The Founding of General Relativity and Its Excellence
Chandrasekhar, Subrahmanyan
It is an old thruth that when scientists get older their interest in the history of science and culture intensifies. When the astrophysicist Subrahmanyan Chandrasekhar gave his first lecture at the Lindau Meetings in 1988, its theme was Einstein’s theory of general relativity from 1916. When six years later, Chandrasekhar returned to give his second and last lecture, the title was Newton and Michelangelo, i.e. something out of the 16th and 17th century! His lecture about the general theory of relativity first gives a pedagogical account of the way that the young patent clerk Albert Einstein first realized the need to enlarge the special theory of relativity. As I remember it from my time at the Nobel Museum, where in 2005 we produced an exhibition about Einstein’s Nobel Prize, this insight came when he in 1906 was asked to write a review article on the special theory. He then saw that all physical laws except gravitation could be included in the special theory. When analyzing the force of gravity, he arrived at his famous principle of equivalence, that gravitation is just a form of acceleration. So, as Chandrasekhar argues, the physical insight came early, but it then took 10 years of work to find the field equations. One particular reason that it took so long was that Einstein, with his tremendous physical insight, was not very good at higher mathematics. Several times he was led astray and it took a long time before he understood how the general covariance needed should be expressed. In the second part of his lecture, Chandrasekhar discusses why physicists today believe in the theory of general relativity. From the historical point of view, this acceptance of the new theory came from the classical observations: The perihelion motion of Mercury and the bending of light close to the sun. But these observations only test small effects in the post-Newtonian approximation. Chandrasekhar argues that the belief in the general theory of relativity comes more from its internal consistency and the fact that it does not contradict other physical theories. He also stresses the fact that there are exactly solvable problems as, e.g., black holes, ”the hydrogen atoms of general relativity”. Today, with the observations of the accelerated expansion of the Universe, we are in the situation that the theory of general relativity is again tested and this time on the most grand scale conceivable! Anders Bárány
 Ladies and gentlemen, Einstein is by all criteria the most distinguished physicist of this century. No physicist in this century has been accorded a greater acclaim. But it is an ironic comment that even though most histories of twenty-eighth century physics starts with the pro-forma statement that this century began with two great revolutions of thought, the general theory of relativity and the quantum theory, the general theory of relativity has not been a staple part of the education of a physicist, certainly not to the extent quantum theory has been. Perhaps on this account, a great deal of mythology has accreted around Einstein's name and the theory of relativity which he founded seventy years ago. And even great physicists are not exempt from making statements which if not downright wrong or at least misleading. Maybe a quote for example, a statement by Durak made in 1979 on an occasion celebrating Einstein's hundredth birthday. This is what he said. When Einstein was working on building up this theory of gravitation, he was not trying to account for some results of observation, far from it. His entire procedure was to search for a beautiful theory, a theory of a type that nature will choose. He was guided only by considerations of the beauty of his equations. Now this contradicts statements made by Einstein himself on more than one occasion. Let me read what he said in 1922 in a lecture he gave titled, How I Came to Discover General Theorem of Relativity. In Larry Einstein's statement, I came to realize that all the natural laws except the law of gravity could be discussed within the framework of the special theory of relativity. I wanted to find out the reason for this but I could not attain this goal easily. The most unsatisfactory point was the following. Although the relationship between inertia and energy was explicitly given by the special theory of relativity, the relationship between inertia and weight or the energy of the gravitational field was not clearly elucidated. I felt that this problem could not be resolved within the framework of the special theory of relativity. The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly a thought struck me. If a man falls freely, he would not feel his weight. I was taken aback. The simple thought experiment made a deep impression on me. This led me to the theory of gravity. I continued my thought. A falling man is accelerated. That then what he feels and judges is happening in the accelerated frame of reference. I decided to extend the theory of relativity to the reference frame with acceleration. I felt that in doing so, I could solve the problem of gravity at the same time. A falling man does not feel his weight because in his reference frame there is a new gravitational field which cancels the gravitational field due to the Earth. In the accelerated frame of reference, we need a new gravitational field. Perhaps that is not quite clear from what I have read precisely what he had in mind. But two things are clear. First, he was guided principally by the equality of the inertial and the gravitational mass in empirical fact which has been very accurately determined and in fact probably the most well-established experimental fact. The second point is that this equality of the inertial and the gravitational mass led him to formulate a principle which he states there very briefly and which has now come to be called the principle of equivalence. Let me try to explain more clearly what is involved in these statements I have just made. The first point in order to do that, I have to go back in time. In fact I have to go back 300 years to the time when Newton wrote the Principia, the fact that the publication of the Principia is 300 years old was celebrated last year in many places. Now Newton notices already, within the first few pages of the Principia, that the notion of mass and weight are two distinct concepts based upon two different notions. The notion of mass follows from his second law of motion which states that if you subject a piece of body to a force then it experiences an acceleration in such a way that the quantity which we call the mass of the body times the acceleration is equal to the force. Precisely, if you apply a force to one cubic centimeter of water and measure the acceleration which it experiences and then you find that another piece of water when subjected to the same force experiences ten times the acceleration then you conclude that the mass of the liquid you have used is one tenth cubic centimeter. In other words then the notion of mass is a consequence of his law of motion. It is a constant of proportionality in the relation that the force is equal to the mass times the acceleration. But the notion of weight comes in a different way. If you take a piece of matter and it is subject to the gravitational field say of the earth then you find that the attraction which it experiences in a given gravitational field is proportional to what one calls the weight. For example if you take a piece of liquid say water and you find that the earth attracts it by a force which you measure and you find that another piece of the same matter experiences a gravitation attraction which is say ten times more then you say the weight is ten times greater. In other words the notion of weight and the notion of mass are derived from two entirely different sets of ideas and Newton goes on to say that the two are the same and in fact as he says as they have found by experiments with pendula made accurately the way he determined the equality of the inertial and the gravitational mass was simply to show that the period of a pendulum, simple pendulum depends only on its length and not upon the weight or the mass of the body or the constitution of it. And he established the equality of the inertial and gravitational mass to a few parts in a thousand. A century later Bessel improved the accuracy to a few parts in several tens of thousands. Where the century yet was showed the equality to one part at ten to the eleven and more recently the experiments of Dickey and Brighinsky have shown that they are equal to one part and ten to the minus thirteen. Now this is a very remarkable fact. The notion of mass and weight are fundamental in physics and when equates them by a fact of experience and this of course is basic to the Newtonian theory. Hermann Weill called it an element of magic in the Newtonian theory and one of the objects of Einstein's theory is to eliminate this magic. But the question of course is you want to eliminate the magic but how? And this Einstein, for this Einstein developed what one might call is what one calls today the principle of equivalence. Let me illustrate his ideas here. Here is the famous experiment with an elevator or a lift which Einstein contemplated. Now the experiment is following. Now here is a lift with a rocket booster and let us imagine that this lift is taken to a region of space which is far from any other external body and if this elevator is accelerated by a value equal to the acceleration of gravity then the observer would find that if he drops a piece of apple or a ball it will fall down towards the bottom with a sudden acceleration. On the other hand if the rockets are shut off and the rocket simply coarsed then if he leaves the same body then it remains where it was. Now you perform the experiment now on the same elevator shaft on the earth and you find that if he leaves the body then it falls down to the ground in the same way as it did when this was accelerated and not subject to gravity. And now suppose this elevator is put in a shaft and falls freely towards the center of the earth then when you leave the body there it remains exactly as it was. In other words in this case the action of gravity