2nd Intl. Conference on History of General Relativity — Session

0:00 I was supposed to arrive this morning by a special train, but instead two imposters appeared. But since we've scheduled the hour anyway, we are going to hear from a duo this morning, not just one speaker. and John Gleeman, as you said, it was, and Clark Guillermo. They will tell us about Einstein's exploration. I should tell you, I talked to John Berman, this one here, just before coming over here. He said he was very worried that question time might be a little stifled because he wouldn't be here. And so he announced that he's actually included a very major error in the paper here. and he's asking everyone to look for it very, very carefully and he says that the first person to find it should rush to the telephone and call him immediately. He has a major prize for the first correct answer. And he says the fact that it's 3 o'clock in the morning now in Pittsburgh should not return you in the least. In fact, he hopes that everyone will bring him up now. Tell him what they think of the paper. He's not about it. He doesn't know. We do that. I think we should elect John Norton to do the job. November 1950 was a very busy month for Einstein. There are four communications to the Berlin Academy, only the fourth and final of which contains what we now call Einstein's gravitational fielding wave. The penultimate paper, 18 November, mentions all three of the theory of relativity. The redshift prediction had already been verified, or so Einstein claimed, in Freundes. However, Freundes' analysis of the spectral lines of stellar sources was criticized by Seidiger. Other experimentalists, such as St. John and the Mount Wilson observatory, were unable to confirm Einstein's redshift prediction for the sun. Thus, this first classical text threatened to be a failure rather than a success. Einstein also stated

2:30 that his theory yielded a deflection value of 1.7 seconds of arc for starlight raising the sun, which was quite the previous 1911 prediction based on the principle of equivalence. Freundlich had set off into Russia in 1914, hoping to use a solar eclipse to test this earlier prediction. But perhaps fortunately for NASA in general theory, the hostilities of the First World War broke out, and Freundlich's equipment was seized, and he was briefly interned by the Russians. Einstein's third prediction was a new one, requiring deduction from a formal theory rather than the heuristic reasoning which led to the predictions of the first two effects. Einstein found that the pyrohelion advance of the planet through orbit should be this quantity here, where A is the semi-major axis of the planet and E is the eccentricity of its orbit. For the planet Mercury, the predicted advance is 43 seconds per century, which is in striking agreement with the actual value of the anomalous advance that had exercised some of the most acute minds in astronomy for over half a century. The resolution of this anomaly was the first solid triumph for Einstein's relativity theory, and after nearly three-quarters of a century of careful and detailed scrutiny, the triumph remains untarnished. Our aims are, first, to discuss the fascinating web of mathematical analysis and physical intuition that led Einstein to his explanation of Mercury's perihelion, and, second, to examine the reactions to his explanation. But first, we need to do some initial stage setting. The anomalous advance of Mercury's perihelion is a theoretical value to be derived by subtracting estimates of the perturbation of Mercury's orbit caused by other planets from the observed advance of some 570 seconds apart per century. The first realization that there was an anomaly came from the French astronomer Le Berrier in 1859, who found an anomalous advance of 38.3 seconds per century. You see all those numbers listed on the slide here. Le Vergeier was able to demonstrate with satisfaction that fiddling with the values of the planetary masses would not produce agreement between observation and Newtonian theory.

