 , is conserved along such geodesics and the another conserved quantity is angular momentum followed by the relation @xmath50 ( where @xmath51 ) . where @xmath52 is the space - like killing vector field due to the rotational isometry .    using these conditions together with the normalization of the four velocity , one can easily derived the radial equation for tn bh on the @xmath53 plane : @xmath54 where the effective potential @xcite is @xmath55 here , @xmath56 for time - like geodesics , @xmath57 for light - like geodesics and @xmath58 for space - like geodesics .      for light rays orbit , the effective potential becomes @xmath59 first we see the behaviour of the test particle in the potential well diagram . in fig . [ nu1 ] , fig . [ nu2 ] , fig . [ nu3 ] , fig . [ nu4 ] , fig . [ nu5 ] , fig . [ nu6 ] , fig . [ nu7 ] and fig . [ nu8 ] , we show how the effective potential for photon changes for different values of angular momentum parameter and nut parameter . from the effective potential diagram , it has been observed that the presence of _ dual mass _ parameter effectively changes the shape of the potential well in comparison with absence of the _ dual mass _ parameter . the structure of the potential well also changes in the presence of adm mass parameter and in the absence of adm mass parameter . with @xmath34 for tn bh and mass - less tn bh . [ nu1],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ nu1],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh for different values of @xmath2 . [ nu2],title="fig:",scaledwidth=45.0% ]    with @xmath34 for tn bh and mass - less tn bh for different values of @xmath2 . [ nu2],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . [ nu3],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ nu3],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . [ nu4],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ nu4],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . [ nu5],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ nu5],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . [ nu6],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ nu6],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh [ nu7],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh [ nu7],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . [ nu8],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ nu8],title="fig:",scaledwidth=45.0% ]    now by introducing the impact parameter @xmath5 , one can reparametrization of any null geodesics described by the parameter @xmath60 and @xmath61 . @xmath62 be the angular momentum of null geodesics when it is reparametrized to have unit energy . now the plot of @xmath62 with @xmath34 gives the information of radial motion for null geodesics . the equations evaluating the radius @xmath63 of the unstable circular photon orbit at @xmath64 and @xmath65 by introducing the impact parameter @xmath66 are @xmath67d_{c}^2 & = & 0 ~.\label{n4}\end{aligned}\ ] ] from eq . ( [ n4 ] ) , we find @xmath68 the behaviour of the impact parameter can be shown from the fig . [ nu10 ] . with @xmath63 for tn bh and mass - less tn bh . [ nu10],title="fig:",scaledwidth=45.0% ]   with @xmath63 for tn bh and mass - less tn bh . [ nu10],title="fig:",scaledwidth=45.0% ]    putting the eq . ( [ n5 ] ) in eq . ( [ n3 ] ) , we obtain the equation of cpo @xcite : @xmath69 for mass - less tn bh @xcite , the root of the eq . becomes @xmath70 the variation of cpo with @xmath63 for tn bh and mass - less tn bh could be found in the fig . [ nu10 ] .    . for mass - less cases , @xmath71 . for @xmath24 this root has the value @xmath72 . [ nu11],title="fig:",scaledwidth=45.0% ] . for mass - less cases , @xmath71 . for @xmath24 this root has the value @xmath72 . [ nu11],title="fig:",scaledwidth=45.0% ]    the gravitational bending of light analyzed in @xcite and the bending angle on the cone is computed in @xcite @xmath73 . in terms of opening angle it should be @xmath74 . for _ massless tn spacetime , this angle reduces to zero value_.      for time - like geodesic we have to set @xmath56 then the effective potential becomes @xmath75 the qualitative behaviour of the test particle can be obtain by studying this potential . first we consider the zero angular momentum geodesics for this the effective potential reduces to @xmath76 one can observe the qualitative behaviour of the geodesics in the presence of dual mass and with out dual mass and also the mass - lees parameter . this can be seen from the fig . [ t1 ] . with @xmath34 for tn bh and mass - less tn bh . [ t1],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ t1],title="fig:",scaledwidth=45.0% ]    in this plot , we can see that the presence of the dual mass parameter deforms the shape of the radial effective potential in comparison with zero dual mass parameter . this behaviour can be seen from the mass - less case also . for @xmath77 , the behaviour of the test particle in the potential well could be seen from the following fig . [ t2 ] , fig . [ t3 ] , fig . [ t4 ] , fig . [ t5 ] , fig . [ t6 ] , fig . [ t7 ] , fig . [ t8 ] and fig . [ t9 ] . with @xmath34 for tn bh and mass - less tn bh . [ t2],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ t2],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . nut parameter . [ t3],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . nut parameter . [ t3],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . [ t4],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ t4],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . [ t5],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ t5],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . [ t6],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ t6],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . [ t7],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ t7],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . [ t8],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ t8],title="fig:",scaledwidth=45.0% ]     with @xmath34 for tn bh and mass - less tn bh . [ t9],title="fig:",scaledwidth=45.0% ]   with @xmath34 for tn bh and mass - less tn bh . [ t9],title="fig:",scaledwidth=45.0% ]    in the above figures , we have seen that the radial dependency of the effective potential with dual mass parameter , without dual mass parameter and @xmath9 . when @xmath24 , the effective potential at large radial distance does not change much more with the increasing of @xmath60 . when we introduced the nut parameter , the shape of the effective potential _ deforms _ in comparison with nut less case and it also changes for different values of angular momentum parameter . finally , when we increase the value of @xmath2 the height of the potential well decreases . this work has been done earlier for the parameter @xmath78 , @xmath79 and @xmath2 @xcite . this work is specially for the parameter values @xmath18 and for mass - less cases which has been not studied previously . now re - write the eq . ( [ tnr ] ) for time - like geodesics as @xmath80 @xmath81 for circular geodesics @xmath45 , we have the following condition : @xmath82 from this condition , we find the energy and angular momentum for circular orbit : @xmath83 and @xmath84 these equations require for energy square and angular momentum square positive definite i.e. @xmath85 . compared with the eq . ( [ n6 ] ) , it implies that the minimum radius for time - like circular orbit is the radius of of the unstable cpo . interestingly for mass - less case , these values are @xmath86 and @xmath87 we have plotted their behaviour in the fig . [ ff ] and fig . [ ff1 ] .     with @xmath88 for tn bh and mass - less tn bh . [ ff],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ ff],title="fig:",scaledwidth=45.0% ]    and     with @xmath88 for tn bh and mass - less tn bh . [ ff1],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ ff1],title="fig:",scaledwidth=45.0% ]    from eq . ( [ n10 ] ) , and for circular orbit one obtains @xmath89 it should be noted that @xmath90 . from this expression , it follows that the angular momentum parameter explicitly dependes upon the energy value . therefore there must be difference in the circular orbits in the @xmath11 plane for @xmath12 , @xmath13 and @xmath14 . we are interested here to look the behaviour of the circular geodesics in this plane by incorporating these energy conditions .    when @xmath13 , we obtain the bound orbits , this can displayed from the @xmath91 diagram . it has been shown in the fig . [ t10 ] . with @xmath88 for tn bh and mass - less tn bh . [ t10],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ t10],title="fig:",scaledwidth=45.0% ]    for @xmath14 , we find the marginally escape orbits . this has been seen from the fig . [ t11 ] . with @xmath88 for tn bh and mass - less tn bh . [ t11],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ t11],title="fig:",scaledwidth=45.0% ]    finally for @xmath12 , we find the unbound orbits . this could be seen from the fig . [ t12 ] , fig . [ t13 ] and fig . [ t14 ] . with @xmath88 for tn bh and mass - less tn bh . [ t12],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ t12],title="fig:",scaledwidth=45.0% ]     with @xmath88 for tn bh and mass - less tn bh . [ t13],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ t13],title="fig:",scaledwidth=45.0% ]     with @xmath88 for tn bh and mass - less tn bh . [ t14],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ t14],title="fig:",scaledwidth=45.0% ]    from fig . [ t15 ] , fig . [ t16 ] , fig . [ t17 ] , fig . [ t18 ] and fig . [ t19 ] , one can observe that the variation of @xmath92 with respect to @xmath88 for various values of nut parameter in the presence of mass parameter and without mass parameter . now we see the behaviour of the @xmath60 with @xmath34 for different values of energy for a fixed value of the dual mass parameter . with @xmath88 for tn bh and mass - less tn bh . [ t15],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ t15],title="fig:",scaledwidth=45.0% ]     with @xmath88 for tn bh and mass - less tn bh . [ t16],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ t16],title="fig:",scaledwidth=45.0% ]     with @xmath88 for tn bh and mass - less tn bh . [ t17],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ t17],title="fig:",scaledwidth=45.0% ]     with @xmath88 for tn bh and mass - less tn bh . [ t18],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ t18],title="fig:",scaledwidth=45.0% ]     with @xmath88 for tn bh and mass - less tn bh . [ t19],title="fig:",scaledwidth=45.0% ]   with @xmath88 for tn bh and mass - less tn bh . [ t19],title="fig:",scaledwidth=45.0% ]    now the isco radius can be obtain by using the condition eq . ( [ ef ] ) with additional condition : @xmath93 although it has been derived earlier in @xcite . we are interested here to see the behaviour of isco in the presence of nut parameter and with out nut parameter in comparison with mass - less cases in more graphically which has been not shown there . therefore the isco equation for tn bh could be obtain by applying the condition given in eq . ( [ ef ] ) and eq . ( [ n15 ] ) ( or by putting @xmath18 in eq . ( 81 ) in @xcite ) : @xmath94 @xmath95 and for the mass - less tn bh , it is @xmath96 and it also indicates there is _ no isco for massless tn bh_. it is a curious result although there exists mass - less cpo . the variation of isco in the presence of the nut parameter and without nut parameter could be found in the fig . [ t20 ] .     in the presence of nut parameter and without nut parameter . where @xmath97 . [ t20],title="fig:",scaledwidth=45.0% ]      another interesting orbit that has not been considered previously in @xcite but considered in @xcite for kntn so we have just set @xmath98 in eq . ( [ ef1 ] ) ( or putting the parameters @xmath18 in eq . ( 98 ) in @xcite ) , we find the mbco for tn bh : @xmath99 @xmath100 and for the mass - less tn bh , it is @xmath101 it implies that there is no existence of mbco for mass - less case . from eq . ( [ n18 ] ) , in the limit @xmath24 , we get the mbco for schwarzschild bh @xcite . in fig . [ t21 ] , we have plotted the mbco with @xmath88 in comparison with schwarzschild bh .    . [ t21],title="fig:",scaledwidth=45.0% ] in the previous section , we have discussed the complete geodesic structure for non - extreme tn bh . moreover , we have clearly explained the difference of geodesic structure between tn spacetime and mass - lees tn spacetime in graphically . in the present section we shall discuss more interesting case i.e. _ extreme tn _ spacetime . what happens the geodesic structure in the extreme limit i.e. @xmath43 . this is the main aim in this section . proceeding similarly , we can write the effective potential for massive particles in the extreme limit [ using eq . ( [ n8 ] ) ] given by @xmath102 the qualitative behaviour of the test particle may be seen from the effective potential diagram ( see fig . [ t22 ] ) . with @xmath34 for extreme tn bh , here @xmath103 . [ t22],title="fig:",scaledwidth=45.0% ]    proceeding analogously , we apply for circular geodesics @xmath45 , one obtains the energy and angular momentum for extreme tn bh [ using eq . ( [ ef1 ] ) and eq . ( [ ef2 ] ) ] : @xmath104 and @xmath105 how @xmath106 and @xmath107 are varied with @xmath88 , it can be seen from the fig . [ t23 ] .     for extreme tn bh , here @xmath103 . [ t23],title="fig:",scaledwidth=45.0% ]   for extreme tn bh , here @xmath103 . [ t23],title="fig:",scaledwidth=45.0% ]    now we have derived the isco equation for non - extreme tn bh but here we will see what would be the isco radius in the extremal limit which is most important class of circular orbit in astrophysics . we find for extreme tn bh the isco radius [ using eq . ( [ n16 ] ) ] should be @xmath108 the corresponding isco energy and isco angular momentum are @xmath109    for photons , the effective potential in the extreme limit [ using eq . ( [ n1 ] ) ] reduces to @xmath110 its behaviour for different values of angular momentum parameter can be seen from the fig . [ t24 ] . with @xmath34 for extreme tn bh , here @xmath103 . [ t24],title="fig:",scaledwidth=45.0% ]    therefore , one obtains the cpo [ using eq . ( [ n6 ] ) ] for extreme tn bh : @xmath111 similarly , we can find the radius of mbco [ using eq . ( [ n18 ] ) ] for extreme tn bh : @xmath112 interestingly , we observed that for extreme tn bh the three radii namely isco , mbco and cpo coincides with the horizon i.e. @xmath113 this has not been previously examined in the literature that for extreme tn bh three orbits are coincident with the killing horizons i.e. null geodesic generators of the horizon . for spherically symmetric extreme string bh , this result has been observed in @xcite . probably , this is a _ first _ time we have seen that such type of features for any spherically symmetric , stationary and non - asymptotic flat extreme bh . in this paper we examined the geodesic motion of test particles in the background geometry of tn spacetime in comparison with mass - less ( zero mass ) tn spacetime . we considered the both massive and mass - less