and the action of acceleration are the same. On the other hand you cannot conclude from this that action of gravity and the action of uniform acceleration are the same. Let us now perform the same experiment in which the observer has two pieces of bodies instead of just one then if the rocket is accelerated then you will find that both of them fall towards the same along parallel lines. And again if the acceleration is stopped then the two bodies will remain at the same point. Now if you go to the earth and similarly you have these two things then the two will fall but not exactly in parallel lines if the curvature of the earth is taken into account. The two lines in which they will drop will intersect at the center of the earth. Now if the same experiment is performed with a lift which is falling freely then as the lift approaches the center of the earth the two objects will come close together. And this is how Einstein showed the equivalence locally of a gravitational field of a uniform gravitational field with a uniform acceleration but showed nevertheless that if the gravitational field is non-uniform then you can no longer make that equivalence. Now in order to show how from this point Einstein derived his principle of equivalence in a form in which he could use it to find gravity I should make a little calculation. Now everyone knows that if you describe the equations of motion in say Cartesian coordinates then the inertial mass times the acceleration is given by the gravitational mass times the gradient of the potential gravitational potential there are similar equations for x and y. Because we want to rewrite this equation in a coordinate system which is not x, y and z but a general covalent coordinates that is instead of x, y and z you change to coordinates q1, q2, q3 and you can associate with the general covalent system a metric in the following way the distance between two neighboring points in Cartesian framework is dx square plus dz square. On the other hand if you find the corresponding distance for general covalent coordinates it will be a certain quantity h alpha beta with a two index quantity which will be functions of the coordinates times dq alpha dq beta for example in spherical polar coordinates will be dr square plus r square d theta square plus r square sin square theta d phi square but more generally that will be the kind of equation you will have. Now let us suppose you write this equation down and ask what the gravitational equations become then you find that m inertial times this quantity q dot beta that is the q double dot beta the acceleration in the coordinate beta times this quantity contracted is equal to the minus the inertial mass times a certain quantity gamma alpha beta gamma we call the Christoffel symbols but it does not matter what they are they are functions of the coordinate functions of the geometry times q dot beta q dot gamma and then minus the same gravitational mass times the gradient of the potential. You see the main point of this equation is to show that the acceleration in the coordinate which is corresponding to that when you write it down in general covalent coordinates the acceleration consists of two terms a term which is geometrical in origin which is a coefficient the inertial mass in a term from the gravitational field and if you accept the equality between the inertial mass and the gravitational mass then the geometrical part of the acceleration and the gravitational part are the same and this is Einstein's remark he said that why make this distinction why not simply say that all acceleration is metrical in origin and that is the starting point of his work he wanted to abolish the distinction between the geometrical part of the acceleration and the gravitational part by saying that all acceleration is metrical in origin. Einstein's conclusion that in the context of gravity all accelerations are metrical in origin is as staggering in its own way as Rutherford's conclusion when Geiger and Marsden first showed him the results of the experiments on the larger angel scattering of alpha rays and Rutherford's remark was it was as though you had fired a 15 inch shell at a piece of tissue paper and it had bounced back and hit you. In the case of Rutherford he was able to derive his law of scattering overnight but he took Einstein many years in fact 10 years almost to obtain his final field equations. The transition from the statement that all acceleration is metrical in origin to the equations of the field in terms of the Riemann tensor is a giant leap and the fact that it took Einstein three or four years to make the transition is understandable. Indeed it is astonishing that he made the transition at all. Of course one can claim that mathematical insight was needed to go from his statement about the metrical origin of gravitational forces to formulating those ideas in terms of Riemannian geometry but Einstein was not particularly well disposed to mathematical treatments and particularly geometrical way of thinking prior in his earlier years. For example when Minkowski wrote a few years after Einstein had formulated his special theory of relativity by describing the special relativity in terms of what we now call Minkowski geometry in which we associate a metric in space time which is dG squared minus dx squared minus dy squared minus dc squared and he showed that rotations in a space time with this metric is equivalent to a special relativity. Einstein's remark on Minkowski's paper was first that well we physicists show how to formulate the laws of physics and mathematicians will come along and say how much better they can do it and indeed he made the remark that Minkowski's work was Yuba Flessiger, Yuba Flessiger-Galer Minkowski, unnecessary learnedness but it is only in 1911 or 1912 that he realized the importance of this geometrical way of thinking and particularly with the aid and assistance of his friend Marcel Grossmann he learned sufficient differential geometry to come to his triumphant conclusion with regard to his field equations in 1915 but even at that time Einstein's familiarity with Riemannian geometry was not sufficiently adequate. He did not realize that the general covariance of his theory required that the field equations must leave four arbitrary functions free. Because of his misunderstanding here he first formulated his field equations by equating the rigid tensor with the energy momentum tensor but then he realized the energy momentum tensor must have its covariant divergence zero but the covariant divergence of the rigid tensor is not zero and he had to modify it to introduce what is the Einstein tensor. Now I do not wish to go into the details more but only to emphasize that the principal motive of the theory was physical insight and it was the strength of his physical insight that led him to the beauty of the formulation of the field equations in terms of Riemannian geometry. Now I want to turn around and say that why is it we believe in the general theory of relativity. Of course there has been a great deal of effort during the past two decades to confirm the predictions of general relativity but these predictions relate to very very small departures from the predictions of the Newtonian theory and in no case more than a few parts in a million. The conformation comes from the deflection of light as light traverses a gravitational field and the consequent time delay, the precession of the Perigalina Mercury and the change in period of double stars, close double stars as pulsars due to the emission of gravitational radiation but in no instance is the effect predicted more than a few parts in a million departures from Newtonian theory and in all instances it is no more than verifying the values of one or two or three parameters in expansion of the equations of general relativity in what one calls the post-Newtonian approximation. But one does not believe in a theory of which only the approximations have been confirmed. For example if you take the Dirac theory of the electron and the only conformation you had was the fine structure of ionized helium in partial experiments of a conviction would not have been as great and suppose there had been no possibility in the laboratory of obtaining energies of a million electron volts then the real experiment, the real verification of Dirac's ideas, prediction of antimatter, the creation of electron-positron pairs would not have been possible and our conviction in the theory would not have been as great. But it must be stated that in the realm of general relativity no phenomenon which requires the full nonlinear aspect of general relativity have been confirmed. Why then do we believe in it? I think of a belief in general relativity comes far more from its internal consistency and from the fact that whenever general relativity has an interface with other parts of physics it does not contradict any of them. Let me illustrate these two things in the following way. We all know that the equations of physics must be causal. Essentially what it means is that if you make a disturbance at one point the disturbance cannot be felt at another point for a time which light will take to go from one point to another. Technically one says that the equations of physics must allow an initial value formulation that is to say you give the initial data on a space like surface and you show that the only part of the space time in which the future can be predicted is that which is determined by sending out light rays from the boundary of the space time region to the point. In other words if for example you... suppose you have a space like slice then you send a light ray here and you light a ray here and it is in that region that the future is defined. Now when Einstein formulated the general theory of relativity he does not seem to have been concerned whether his equations allowed an initial value formulation and in fact to prove in spite of the non-linearity of the equations the initial value formulation is possible in general relativity was proved only in the early 40s by Lechnerovitz in France so that even though when in formulating the general theory of relativity the requirement that it satisfy the laws of causality was not included in fact it was consistent