5:00 For example, increasing Venus' mass estimate by some 10% would explain away the anomaly, but such an increase would have, according to the Newtonian laws of gravitation, other consequences that contradicted observation. The next major advance in the analysis of the observation was to the American astronomer Simon Newcomb. Whereas Le Verrier based computations of secular perturbations on different mass values for the same planets in different parts of the computation, Newcomb sought a consistent set of planetary masses. He estimated masses independently of the problem of secular perturbations perturbations, by means of observations of satellites on the planet, deflection of facts and comments, and periodic perturbations on other planets. Einstein, though probably not familiar with the primary literature, was well aware of Newcombe's accomplishments. Writing to his friend Michele Besso on December 10, 1915, Einstein explained that the, quote, splendid precision, unquote, of the Piraeus events is, quote, perfectly assured from the point of view of astronomy, because the determination of the masses of the inner planets, had been made by Newcombe from the periodic perturbations and not the secular." Using his mass estimates for the planets, Newcombe in 1895 arrived at the value list of transparency for the product of Mercury's eccentricity and its anomalous perihelion advanced for centuries. That's those two figures here. A figure repeatedly cited in the literature. What is more than a little puzzling are the different values for the advance bandied about in the literature from the period immediately surrounding Einstein's period helium paper, as indicated by the list of the transparency. Jeffrey's value is understandable if the favored modern value of E equals 0.2056 is used in conjunction with Newcombe's product figure, yielding a centennial advance of 41.24 cycles he's presumably rounded off to 41 seconds. The most plausible explanation of Silverstein's value is that he confused the observed value of the anomaly with the theoretical value predicted by a general relativity. We find Drost's value of 44 seconds inexplicable. And Einstein's value of 45 seconds plus or minus 5 seconds is even more mysterious, since he gives 43 seconds as a theoretical prediction, which he presumably arrives at by using E equals 0.2056.

7:30 Nor is the 45-second figure due to a slip of the pen, for Einstein repeats it in a letter to Sommerfeld, written a week after his perihelion paper. There are similar, but less severe, discrepancies in the quoted values of the theoretical prediction of the general relativity, the most popular value being 43.03 seconds, while 42.9 and 42.95 had also cited those values you can see here. It is somewhat disconcerting to find such divergences on both theory and observation in a topic that turns on a small handful of seconds of r per century. To set the record straight, the correct theoretical value is 42.98. The last value on the sheet here. The observational... The observational estimates have remained remarkably consistent, although there's a slight upward tendency from Newcom's 41.24, to Clemence's 1947 value of 42.56, to Shapiro, et al. 1976 value of 43.11 plus or minus 0.21. More to the point of the present work, when Einstein offered his explanation of the motion of Booker-Espirian helium, Newcomb's work was generally regarded as reliable, and thus there was general agreement both that there was an anomaly, and that plus or minus 0.43 seconds, the product of the subdenial advance by the eccentricity of the orbit, was an accurate measure of the anomaly. The next section deals with the theory of the perihelion motion of Mercury. And then we have these in the end. Yes, over there. The anomaly of Mercury is perihelion, documented by Newcomb, left astronomers with two main choices. Either continue to maintain Newton's laws of motion, and this one over R-squared law of gravitational attraction, of perturbation of Mercury's orbit, or else modify Newton's second law, Earth's law of gravitation, of both. The most obvious candidates for additional sources of perturbation were solar of lateness and inter-mercureal matter.

10:00 The results of 19th century optical measurements of the photosphere of the sun were summarized in 1895 by Newcomb, and I quote, the general result is that the mean equatorial measures are slightly less than the mean of the polar measures. The differences, however, being within the probable errors of the results. I conclude that there can be no such non-symmetrical distribution of matter in the interior of the Sun as would produce the observed effect, and of course. As for intramercurial matter, the simplest hypothesis would have been an extra planet in an orbit between the Sun and Mercury. Various claims to have observed such a planet, Vulcan, were reported in the 19th century, and astronomers continued searching for the hypothetical Vulcan well into the 20th century. But by 1915, this hypothesis was not taken seriously, both because the probability was low that a Vulcan sufficient to account for the Piri-Helian anomaly could have eluded observation, and also because theorists had convinced themselves that this Vulcan would engender other anomalies. This left Seidegger's 1906 hypothesis of bands of diffused intermercural matter, which received independent observational support from the existence of zodiacal light, and which was flexible enough to hold out the promise of a consistent anomaly-free account of planetary motions in Newtonian terms. Zelliger's account will receive more attention below, but we now turn to a summary of attempts to deal with the perihelion anomaly in non-Newtonian terms. Proposed modification of Newtonian theory can be conveniently divided into two or classes. The first one is non-realistic theories. At least two modifications of Newton's 1 over R-square law received serious attention. The first initially proposed by Cleo in 1745, in connection with the moon's perigee, would add to Newton's 1 over R-square a term C over R-4. To understand this proposal, recall that one of Newton's demonstrations of his law of gravitational attraction combined the assertion that the epsides of the planets are quiescent with the proof that if the central force law differed from 1 over r squared, the epsides would rotate. This was a double-edged argument. The moon's perigee was known to rotate some 3 degrees per orbit, the obvious cause being the attraction exerted by the sun. However, when Clairot, d'Alembert, and Euler tried to calculate the influence of the sun, they could only account