cases . we differentiated the isco , mbco and cpo in graphically between tn spacetime and mass - less tn spacetime . from effective potential diagram , we showed the presence of the nut parameter changes the shape of the potential well in comparison with zero nut paprameter and mass - less spacetime . we studied the circular orbits in the @xmath11 plane for different values of energy i.e. @xmath12 , @xmath13 and @xmath14 in case of tn and mass - less tn spacetime . we also derived the geodesic motion of test particles ( both massive and mass - less ) in case of extreme tn bh . interestingly , we showed that the three radii i.e. the radius of isco , the radius of mbco and the radius of cpo are coincident with the killing horizon radius . this is a very surprising result because probably we first obtained this result for any spherically symmetric , stationary and non - asymptotic flat spacetime . in appendix - a , we derived the cm energy for extreme tn bh and we found that the diverging value of cm energy . in this section , we should study what is the role of nut parameter in the baados , silk and west ( bsw ) effect which was predicted by baados , silk and west @xcite several years ago . does tn bh could be act as a particle accelerator with arbitrarily high energy when the bh is extremal . this is the main aim in this section and motivated by the previous section from the analysis of geodesic structure . we have considered the particle acceleration and collision in the cm frame . to determine the cm energy , we consider first two particles coming from infinity with @xmath114 approaching the tn bh with different angular momenta @xmath115 and @xmath116 and colliding at some radius @xmath34 . later , we choose the collision point is at @xmath34 to approach the horizon @xmath117 . also we have assumed that the particles are initially to be at rest at infinity . the formula that we have used here suggested first by bsw @xcite should read @xmath118 since in the previous section we have calculated total geodesic structure confined in the equatorial plane for tn bh , so we should not repeat it again . due to the time - like isometry and space - like isometry the space - time possesses two conserved quantities one is the energy and other is the angular momentum . thus for massive particles , the components of the four velocity are @xmath119 and      putting these values in ( [ cme ] ) , we find the expression for cm energy : @xmath121 where , @xmath122 for simplicity , @xmath123 and reverting back the value of @xmath124 , we obtain the cm energy near the event horizon ( @xmath22 ) of tn space - time : @xmath125 and for cauchy horizon ( @xmath23 ) the cm energy is @xmath126 where @xmath127 is defined previously . it implies that the cm energy is finite and depends upon the values of angular momentum parameter . it also suggests that the cm energy depends upon the nut parameter . it may also playing a key role in the bsw effect as we have seen from the above expression . now see what happens in case of mass - less case and _ extreme _ case ? first we consider the mass - less case , in this case the cm energy is given by @xmath129 it indicates that the cm energy is finite depends on the nut parameter . more interesting case i.e. the extreme case where the cm energy is given by @xmath130 since we know for extreme case , @xmath131 and @xmath132 . therefore we find for extreme tn bh : @xmath133 i.e the cm energy is _ divergent_. this is an amusing result . because we first reported a non - asymptotic flat , spherically symmetric and stationary extreme bh spacetime possesses such properties .
in this work we study the equatorial causal geodesics of the taub - nut ( tn ) space - time in comparison with _ mass - less _ tn space - time . 
 we emphasized both on the null circular geodesics and time - like circular geodesics . from the effective potential diagram of 
 null and time - like geodesics , we differentiate the geodesics structure between tn spacetime and mass - less tn space - time . 
 it has been shown that there is a key role of the nut parameter to changes the shape of pattern of the potential well in the nut spacetime in comparison with _ mass - less _ nut space - time . 
 we compared the isco ( innermost stable circular orbit ) , mbco ( marginally bound circular orbit ) and cpo ( circular photon orbit ) of the said space - time with graphically in comparison with mass - less cases . 
 moreover , we compute the radius of isco , mbco and cpo for _ extreme _ tn black hole ( bh ) . 
 interestingly , we show that these _ three radii _ coincides with the killing horizon i.e. the null geodesic generators of the horizon . finally in appendix - a , we compute the center - of - mass ( cm ) energy for tn bh and mass - less tn bh . 
 we show that in both cases the cm energy is finite . for _ extreme _ nut bh 
 , we have found that the diverging nature of cm energy . _ first _ , we have observed that a non - asymptotic flat , spherically symmetric and stationary extreme bh is showing such feature .