with it or let us take the notion of energy. In physics the notion of energy is of course central we define it locally and it is globally concerned. In general relativity for a variety of reasons I cannot go into you cannot define a local energy. On the other hand you should expect on physical grounds that you have an isolated matter and even if it radiates energy then globally you ought to be able to define a quantity which you could call the energy of the system and that if the energy varies it can only be because gravitational waves cross the boundary at a sufficiently large distance and that is the second point. Of course the energy of a gravitating system must include the potential energy of the field itself but the potential energy in the Newtonian theory has no lower bound by bringing two points sufficiently close together you can have an infinite negative energy but in general relativity you must expect that there is a lower bound to the energy of any gravitating system and if you take a reference with this lower bound as the origin of measuring the energy then the energy must always be positive. In other words if general relativity is to be consistent with other laws of physics you ought to be able to define for an isolated system a global meaning for its energy and you must also be able to show that the energy is positive but actually this has been the so-called positive energy conjecture for more than 16 years and only a few years ago it has proved rigorously by Ed Witten and Yoham. Now in other words then that even though Einstein formulated the theory from very simple considerations like all accelerations must be metrically in origin and putting it in the mathematical framework of Riemannian geometry it nevertheless is consistent in a way in which its originator could not have contemplated but what is even more remarkable is that general relativity does have interfaces with other branches of physics. I cannot go into the details but one can show that if you take a black hole and have the direct waves reflected and scattered by a black hole then there are certain requirements of the nature of scattering which the quantum theory requires but even though in the formulating this problem in general relativity no aspect of quantum theory is included the results one gets are entirely consistent with the requirements of the quantum theory in exactly the same way general relativity has interfaces with thermodynamics and it is possible to introduce the notion of entropy for example in the context of what one generally calls Hawking radiation. Now certainly thermodynamics must not incorporate in forming general relativity but one finds that when you find the need to include concepts from other branches of physics in consequences of general relativity then all these consequences do not contradict branches of other parts of physics and it is this consistency with physical requirements this lack of contradiction with other branches of physics which was not contemplated in its founding and it is these which gives one confidence in the theory. Well I am afraid I do not have too much time to go into the other aspect of my talk namely why is the general theory of an excellent theory well let me just make one comment if you take any new physical theory then it is characteristic of a good physical theory that it isolates a physical problem which incorporates the essential features of that theory and for which the theory gives an exact solution for the Newtonian theory of gravitation you have the solution to the Kepler problem for quantum mechanics relativistic or non relativistic you have the predictions of the energy of the hydrogen atom and in the case of the Dirac theory I suppose the Klein-Lichino formula and the Pair production. Now in the case of the general theory of relativity you can ask is there a problem which incorporates the basic concepts of general relativity in its purest form. In its purest form the general theory of relativity is a theory of space and time. Now a black hole is one whose construction is based only on the notions of space and time. A black hole is an object which divides the three dimensional space into two parts an interior part and an exterior part bound by a certain surface which one calls a horizon and the reason for calling that the horizon is that no person, no observer in the interior of the horizon can communicate with the space outside. So a black hole is defined as a solution of Einstein's vacuum equations which has a horizon which is convex and which is asymptotically flat in the sense that the space time is Minkowski at sufficiently large distances. It is a remarkable fact that these two simple requirements provide in the basis of general relativity a unique solution to the problem. A solution which has just two parameters the mass and the angular momentum. This is a solution of Kerr discovered in 1962. The point is that if you ask what a black hole solution consistent with general relativity is you find that there is only one simple solution with two parameters and all black holes which occur in nature must belong to it. And one can say the following if you see macroscopic objects then we see macroscopic objects all around us and if you want to understand them it depends upon a variety of physical theories with a variety of approximations and you understand it approximately. There is no example in macroscopic physics of an object which is described exactly and with only two parameters. In other words one could say that almost by definition the black holes are the most perfect objects in the universe because their construction requires only the notions of space and time. It is not vulgarized by any other part of physics with which we are mostly dealing with and one can go on and point out the exceptional mathematical perfectness of the theory of black holes. Einstein when he wrote his last paper his first paper announcing his field equations stated that anyone scarcely anyone who understands my theory can escape its magic. For one practitioner at least the magic of the general theory of relativity is in its harmonious mathematical character and the harmonious structure of its consequences. Thank you.
 Ladies and gentlemen, Einstein is by all criteria the most distinguished physicist of this century.  No physicist in this century has been accorded a greater acclaim.  But it is an ironic comment that even though most histories of twenty-eighth century physics  starts with the pro-forma statement that this century began with two great revolutions  of thought, the general theory of relativity and the quantum theory, the general theory  of relativity has not been a staple part of the education of a physicist, certainly  not to the extent quantum theory has been.  Perhaps on this account, a great deal of mythology has accreted around Einstein's name  and the theory of relativity which he founded seventy years ago.  And even great physicists are not exempt from making statements which if not downright wrong  or at least misleading.  Maybe a quote for example, a statement by Durak made in 1979 on an occasion celebrating  Einstein's hundredth birthday.  This is what he said.  When Einstein was working on building up this theory of gravitation, he was not trying to  account for some results of observation, far from it.  His entire procedure was to search for a beautiful theory, a theory of a type that nature will  choose.  He was guided only by considerations of the beauty of his equations.  Now this contradicts statements made by Einstein himself on more than one occasion.  Let me read what he said in 1922 in a lecture he gave titled, How I Came to Discover General  Theorem of Relativity.  In Larry Einstein's statement, I came to realize that all the natural laws except the law of  gravity could be discussed within the framework of the special theory of relativity.  I wanted to find out the reason for this but I could not attain this goal easily.  The most unsatisfactory point was the following.  Although the relationship between inertia and energy was explicitly given by the special  theory of relativity, the relationship between inertia and weight or the energy of the gravitational  field was not clearly elucidated.  I felt that this problem could not be resolved within the framework of the special theory  of relativity.  The breakthrough came suddenly one day.  I was sitting on a chair in my patent office in Bern.  Suddenly a thought struck me.  If a man falls freely, he would not feel his weight.  I was taken aback.  The simple thought experiment made a deep impression on me.  This led me to the theory of gravity.  I continued my thought.  A falling man is accelerated.  That then what he feels and judges is happening in the accelerated frame of reference.  I decided to extend the theory of relativity to the reference frame with acceleration.  I felt that in doing so, I could solve the problem of gravity at the same time.  A falling man does not feel his weight because in his reference frame there is a new gravitational  field which cancels the gravitational field due to the Earth.  In the accelerated frame of reference, we need a new gravitational field.  Perhaps that is not quite clear from what I have read precisely what he had in mind.  But two things are clear.  First, he was guided principally by the equality of the inertial and the gravitational mass  in empirical fact which has been very accurately determined and in fact probably the most well-established  experimental fact.  The second point is that this equality of the inertial and the gravitational mass led  him to formulate a principle which he states there very briefly and which has now come  to be called the principle of equivalence.  Let me try to explain more clearly what is involved in these statements I have just made.  The first point in order to do that, I have to go back in time.  In fact I have to go back 300 years to the time when Newton wrote the Principia, the  fact that the publication of the Principia is 300 years old was celebrated last year  in many places.  Now Newton notices already, within the first few pages of the Principia, that the notion  of mass and weight are two distinct concepts based upon two different notions.  