12:30 for half of the observed 3 degrees of X. Newton's demonstration would then seem to imply that the departure from the 1 over R squared law was involved. The anomalous motion of Mercury's perihelion can be accounted for by inserting Clairvaux's proposed 4-4 into Newton's second law and adjusting the value of C. As Newcomb noted, however, at small distances, the 1 over R to the power 4 term would dominate, producing effects that would contradict advantage-type experiments. In a sense, Einstein's general theory of relativity revives Clairvaux's law, as we'll see you later on. Another ad hoc modification, proposed by Hall in 1894, replaced Newton's 1 over R squared by 1 over R to the power 2 plus delta. Newton found that the value of delta, listed here, which I don't care to pronounce, would account for an advantage of 42.4 seconds per century, where it is very helium. But it was difficult to believe that such an ugly law could be true, and in any case, the sitter, in 1913, showed that such a law would lead to problems with the motion of the moon's peri-gene. More radical modifications of Newtonian gravitational theory were inspired by the work of Weber, Gauss, Riemann, and Clausius on action at the distance electrodynamics. Applying their velocity-dependent force laws to gravitation gives values for Mercury's period healing advance ranging from 7 to 14 seconds per century. Gerber, in 1898 and 1902, also published the velocity-dependent force law. And the right chromatic formula was a very revealing advance, given on the transparency, with t in this formula, the orbital period of the planet, and v, the velocity of propagation of the gravitational potential. If v is identified with the velocity of light c, Gerber's formula is exactly the one that appears in Einstein's paper of 18 November. No one, however, pretended to be able to find a coherent physical foundation for Gerber's theory. The second class of theories are so-called transitional theories. If Newtonian becomes modified by adding Lorentz's mass transformation here, the result is somewhat analogous to the velocity dependent loss management in the first category. The upshot

15:00 is an advance in Mercury-Spury-Helion of seven seconds per century. In the third class, called Special Relativistic Theories, both Poincaré and Minkowski offered Lorentz invariant forms of Newton's 1 over R-squared law, using a retarded action-at-the-distance scheme. The sitter in 1911 found that Minkowski's version of his scheme gave no secular advance to Mercury-Spury-Helion, while Poincaré's gave 7.15 seconds per century. He also found that Poincaré's law could easily be generalized by multiplying by an integral power of a certain factor, in which case the secular advance would be n times 7.15 seconds. Choosing n equals 6 gives 42.9 seconds. However, De Sitter mentioned the generalization only in passing, and neither he nor anyone else thought n equals 6 as an explanation of the anomaly, at least this until Silverstein in 1917, rediscovered a version of the Sitter's finding. The fourth category, called post-special relativistic theories. Einstein's first formal theory of gravitation, dated from 1912, relied upon the notion of a variable speed of light, a notion that from the modern perspective either involves a coordinate-dependent effect or else implicitly involves a biometric approach, with a curved space diametric on top of a flat background metric. On one analysis, this theory gives a perihelion advance for Mercury of 28 seconds per century. In the same year, Max Abraham also concocted two theories involving a variable speed of light. The first gives an advance of 14.52 seconds. The second yields three seconds. Not strong likewise over the pair of theories, both of which yielded a retrogression for Mercury's perihelion. prediction was worked out in Mee's theory, but presumably it does not give a secular advance since it posits a Newtonian attraction. Finally, the Einstein-Grossman theory predicts 18 seconds per century. Though brief to the point of superficiality, this summary serves to establish that prior to Einstein's general theory, no modification of Newtonian theory, which did not contemplate additional sources of perturbation, offered a non-problematic resolution of the Pirehelian anomaly. may I make a brief comment I mentioned that the round table

