Subrahmanyan Chandrasekhar (1988) - The founding of general relativity and its excellence

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 20th century physics starts with the proforma 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 stable part of the education of a physicist, certainly not to the extent quantum theory has been. Perhaps on this account a great deal of my ethology has created an own Einstein’s name and the theory of relativity which he founded seventy years ago. Even great physicists are not exempt from making statements which, if not downright wrong, are at least misleading. Let me quote for example a statement by Dirac made in 1979 and on occasion celebrating Einstein’s 100th birthday. This is what he said: he was not claimed 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 the General Theory of Relativity”. I read Einstein’s statement: 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 a patent office in Bern, suddenly a thought struck me. If a man falls freely, he would not feel his weight. I was taken aback, this 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, 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 it 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 gravitational mass, an empirical fact which has been really 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 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 the subject is a body till it falls, 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 centimetre 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 can conclude that the mass of the liquid you have used is one tenth cubic centimetre. In other words then, the notion of mass is a consequence of his law of motion, it is a consequence 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 force, which you measure and you find that another piece of the same matter experiences gravitational attraction which is, say ten times more, then you say the rate 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 I have found experiments with pendula made accurately. The way he determined the quality of the inertial and the gravitational mass was simply to show that a period of a pendulum, a certain pendulum, depends only on its length and not upon the weight or the mass or 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. Essentially later Wentzel improved the accuracy to a few parts in several tens of thousands. Earlier in this century yet was shown the equality to one part at ten to the eleven. And more recently the experiments of Dicke and Braginskii have shown that they are equal to one part at ten to the minus thirteen. Now this is a very remarkable fact, the notion of mass and weight are fundamental in physics. And when one equates them by the fact of experience and this of course is basic to the Newtonian theory. Herman Wilde called it an element of magic in the Newtonian theory. And one of the objects of Einstein’s theory is to illuminate this magic. But the question of course is, you want to illuminate the magic, but how? For this Einstein developed what one might call today the principle of equivalence. Now 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, here is a lift with a rocket booster. Let us imagine that this lift is taken to a region of space which is far from any other external body. If this elevator is accelerated by a value equal to the acceleration of gravity, then the observer will find that if he drops a piece of apple or a ball, it will fall down towards the bottom with a certain acceleration. On the other hand, if the rockets are shut off and the rocket simply cools, 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. Now suppose this elevator is put in a shaft and falls freely towards the centre 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 he will find that both of them fall 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 centre of the earth. Now if the same experiment is performed with a lift which is falling freely, then as the lift approaches the centre 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 not 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. Suppose we want to realign this equation in a coordinate system which is not XYZ, but a general curvilinear coordinates. That is, instead of XYZ you change to coordinates Q1, Q2, Q3. And you can associate with the general curvilinear system in metric in the following way: The distance between two neighbouring points in Cartesian framework is the DX2 and DZ2. On the other hand, if you find the corresponding distance for general curvilinear coordinates, it will be a certain quantity each alpha beta with the two index quantity which will be functions of the coordinates times DQ Alpha, DQ Beta. For example in spherical polar coordinates it will be DR2 + R2 D Theta2 + R2Psi2Theta D Phi2. 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.Beta that is the Q..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, called the Christoffel symbols, but it doesn’t matter what they are, they are functions of the coordinate, functions of the geometry, times Q.Beta, Q.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 curvilinear coordinates, the acceleration consists of two terms. A term which is geometrical in origin which is a co-efficient 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 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 rather for its conclusion when Geiger and Marsden first showed him the result of the experiments on the large angle scattering of alpha-rays, another remark was it was as though you had fired a fifteen 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 scattering over night but it took Einstein many years, in fact ten 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 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 geometric in spacetime, which is DT2– DX2- DY 2- DC2. And he showed that rotations in a spacetime 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 “überflüssige Gelehrsamkeit” -“unnecessary learnedness”. But it was only in 1911 or 1912 that he realised the importance of this geometrical way of thinking and particularly with the aide and assistance of his friend Marcel Grossmann, he learned sufficient differential geometry to come to his triumph and 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 realise that the general co-variants 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 realised that the energy-momentum tensor must have its co-variant divergence zero, but the covariant divergence of the digitants 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 emphasise that the principle motive of the theory was a physical insight and it was the strength of this 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 that 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 millionth, the confirmation comes from the reflection of light, as light traverses a gravitational field and the consequent time delay. The procession of the perihelion of Mercury and the changing period of double stars, the 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 expression of the equations of general relativity, in what one calls the post-Newtonian approximation. But one does not believe in a theory in which only the approximations have been confirmed. For example, if you take the Dirac theory of the electron and the only confirmation you had was the fine structure of ionised helium in partial experiments, 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 anti-matter, the creation of electron-positron-paths would not have been possible. And of a 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 non-linear aspects 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 followed on another point for a time light will take it 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 spacetime in which the future can be predicted is that which is determined by sending out light rays from the boundary of the spacetime region to the point. In other words, if for example, suppose you have a space like slice, then you send a light ray here and you light a region here, 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-numerity of the equations, the initial value formulation is possible in general relativity, was proved only in the early ‘40s by Lichnerowicz in France. So that even though, when formulating the general theory of relativity, the requirement that satisfied 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 really emits 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. There’s a 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 bar to the energy of any gravitating system. And if you take a reference with this lower bar 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, yet 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 sixty years. And only a few years ago it was proved rigorously by Ed Witten and Yau. Now, in other words then that, even though Einstein formulated the theory from very simple considerations, like all accelerations must be metrical 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 Dirac waves reflected and scattered by a black hole, then there are some requirements of the nature of scattering which the quantum theory requires. But even though in 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 be incorporated in founding 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´s these which gives one confidence in the theory. Now, I'm afraid I do not have too much time to go into the other aspect of my talk namely why is the general theory an excellent theory. Well, let me just make one comment. If you take a 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 creation of formula and the pair production. Now, in the case of the general theory of relativity you get asked, 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 notion of space and time. The 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 it that “the horizon” is that no person, no observer in the interior of the horizon can communicate with the space outside. So your black hole is defined as a solution of Einstein’s vacuum equations which has a horizon, which his convex and which is asymptotically flat, in the sense that this spacetime is minkowskian 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 discovered in 1962. The point is that if you ask what a black hole solution consistent to general relativity is, you find that there’s only one simple solution, its two parameters and all black holes which occur in nature must belong to it. One can say the following: If you see macroscopic objects, then you see microscopic objects all around us, if you want to understand them it depends upon a variety of physical theories, 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 vulgarised 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.

Subrahmanyan Chandrasekhar (1988)

The founding of general relativity and its excellence

Subrahmanyan Chandrasekhar (1988)

The founding of general relativity and its excellence

Comment

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

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