The notion of mass follows from his second law of motion which states that if you subject  a piece of body to a force then it experiences an acceleration in such a way that the quantity  which we call the mass of the body times the acceleration is equal to the force.  Precisely, if you apply a force to one cubic centimeter of water and measure the acceleration  which it experiences and then you find that another piece of water when subjected to the  same force experiences ten times the acceleration then you conclude that the mass of the liquid  you have used is one tenth cubic centimeter.  In other words then the notion of mass is a consequence of his law of motion.  It is a constant of proportionality in the relation that the force is equal to the mass  times the acceleration.  But the notion of weight comes in a different way.  If you take a piece of matter and it is subject to the gravitational field say of the earth  then you find that the attraction which it experiences in a given gravitational field  is proportional to what one calls the weight.  For example if you take a piece of liquid say water and you find that the earth attracts  it by a force which you measure and you find that another piece of the same matter experiences  a gravitation attraction which is say ten times more then you say the weight is ten  times greater.  In other words the notion of weight and the notion of mass are derived from two entirely  different sets of ideas and Newton goes on to say that the two are the same and in fact  as he says as they have found by experiments with pendula made accurately the way he determined  the equality of the inertial and the gravitational mass was simply to show that the period of  a pendulum, simple pendulum depends only on its length and not upon the weight or the  mass of the body or the constitution of it.  And he established the equality of the inertial and gravitational mass to a few parts in a  thousand.  A century later Bessel improved the accuracy to a few parts in several tens of thousands.  Where the century yet was showed the equality to one part at ten to the eleven and more  recently the experiments of Dickey and Brighinsky have shown that they are equal to one part  and ten to the minus thirteen.  Now this is a very remarkable fact.  The notion of mass and weight are fundamental in physics and when equates them by a fact  of experience and this of course is basic to the Newtonian theory.  Hermann Weill called it an element of magic in the Newtonian theory and one of the objects  of Einstein's theory is to eliminate this magic.  But the question of course is you want to eliminate the magic but how?  And this Einstein, for this Einstein developed what one might call is what one calls today  the principle of equivalence.  Let me illustrate his ideas here.  Here is the famous experiment with an elevator or a lift which Einstein contemplated.  Now the experiment is following.  Now here is a lift with a rocket booster and let us imagine that this lift is taken to  a region of space which is far from any other external body and if this elevator is accelerated  by a value equal to the acceleration of gravity then the observer would find that if he drops  a piece of apple or a ball it will fall down towards the bottom with a sudden acceleration.  On the other hand if the rockets are shut off and the rocket simply coarsed then if he  leaves the same body then it remains where it was.  Now you perform the experiment now on the same elevator shaft on the earth and you find  that if he leaves the body then it falls down to the ground in the same way as it did when  this was accelerated and not subject to gravity.  And now suppose this elevator is put in a shaft and falls freely towards the center of the  earth then when you leave the body there it remains exactly as it was.  In other words in this case the action of gravity and the action of acceleration are  the same.  On the other hand you cannot conclude from this that action of gravity and the action  of uniform acceleration are the same.  Let us now perform the same experiment in which the observer has two pieces of bodies instead  of just one then if the rocket is accelerated then you will find that both of them fall  towards the same along parallel lines.  And again if the acceleration is stopped then the two bodies will remain at the same point.  Now if you go to the earth and similarly you have these two things then the two will fall  but not exactly in parallel lines if the curvature of the earth is taken into account.  The two lines in which they will drop will intersect at the center of the earth.  Now if the same experiment is performed with a lift which is falling freely then as the  lift approaches the center of the earth the two objects will come close together.  And this is how Einstein showed the equivalence locally of a gravitational field of a uniform  gravitational field with a uniform acceleration but showed nevertheless that if the gravitational  field is non-uniform then you can no longer make that equivalence.  Now in order to show how from this point Einstein derived his principle of equivalence in a  form in which he could use it to find gravity I should make a little calculation.  Now everyone knows that if you describe the equations of motion in say Cartesian coordinates  then the inertial mass times the acceleration is given by the gravitational mass times the  gradient of the potential gravitational potential there are similar equations for x and y.  Because we want to rewrite this equation in a coordinate system which is not x, y and  z but a general covalent coordinates that is instead of x, y and z you change to coordinates  q1, q2, q3 and you can associate with the general covalent system a metric in the following  way the distance between two neighboring points in Cartesian framework is dx square plus dz  square.  On the other hand if you find the corresponding distance for general covalent coordinates  it will be a certain quantity h alpha beta with a two index quantity which will be functions  of the coordinates times dq alpha dq beta for example in spherical polar coordinates  will be dr square plus r square d theta square plus r square sin square theta d phi square  but more generally that will be the kind of equation you will have.  Now let us suppose you write this equation down and ask what the gravitational equations  become then you find that m inertial times this quantity q dot beta that is the q double  dot beta the acceleration in the coordinate beta times this quantity contracted is equal  to the minus the inertial mass times a certain quantity gamma alpha beta gamma we call the  Christoffel symbols but it does not matter what they are they are functions of the coordinate  functions of the geometry times q dot beta q dot gamma and then minus the same gravitational  mass times the gradient of the potential.  You see the main point of this equation is to show that the acceleration in the coordinate  which is corresponding to that when you write it down in general covalent coordinates the  acceleration consists of two terms a term which is geometrical in origin which is a coefficient  the inertial mass in a term from the gravitational field and if you accept the equality between  the inertial mass and the gravitational mass then the geometrical part of the acceleration  and the gravitational part are the same and this is Einstein's remark he said that why  make this distinction why not simply say that all acceleration is metrical in origin and  that is the starting point of his work he wanted to abolish the distinction between  the geometrical part of the acceleration and the gravitational part by saying that all  acceleration is metrical in origin. Einstein's conclusion that in the context of gravity  all accelerations are metrical in origin is as staggering in its own way as Rutherford's  conclusion when Geiger and Marsden first showed him the results of the experiments on the  larger angel scattering of alpha rays and Rutherford's remark was it was as though you had fired  a 15 inch shell at a piece of tissue paper and it had bounced back and hit you. In the  case of Rutherford he was able to derive his law of scattering overnight but he took Einstein  many years in fact 10 years almost to obtain his final field equations. The transition from  the statement that all acceleration is metrical in origin to the equations of the field in  terms of the Riemann tensor is a giant leap and the fact that it took Einstein three or  four years to make the transition is understandable. Indeed it is astonishing that he made the  transition at all. Of course one can claim that mathematical insight was needed to go  from his statement about the metrical origin of gravitational forces to formulating those  ideas in terms of Riemannian geometry but Einstein was not particularly well disposed  to mathematical treatments and particularly geometrical way of thinking prior in his earlier  years. For example when Minkowski wrote a few years after Einstein had formulated his  special theory of relativity by describing the special relativity in terms of what we  now call Minkowski geometry in which we associate a metric in space time which is dG squared  minus dx squared minus dy squared minus dc squared and he showed that rotations in a  space time with this metric is equivalent to a special relativity. Einstein's remark on  Minkowski's paper was first that well we physicists show how to formulate the laws  of physics and mathematicians will come along and say how much better they can do it and  indeed he made the remark that Minkowski's work was Yuba Flessiger, Yuba Flessiger-Galer  Minkowski, unnecessary learnedness but it is only in 1911 or 1912 that he realized the  importance of this geometrical way of thinking and particularly with the aid and assistance  of his friend Marcel Grossmann he learned sufficient differential geometry to come to his triumphant  conclusion with regard to his field equations in 1915 but even at that time Einstein's familiarity  with Riemannian geometry was not sufficiently adequate. He did not realize that the general  covariance of his theory required that the field equations must leave four arbitrary  functions free. Because of his misunderstanding here he first formulated his field equations  by equating the rigid tensor with the energy momentum tensor but then he realized the energy  momentum tensor must have its covariant divergence zero but the covariant divergence of the rigid  tensor is not zero and he had to modify it to introduce what is the Einstein tensor.  