17:30 that both Benin Trante and Berkhoff and Kustenheimer had generalized the possibility of reducing theories that you can match the advanced period with either one we are lucky that this was not done earlier it's not a problem with the other test with the series the test of the deviation of flour. So the, with Birkhoff theory, you get it is a problem. I mean, that in all of them, but of course it's... Can I ask, when you say that they can match it, does this mean that you can match it by setting a parameter to justify, so there's an ad-hoc flavor to it then? Right. Yeah, okay. Einstein's paper of 18 November marks the first occasion on which the Perylian problem appears in Einstein's published work. The first known mention occurs in 1907, the year in which he began to think seriously about the relativistic theory of gravitation. In a letter to Konrad Habitsch, he writes, I quote, I'm busy on the relativistic theory of the gravitational law with which I hope to account for the still unexplained secular change of the Perylian motion of Mercury. So far, I have not managed to succeed, end quote. but as far as we are aware there is no other mention of the period helium problem in Einstein's correspondence prior to November 1950 two questions thus naturally suggest themselves, the first one is why did Einstein neglect the period helium problem between 1907 and 1915, the second one is why did the problem suddenly come to the fore in November The first question may be general, since of the seven post-special relativistic theories mentioned in the preceding section, a formula for the perihelion advance of a planet is worked out in only one case by the primary author, Nordstrom in the case of his second theory. And in that case, no numerical value for the predicted perihelion shift of world theory is given. A cynical explanation is suggested by the fact that none of these theories gives anything like the correct value. is perhaps healthy, a much more plausible explanation has been offered by Rose Bear. Namely, while the advance of the perihelion of Mercury was an anomaly for Newtonian gravitational theory, it was not generally regarded as an effect that had to be explained purely on the basis of new principles, since Sadeger's hypothesis was widely accepted as offering the means of a satisfactory revolution.

20:00 To support this interpretation, it suffices to quote from a 1930 article, 13 article of the Sitter, who was the leading authority on the matter. is proposing that Zelliger's hypothesis be adopted as a working hypothesis, and he writes a quote, can we then consider the problem as finally solved, meaning solved by Zelliger? I think not. One more step remains to be done. The fate of Hall's hypothesis should be remembered. It is true that Zelliger's explanation differs from Hall's hypothesis in being vastly less hypothetical. In fact, it may be considered as nothing more nor less than a determination of mass of a material body whose existence is known beforehand. But, taking this point of view, we cannot consider that determination is final before it has been certain that it is not in contradiction with other possible determinations. In other words, before it has been verified that the attraction of the zodiacal masses does not give rise to other effects, which might be in contradiction with observation. End of quote. But if the suggested answer to the first question is correct, then the second question poses itself again, this time with more force. If Zelliger's hypothesis was generally accepted as the working hypothesis, why did Einstein concern himself with the Piraeus-Elean problem in November 1915? No doubt, part of the answer lies in Einstein's healthy disrespect for generally accepted wisdom. Nor was he fond of Zelliger, who had attacked Freud's in very nasty terms. But that part of the answer does not speak for the timing factor. To deal with that factor, we need to remind ourselves that for the past two years, Einstein had been wandering in the wilderness, quite lost in the denser. By means of a mistaken but ingenious and ultimately profound argument, Einstein had managed to convince himself that suitable gravitational field equations would not be generally covariant. According to a letter to Sommerfeld, dated November 28, 1915, Einstein's dissatisfaction with the results of his experimentation with non-generally covariant equations was connected with the peri-helion problem. One of the three reasons he gives for abandoning the Einstein-Grossman theory is that, I quote, the motion of the perihelion of mercury yields 18 seconds instead of 45 seconds. One should perhaps not attach too much weight to this retrospective assessment made in the flight of the success of the deduction of the 43 seconds, especially since there is no evidence that Einstein had tried to calculate the perihelion advance for the Einstein-Grossman theory, and the 18-second figure he quotes of Sommerfeld