Now I do not wish to go into the details more but only to emphasize that the principal  motive of the theory was physical insight and it was the strength of his physical insight  that led him to the beauty of the formulation of the field equations in terms of Riemannian  geometry. Now I want to turn around and say that why is it we believe in the general theory  of relativity. Of course there has been a great deal of effort during the past two decades  to confirm the predictions of general relativity but these predictions relate to very very small  departures from the predictions of the Newtonian theory and in no case more than a few parts  in a million. The conformation comes from the deflection of light as light traverses  a gravitational field and the consequent time delay, the precession of the Perigalina  Mercury and the change in period of double stars, close double stars as pulsars due to  the emission of gravitational radiation but in no instance is the effect predicted more  than a few parts in a million departures from Newtonian theory and in all instances it is  no more than verifying the values of one or two or three parameters in expansion of the  equations of general relativity in what one calls the post-Newtonian approximation. But  one does not believe in a theory of which only the approximations have been confirmed.  For example if you take the Dirac theory of the electron and the only conformation you  had was the fine structure of ionized helium in partial experiments of a conviction would  not have been as great and suppose there had been no possibility in the laboratory of obtaining  energies of a million electron volts then the real experiment, the real verification  of Dirac's ideas, prediction of antimatter, the creation of electron-positron pairs would  not have been possible and our conviction in the theory would not have been as great.  But it must be stated that in the realm of general relativity no phenomenon which requires  the full nonlinear aspect of general relativity have been confirmed. Why then do we believe  in it? I think of a belief in general relativity comes far more from its internal consistency  and from the fact that whenever general relativity has an interface with other parts of physics  it does not contradict any of them. Let me illustrate these two things in the following  way. We all know that the equations of physics must be causal. Essentially what it means  is that if you make a disturbance at one point the disturbance cannot be felt at another  point for a time which light will take to go from one point to another. Technically  one says that the equations of physics must allow an initial value formulation that is  to say you give the initial data on a space like surface and you show that the only part  of the space time in which the future can be predicted is that which is determined by  sending out light rays from the boundary of the space time region to the point. In other  words if for example you... suppose you have a space like slice then you send a light ray  here and you light a ray here and it is in that region that the future is defined. Now  when Einstein formulated the general theory of relativity he does not seem to have been  concerned whether his equations allowed an initial value formulation and in fact to  prove in spite of the non-linearity of the equations the initial value formulation is  possible in general relativity was proved only in the early 40s by Lechnerovitz in France  so that even though when in formulating the general theory of relativity the requirement  that it satisfy the laws of causality was not included in fact it was consistent with  it or let us take the notion of energy. In physics the notion of energy is of course  central we define it locally and it is globally concerned. In general relativity for a variety  of reasons I cannot go into you cannot define a local energy. On the other hand you should  expect on physical grounds that you have an isolated matter and even if it radiates energy  then globally you ought to be able to define a quantity which you could call the energy  of the system and that if the energy varies it can only be because gravitational waves  cross the boundary at a sufficiently large distance and that is the second point. Of  course the energy of a gravitating system must include the potential energy of the field  itself but the potential energy in the Newtonian theory has no lower bound by bringing two  points sufficiently close together you can have an infinite negative energy but in general  relativity you must expect that there is a lower bound to the energy of any gravitating  system and if you take a reference with this lower bound as the origin of measuring the  energy then the energy must always be positive. In other words if general relativity is to  be consistent with other laws of physics you ought to be able to define for an isolated  system a global meaning for its energy and you must also be able to show that the energy  is positive but actually this has been the so-called positive energy conjecture for more  than 16 years and only a few years ago it has proved rigorously by Ed Witten and Yoham.  Now in other words then that even though Einstein formulated the theory from very simple considerations  like all accelerations must be metrically in origin and putting it in the mathematical  framework of Riemannian geometry it nevertheless is consistent in a way in which its originator  could not have contemplated but what is even more remarkable is that general relativity  does have interfaces with other branches of physics. I cannot go into the details but  one can show that if you take a black hole and have the direct waves reflected and scattered  by a black hole then there are certain requirements of the nature of scattering which the quantum  theory requires but even though in the formulating this problem in general relativity no aspect  of quantum theory is included the results one gets are entirely consistent with the requirements  of the quantum theory in exactly the same way general relativity has interfaces with  thermodynamics and it is possible to introduce the notion of entropy for example in the context  of what one generally calls Hawking radiation. Now certainly thermodynamics must not incorporate  in forming general relativity but one finds that when you find the need to include concepts  from other branches of physics in consequences of general relativity then all these consequences  do not contradict branches of other parts of physics and it is this consistency with  physical requirements this lack of contradiction with other branches of physics which was not  contemplated in its founding and it is these which gives one confidence in the theory.  Well I am afraid I do not have too much time to go into the other aspect of my talk namely  why is the general theory of an excellent theory well let me just make one comment if  you take any new physical theory then it is characteristic of a good physical theory that  it isolates a physical problem which incorporates the essential features of that theory and  for which the theory gives an exact solution for the Newtonian theory of gravitation you  have the solution to the Kepler problem for quantum mechanics relativistic or non relativistic  you have the predictions of the energy of the hydrogen atom and in the case of the Dirac  theory I suppose the Klein-Lichino formula and the Pair production. Now in the case of  the general theory of relativity you can ask is there a problem which incorporates the  basic concepts of general relativity in its purest form. In its purest form the general  theory of relativity is a theory of space and time. Now a black hole is one whose construction  is based only on the notions of space and time. A black hole is an object which divides  the three dimensional space into two parts an interior part and an exterior part bound  by a certain surface which one calls a horizon and the reason for calling that the horizon  is that no person, no observer in the interior of the horizon can communicate with the space  outside. So a black hole is defined as a solution of Einstein's vacuum equations which has a  horizon which is convex and which is asymptotically flat in the sense that the space time is  Minkowski at sufficiently large distances. It is a remarkable fact that these two simple  requirements provide in the basis of general relativity a unique solution to the problem.  A solution which has just two parameters the mass and the angular momentum. This is a solution  of Kerr discovered in 1962. The point is that if you ask what a black hole solution consistent  with general relativity is you find that there is only one simple solution with two parameters  and all black holes which occur in nature must belong to it. And one can say the following  if you see macroscopic objects then we see macroscopic objects all around us and if you  want to understand them it depends upon a variety of physical theories with a variety  of approximations and you understand it approximately. There is no example in macroscopic physics  of an object which is described exactly and with only two parameters. In other words one  could say that almost by definition the black holes are the most perfect objects in the  universe because their construction requires only the notions of space and time. It is  not vulgarized by any other part of physics with which we are mostly dealing with and  one can go on and point out the exceptional mathematical perfectness of the theory of  black holes. Einstein when he wrote his last paper his first paper announcing his field  equations stated that anyone scarcely anyone who understands my theory can escape its magic.  For one practitioner at least the magic of the general theory of relativity is in its  harmonious mathematical character and the harmonious structure of its consequences.  Thank you.
1988_chandrasekhar_996x560_0_1667.png 1988_chandrasekhar_996x560_1668_5231.png 1988_chandrasekhar_996x560_5232_6599.png 1988_chandrasekhar_996x560_6600_7151.png 1988_chandrasekhar_996x560_7152_7967.png 1988_chandrasekhar_996x560_7968_8243.png 1988_chandrasekhar_996x560_8244_8519.png 1988_chandrasekhar_996x560_8520_9887.png 1988_chandrasekhar_996x560_9888_13451.png 1988_chandrasekhar_996x560_13452_14819.png 1988_chandrasekhar_996x560_14820_16187.png 1988_chandrasekhar_996x560_16188_16463.png 1988_chandrasekhar_996x560_16464_16739.png 1988_chandrasekhar_996x560_16740_19751.png 1988_chandrasekhar_996x560_19752_20027.png 1988_chandrasekhar_996x560_20028_21671.png 1988_chandrasekhar_996x560_21672_23039.png 1988_chandrasekhar_996x560_23040_24407.png 1988_chandrasekhar_996x560_24408_24959.png 1988_chandrasekhar_996x560_24960_26327.png 1988_chandrasekhar_996x560_26328_27971.png 1988_chandrasekhar_996x560_27972_29891.png 1988_chandrasekhar_996x560_29892_31811.png 1988_chandrasekhar_996x560_31812_32627.png 1988_chandrasekhar_996x560_32628_33179.png 1988_chandrasekhar_996x560_33180_34271.png 1988_chandrasekhar_996x560_34272_34547.png 1988_chandrasekhar_996x560_34548_36191.png 1988_chandrasekhar_996x560_36192_36467.png 1988_chandrasekhar_996x560_36468_37007.png 1988_chandrasekhar_996x560_37008_38111.png 1988_chandrasekhar_996x560_38112_41123.png 1988_chandrasekhar_996x560_41124_41399.png 1988_chandrasekhar_996x560_41400_42767.png 1988_chandrasekhar_996x560_42768_44411.png 1988_chandrasekhar_996x560_44412_44687.png 1988_chandrasekhar_996x560_44688_46331.png 1988_chandrasekhar_996x560_46332_48251.png 1988_chandrasekhar_996x560_48252_49343.png 1988_chandrasekhar_996x560_49344_49619.png 1988_chandrasekhar_996x560_49620_50987.png 1988_chandrasekhar_996x560_50988_52631.png 1988_chandrasekhar_996x560_52632_54551.png 1988_chandrasekhar_996x560_54552_55367.png
Ladies and gentlemen, Einstein is by all criteria the most distinguished physicist of this century. No physicist in this century has been accorded a greater acclaim. But it is an ironic comment that even though most histories of twenty-eighth century physics starts with the pro-forma statement that this century began with two great revolutions of thought, the general theory of relativity and the quantum theory, the general theory of relativity has not been a staple part of the education of a physicist, certainly not to the extent quantum theory has been. Perhaps on this account, a great deal of mythology has accreted around Einstein's name and the theory of relativity which he founded seventy years ago. And even great physicists are not exempt from making statements which if not downright wrong or at least misleading. Maybe a quote for example, a statement by Durak made in 1979 on an occasion celebrating Einstein's hundredth birthday. This is what he said. When Einstein was working on building up this theory of gravitation, he was not trying to account for some results of observation, far from it. His entire procedure was to search for a beautiful theory, a theory of a type that nature will choose. He was guided only by considerations of the beauty of his equations. Now this contradicts statements made by Einstein himself on more than one occasion. Let me read what he said in 1922 in a lecture he gave titled, How I Came to Discover General Theorem of Relativity. In Larry Einstein's statement, I came to realize that all the natural laws except the law of gravity could be discussed within the framework of the special theory of relativity. I wanted to find out the reason for this but I could not attain this goal easily. The most unsatisfactory point was the following. Although the relationship between inertia and energy was explicitly given by the special theory of relativity, the relationship between inertia and weight or the energy of the gravitational field was not clearly elucidated. I felt that this problem could not be resolved within the framework of the special theory of relativity. The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly a thought struck me. If a man falls freely, he would not feel his weight. I was taken aback. The simple thought experiment made a deep impression on me. This led me to the theory of gravity. I continued my thought. A falling man is accelerated. That then what he feels and judges is happening in the accelerated frame of reference. I decided to extend the theory of relativity to the reference frame with acceleration. I felt that in doing so, I could solve the problem of gravity at the same time. A falling man does not feel his weight because in his reference frame there is a new gravitational field which cancels the gravitational field due to the Earth. In the accelerated frame of reference, we need a new gravitational field. Perhaps that is not quite clear from what I have read precisely what he had in mind. But two things are clear. First, he was guided principally by the equality of the inertial and the gravitational mass in empirical fact which has been very accurately determined and in fact probably the most well-established experimental fact. The second point is that this equality of the inertial and the gravitational mass led him to formulate a principle which he states there very briefly and which has now come to be called the principle of equivalence. Let me try to explain more clearly what is involved in these statements I have just made. The first point in order to do that, I have to go back in time. In fact I have to go back 300 years to the time when Newton wrote the Principia, the fact that the publication of the Principia is 300 years old was celebrated last year in many places. Now Newton notices already, within the first few pages of the Principia, that the notion of mass and weight are two distinct concepts based upon two different notions. The notion of mass follows from his second law of motion which states that if you subject a piece of body to a force then it experiences an acceleration in such a way that the quantity which we call the mass of the body times the acceleration is equal to the force. Precisely, if you apply a force to one cubic centimeter of water and measure the acceleration which it experiences and then you find that another piece of water when subjected to the same force experiences ten times the acceleration then you conclude that the mass of the liquid you have used is one tenth cubic centimeter. In other words then the notion of mass is a consequence of his law of motion. It is a constant of proportionality in the relation that the force is equal to the mass times the acceleration. But the notion of weight comes in a different way. If you take a piece of matter and it is subject to the gravitational field say of the earth then you find that the attraction which it experiences in a given gravitational field is proportional to what one calls the weight. For example if you take a piece of liquid say water and you find that the earth attracts it by a force which you measure and you find that another piece of the same matter experiences a gravitation attraction which is say ten times more then you say the weight is ten times greater. In other words the notion of weight and the notion of mass are derived from two entirely different sets of ideas and Newton goes on to say that the two are the same and in fact as he says as they have found by experiments with pendula made accurately the way he determined the equality of the inertial and the gravitational mass was simply to show that the period of a pendulum, simple pendulum depends only on its length and not upon the weight or the mass of the body or the constitution of it. And he established the equality of the inertial and gravitational mass to a few parts in a thousand. A century later Bessel improved the accuracy to a few parts in several tens of thousands. Where the century yet was showed the equality to one part at ten to the eleven and more recently the experiments of Dickey and Brighinsky have shown that they are equal to one part and ten to the minus thirteen. Now this is a very remarkable fact. The notion of mass and weight are fundamental in physics and when equates them by a fact of experience and this of course is basic to the Newtonian theory. Hermann Weill called it an element of magic in the Newtonian theory and one of the objects of Einstein's theory is to eliminate this magic. But the question of course is you want to eliminate the magic but how? And this Einstein, for this Einstein developed what one might call is what one calls today the principle of equivalence. Let me illustrate his ideas here. Here is the famous experiment with an elevator or a lift which Einstein contemplated. Now the experiment is following. Now here is a lift with a rocket booster and let us imagine that this lift is taken to a region of space which is far from any other external body and if this elevator is accelerated by a value equal to the acceleration of gravity then the observer would find that if he drops a piece of apple or a ball it will fall down towards the bottom with a sudden acceleration. On the other hand if the rockets are shut off and the rocket simply coarsed then if he leaves the same body then it remains where it was. Now you perform the experiment now on the same elevator shaft on the earth and you find that if he leaves the body then it falls down to the ground in the same way as it did when this was accelerated and not subject to gravity. And now suppose this elevator