22:30 taken from Drost's poet's calculation. Of course, once Einstein had abandoned his previous efforts, there would have been an understandable desire to convince himself that the new approach he was exploring towards the end of 1915 was not also heading up a blind alley. Meeting the challenge he had set for himself in 1907, accounting for this still unexplained secular change in the period alien motion of Mercury, would restore confidence. Meeting this challenge would also serve to establish priority for the still-developing theory, a concern one can reasonably attribute to Einstein since he knew that Hilbert was also at work on the same problem. Another possible factor relevant to the timing issue was the appearance of a paper by Freudlich in 1915, attacking Zaliger's zodiacal matter hypothesis. Freudlich's paper is cited as a quote, noteworthy article, and quote, in a footnote in Einstein's very alien paper. In a letter to Sommerfeld, written in February 1960, Einstein gives the impression that he was not much influenced by frontal expertise, which he likened to, and I quote, pushing against an open door. What this intriguing phrase means is not made perlucid by the context, but a natural interpretation suggests itself in an analogy with the ether problem. Ten years earlier, in 1905, Einstein would surely have regarded the strategy of attacking the luminiferous aether by objecting to aspects of its supposed physical composition as pushing against an open door, for in his mind, the way was open to show that the aether was superfluous by giving a principle theory of electromagnetism that had no need of an ethereal medium. Similarly, the principle theory of gravitation would render the hypothesized zodiacal matter hyperfluous, at least insofar as it affected the helium problem. But if this is the correct reading, it is difficult to see how Einstein could have been confident that the door was opened until he had a 40-feet second in hand. By the same token, if the other two hypothetical motivations of restoring confidence in establishing priority were to be served, it was essential to predict an advance close to the observed value. In the next section we will see another equally compelling reason why Einstein had to come close to the accepted value of the advance. In his communication of 11 November 1915, Einstein had taken the field equations given

25:00 on the sheet as Formula 1, where GIA is not the Einstein tensor but the Ricci tensor. For the exterior field of a massive body, such as the sun, one reduces to two, which coincides with the implication of the final Einstein field equation. As Einstein had shown in his communication of November 4, the Ricci tensor can be split into two parts, denoted by Rij and Sij, where R and S are given by these expressions. Now, Rij is not a tensor, but it transforms like a tensor under coordinate transformations of determinant unity, which is through a coordinate system satisfying the coordinate condition Since we also have this relation here, it follows that in a coordinate system satisfying the coordinate condition 4, Sij equals 0. that's the November 11th equations reduce to equations by Rij equals 0 or I should say Rij equals K times the but for the exterior solutions the right and side becomes 0 now 4 and 5 equations 4 and 5 I can use in his explanation of work very early in motion the unphysical coordinate condition for not only simplified the field equations but it was stirred to other advantages that would become happen to be low now what I said was he saw the space diametric GIJ satisfying the coordinate the condition core, just on the previous slide, and also the field equation 5, the field equation at ij equals 0, and also the following conditions listed here as C1 to C4, namely space-time metric is stationary, it is time orthogonal, it is spherically symmetric, and it is asymptotically ideally one would like to find to first find an exact solution to the field equations

27:30 or rather a family of exact solutions, parametrized by the value of the central mass, and then to demonstrate that this is the unique family of solutions satisfying condition C. Einstein did not attempt to produce an exact solution, and he put aside the uniqueness problem with the remark, I quote I am satisfied, for the time being, with deriving here a solution without discussing the question whether the solution might be unique Two difficulties are generated by the two ways in which Einstein's treatment falls short of the ideal. In the first place, the periodic summary has not been resolved if the field equations admit for the same value of the central mass different solutions which satisfy condition C, and yet which give different periodian predictions. Now assume, as it's actually the case, that there is a unique solution, or family of solutions, parametrized by the mass of the sun. But of course, approximations to the unique solution are not unique, and in November 1915 Einstein did not know what the unique solution was, and therefore could not know how well an approximate solution represented the exact solution. Consequently, they had no sure way to know how much faith to put in the Piri-Helian advanced value derived from some approximate solution, but if the approximate solution yields the magic 43 seconds per century, one simply ceases to worry since a stupendous stroke of ill fortune would seem to be needed to make the approximate solution yield exactly the correct value, and yet disagree strongly with the exact solution. And in the brilliance of the magic 43 seconds, the uniqueness problem recedes into the background, where it can be left until a better opportunity arises to tackle it. We can only guess at the psychological drama of the moment when Einstein inserted the numbers into his spirulence formula and had the 43 seconds pop out. But some sense of the intensity of the drama Hawker's report that Einstein suffered heart palpitations, and by De Haas' report that Einstein had the feeling that I thought something actually snapped in him, and it broke. Little wonder then that the psychological resolution of the above problems sufficed for the time being. Now, to the more technical details of finding the possible solution. In spherical coordinates, the most general line element satisfying Einstein's condition C C1 to, or rather C1 to C3