is put in a shaft and falls freely towards the center of the earth then when you leave the body there it remains exactly as it was. In other words in this case the action of gravity and the action of acceleration are the same. On the other hand you cannot conclude from this that action of gravity and the action of uniform acceleration are the same. Let us now perform the same experiment in which the observer has two pieces of bodies instead of just one then if the rocket is accelerated then you will find that both of them fall towards the same along parallel lines. And again if the acceleration is stopped then the two bodies will remain at the same point. Now if you go to the earth and similarly you have these two things then the two will fall but not exactly in parallel lines if the curvature of the earth is taken into account. The two lines in which they will drop will intersect at the center of the earth. Now if the same experiment is performed with a lift which is falling freely then as the lift approaches the center of the earth the two objects will come close together. And this is how Einstein showed the equivalence locally of a gravitational field of a uniform gravitational field with a uniform acceleration but showed nevertheless that if the gravitational field is non-uniform then you can no longer make that equivalence. Now in order to show how from this point Einstein derived his principle of equivalence in a form in which he could use it to find gravity I should make a little calculation. Now everyone knows that if you describe the equations of motion in say Cartesian coordinates then the inertial mass times the acceleration is given by the gravitational mass times the gradient of the potential gravitational potential there are similar equations for x and y. Because we want to rewrite this equation in a coordinate system which is not x, y and z but a general covalent coordinates that is instead of x, y and z you change to coordinates q1, q2, q3 and you can associate with the general covalent system a metric in the following way the distance between two neighboring points in Cartesian framework is dx square plus dz square. On the other hand if you find the corresponding distance for general covalent coordinates it will be a certain quantity h alpha beta with a two index quantity which will be functions of the coordinates times dq alpha dq beta for example in spherical polar coordinates will be dr square plus r square d theta square plus r square sin square theta d phi square but more generally that will be the kind of equation you will have. Now let us suppose you write this equation down and ask what the gravitational equations become then you find that m inertial times this quantity q dot beta that is the q double dot beta the acceleration in the coordinate beta times this quantity contracted is equal to the minus the inertial mass times a certain quantity gamma alpha beta gamma we call the Christoffel symbols but it does not matter what they are they are functions of the coordinate functions of the geometry times q dot beta q dot gamma and then minus the same gravitational mass times the gradient of the potential. You see the main point of this equation is to show that the acceleration in the coordinate which is corresponding to that when you write it down in general covalent coordinates the acceleration consists of two terms a term which is geometrical in origin which is a coefficient the inertial mass in a term from the gravitational field and if you accept the equality between the inertial mass and the gravitational mass then the geometrical part of the acceleration and the gravitational part are the same and this is Einstein's remark he said that why make this distinction why not simply say that all acceleration is metrical in origin and that is the starting point of his work he wanted to abolish the distinction between the geometrical part of the acceleration and the gravitational part by saying that all acceleration is metrical in origin. Einstein's conclusion that in the context of gravity all accelerations are metrical in origin is as staggering in its own way as Rutherford's conclusion when Geiger and Marsden first showed him the results of the experiments on the larger angel scattering of alpha rays and Rutherford's remark was it was as though you had fired a 15 inch shell at a piece of tissue paper and it had bounced back and hit you. In the case of Rutherford he was able to derive his law of scattering overnight but he took Einstein many years in fact 10 years almost to obtain his final field equations. The transition from the statement that all acceleration is metrical in origin to the equations of the field in terms of the Riemann tensor is a giant leap and the fact that it took Einstein three or four years to make the transition is understandable. Indeed it is astonishing that he made the transition at all. Of course one can claim that mathematical insight was needed to go from his statement about the metrical origin of gravitational forces to formulating those ideas in terms of Riemannian geometry but Einstein was not particularly well disposed to mathematical treatments and particularly geometrical way of thinking prior in his earlier years. For example when Minkowski wrote a few years after Einstein had formulated his special theory of relativity by describing the special relativity in terms of what we now call Minkowski geometry in which we associate a metric in space time which is dG squared minus dx squared minus dy squared minus dc squared and he showed that rotations in a space time with this metric is equivalent to a special relativity. Einstein's remark on Minkowski's paper was first that well we physicists show how to formulate the laws of physics and mathematicians will come along and say how much better they can do it and indeed he made the remark that Minkowski's work was Yuba Flessiger, Yuba Flessiger-Galer Minkowski, unnecessary learnedness but it is only in 1911 or 1912 that he realized the importance of this geometrical way of thinking and particularly with the aid and assistance of his friend Marcel Grossmann he learned sufficient differential geometry to come to his triumphant conclusion with regard to his field equations in 1915 but even at that time Einstein's familiarity with Riemannian geometry was not sufficiently adequate. He did not realize that the general covariance of his theory required that the field equations must leave four arbitrary functions free. Because of his misunderstanding here he first formulated his field equations by equating the rigid tensor with the energy momentum tensor but then he realized the energy momentum tensor must have its covariant divergence zero but the covariant divergence of the rigid tensor is not zero and he had to modify it to introduce what is the Einstein tensor. Now I do not wish to go into the details more but only to emphasize that the principal motive of the theory was physical insight and it was the strength of his physical insight that led him to the beauty of the formulation of the field equations in terms of Riemannian geometry. Now I want to turn around and say that why is it we believe in the general theory of relativity. Of course there has been a great deal of effort during the past two decades to confirm the predictions of general relativity but these predictions relate to very very small departures from the predictions of the Newtonian theory and in no case more than a few parts in a million. The conformation comes from the deflection of light as light traverses a gravitational field and the consequent time delay, the precession of the Perigalina Mercury and the change in period of double stars, close double stars as pulsars due to the emission of gravitational radiation but in no instance is the effect predicted more than a few parts in a million departures from Newtonian theory and in all instances it is no more than verifying the values of one or two or three parameters in expansion of the equations of general relativity in what one calls the post-Newtonian approximation. But one does not believe in a theory of which only the approximations have been confirmed. For example if you take the Dirac theory of the electron and the only conformation you had was the fine structure of ionized helium in partial experiments of a conviction would not have been as great and suppose there had been no possibility in the laboratory of obtaining energies of a million electron volts then the real experiment, the real verification of Dirac's ideas, prediction of antimatter, the creation of electron-positron pairs would not have been possible and our conviction in the theory would not have been as great. But it must be stated that in the realm of general relativity no phenomenon which requires the full nonlinear aspect of general relativity have been confirmed. Why then do we believe in it? I think of a belief in general relativity comes far more from its internal consistency and from the fact that whenever general relativity has an interface with other parts of physics it does not contradict any of them. Let me illustrate these two things in the following way. We all know that the equations of physics must be causal. Essentially what it means is that if you make a disturbance at one point the disturbance cannot be felt at another point for a time which light will take to go from one point to another. Technically one says that the equations of physics must allow an initial value formulation that is to say you give the initial data on a space like surface and you show that the only part of the space time in which the future can be predicted is that which is determined by sending out light rays from the boundary of the space time region to the point. In other words if for example you... suppose you have a space like slice then you send a light ray here and you light a ray here and it is in that region that the future is defined. Now when Einstein formulated the general theory of relativity he does not seem to have been concerned whether his equations allowed an initial value formulation and in fact to prove in spite of the non-linearity of the equations the initial value formulation is possible in general relativity was proved only in the early 40s by Lechnerovitz in France so that even though when in formulating the general theory of relativity the requirement that it satisfy the laws of causality was not included in fact it was consistent with it or let us take the notion of energy. In physics the notion of energy is of course central we define it locally and it is globally concerned. In general relativity for a variety of reasons I cannot go into you cannot define a local energy. On the other hand you should expect on physical grounds that you have an isolated matter and even if it radiates energy then globally you ought to be able to define a quantity which you could call the energy of the system and that if the energy varies it can only be because gravitational waves cross the boundary at a sufficiently large distance and that is the second point. Of course the energy of a gravitating system must include the potential energy of the field itself but the potential energy in the Newtonian theory has no lower bound by bringing two points sufficiently close together you can have an infinite negative energy but in general relativity you must expect that there is a lower bound to the energy of any gravitating system and if you take a reference with this lower bound as the origin of measuring the energy then the energy must always be positive. In other words if general relativity is to be consistent with other laws of physics you ought to be able to define for an isolated system a global meaning for its energy and you must also be able to show that the energy is positive but actually this has been the so-called positive energy conjecture for more than 16 years and only a few years ago it has proved rigorously by Ed Witten and Yoham. Now in other words then that even though Einstein formulated the theory from very simple considerations like all accelerations must be metrically in origin and putting it in the mathematical framework of Riemannian geometry it nevertheless is consistent in a way in which its originator could not have contemplated but what is even more remarkable is that general relativity does have interfaces with other branches of physics. I cannot go into the details but one can show that if you take a black hole and have the direct waves reflected and scattered by a black hole then there are certain requirements of the nature of scattering which the quantum theory requires but even though in the formulating this problem in general relativity no aspect of quantum theory is included the results one gets are entirely consistent with the requirements of the quantum theory in exactly the same way general relativity has interfaces with thermodynamics and it is possible to introduce the notion of entropy for example in the context of what one generally calls Hawking radiation. Now certainly thermodynamics must not incorporate in forming general relativity but one finds that when you find the need to include concepts from other branches of physics in consequences of general relativity then all these consequences do not contradict branches of other parts of physics and it is this consistency with physical requirements this lack of contradiction with other branches of physics which was not contemplated in its founding and it is these which gives one confidence in the theory. Well I am afraid I do not have too much time to go into the other aspect of my talk namely why is the general theory of an excellent theory well let me just make one comment if you take any new physical theory then it is characteristic of a good physical theory that it isolates a physical problem which incorporates the essential features of that theory and for which the theory gives an exact solution for the Newtonian theory of gravitation you have the solution to the Kepler problem for quantum mechanics relativistic or non relativistic you have the predictions of the energy of the hydrogen atom and in the case of the Dirac theory I suppose the Klein-Lichino formula and the Pair production. Now in the case of the general theory of relativity you can ask is there a problem which incorporates the basic concepts of general relativity in its purest form. In its purest form the general theory of relativity is a theory of space and time. Now a black hole is one whose construction is based only on the notions of space and time. A black hole is an object which divides the three dimensional space into two parts an interior part and an exterior part bound by a certain surface which one calls a horizon and the reason for calling that the horizon is that no person, no observer in the interior of the horizon can communicate with the space outside. So a black hole is defined as a solution of Einstein's vacuum equations which has a horizon which is convex and which is asymptotically flat in the sense that the space time is Minkowski at sufficiently large distances. It is a remarkable fact that these two simple requirements provide in the basis of general relativity a unique solution to the problem. A solution which has just two parameters the mass and the angular momentum. This is a solution of Kerr discovered in 1962. The point is that if you ask what a black hole solution consistent with general relativity is you find that there is only one simple solution with two parameters and all black holes which occur in nature must belong to it. And one can say the following if you see macroscopic objects then we see macroscopic objects all around us and if you want to understand them it depends upon a variety of physical theories with a variety of approximations and you understand it approximately. There is no example in macroscopic physics of an object which is described exactly and with only two parameters. In other words one could say that almost by definition the black holes are the most perfect objects in the universe because their construction requires only the notions of space and time. It is not vulgarized by any other part of physics with which we are mostly dealing with and one can go on and point out the exceptional mathematical perfectness of the theory of black holes. Einstein when he wrote his last paper his first paper announcing his field equations stated that anyone scarcely anyone who understands my theory can escape its magic. For one practitioner at least the magic of the general theory of relativity is in its harmonious mathematical character and the harmonious structure of its consequences. Thank you.
building up this theory of gravitation, he was statement, I came to realize that all the natural laws except the law of gravity could be discussed relativity. I wanted to find out the reason for framework of the special theory of relativity. solve the problem of gravity at the same time. A to the Earth. In the accelerated frame of fact. The second point is that this equality of Newton wrote the Principia, the fact that the from his second law of motion which states that experiences an acceleration in such a way that times the acceleration is equal to the force. his law of motion. It is a constant of A century later Bessel improved the accuracy to a of the objects of Einstein's theory is to principle of equivalence. Let me illustrate his there it remains exactly as it was. In other that is the starting point of his work he wanted geometrical part of the acceleration and the years in fact 10 years almost to obtain his final field equations. The transition from the that it took Einstein three or four years to make was needed to go from his statement about the that well we physicists show how to formulate the covariance of his theory required that the field but then he realized the energy momentum tensor insight and it was the strength of his physical but these predictions relate to very very small deflection of light as light traverses a expansion of the equations of general relativity in what one calls the post-Newtonian approximation. But one does not believe in a in general relativity comes far more from its internal consistency and from the fact that other parts of physics it does not contradict any causality was not included in fact it was find the need to include concepts from other aspect of my talk namely why is the general theory then it is characteristic of a good of that theory and for which the theory gives an energy of the hydrogen atom and in the case of form. In its purest form the general theory of only on the notions of space and time. A black communicate with the space outside. So a black relativity is you find that there is only one physical theories with a variety of holes are the most perfect objects in the perfectness of the theory of black holes. Einstein when he wrote his last paper his first