30:00 is given by expression 6 here where A, B, C, and D are functions of R alone It can be shown that without loss of generality we can make C equals D equals 1 leaving only A of R and B of R to be determined Note, however, that in these coordinates, the determinant of the metric equals, is going to be A, B, R to the power of 4 sine square of theta, making it impossible to satisfy Einstein's coordinate condition, the determinant of G equals 1, or equals minus 1, I should say. So, to satisfy four, we have to shift to Cartesian coordinates in which line element becomes expression seven. From his previous investigation of the redshift, Einstein knew that he wanted the R dependence in A of R to go as alpha over R. with alpha equal to 2gm and this should also be a c squared in the denominator combining this with condition c4 which is the condition of metric being asymptotically Minkowskian, indicates that the choice a of r equals 1 minus alpha over r given in equation 8 there indicates that choice Condition A is also confirmed by the fact that it gives the correct Newtonian slow motion approximation. Now impose the coordinate condition 4, the determinant of G equals minus 1. With A of R fixed by 8 here, 4, condition 4, forces out the first order in alpha over R, this result for B. Now the upshot is given on the next line, where the line element is given using the results derived from the previous line. And to make this result more perspicuous, we transform back to a spherical coordinates, which equals 11 for the line element. And this is the first order, the first order A over R approximation, alpha over R approximation to the exact swatch shield line element given as expression 12 of this transparency.

32:30 Now, it should be noted that the field equations do not play any role in this construction. Einstein's guess as to what A of R or G44 should be together with this coordinate condition leads directly to expression 10, line element 10. gives no details in this paper, we supposed to theorize at 10 by similar reasoning. After presenting 10, Einstein then goes on to note that the field equations are, and I quote, fulfilled in the first approximation. Since Einstein's all derivation of the geodesic equation is fraught with uncertainties to be discussed below, we first present a clean derivation of the equation of motion. On the next slide, So the point is to derive the equation of motion, of course the derivation is what you often call a lean derivation, and then we will go and look at Einstein's own derivation. The geodesics, time-wise geodesics are derived using the variational principle 13, where the Dottie-Nose differentiation with respect to proper time S. Since the motion takes place in the plane, we can choose theta equals pi over 2, in which case we have relations 14, 15, and 16. Combining these results and introducing a new notation, r prime equals dr by d5 and u equals 1 over r, we get expression 17. and then by neglecting the U squared on the left hand side of 17 we get equation 18 now this is the exact equation of motion for the Schwarzschild line element apart from the values of the constant it is also the equation of motion Einstein derives in his period helium paper so working with Einstein's approximate solution and making the natural approximations yields exactly the right result however There's no reason to think that such happy results will emerge in all coordinate systems. To emphasize the point, rewrite the structure of the line element in isotropic coordinates, which are arguably the most natural coordinates to use to express the condition of spherical symmetry. And on the next transparency, we see the form of the line element in these coordinates to first order in alpha over R.

35:00 That's expression 19. What is led naturally to 19 by solving the linearized Einstein field equations for a static source. But 19 takes a null-peri-helion shift. Writing in the philosophical magazine in 1920, Professor Anderson of University College concluded from his analysis of motion in Einstein's general theory that, and I quote, Mercury, unfortunately, is left with the advance of his peri-helion unexplained, end of quote. Anderson's mistake lay in trying to draw conclusions from an unfortunate approximation similar to the one above, or the one on the transparency. It is worth mentioning parenthetically that in an otherwise worthless paper, Anderson gives what is perhaps the first prediction of black hole formation in general relativity. I-Fan's coordinate condition, the determinant of g equals minus 1, blocks the particular a form of disastrous pseudo-result following from 19. Since 19, even when rewritten in Cartesian coordinates, violates Einstein's coordinate conditions. But the existence of approximations that give no clear helium shift serves to bring into question all approximations. And the unphysical coordinate condition, while simplifying expedients, is no asset to such Now we turn to Einstein's own derivation of the equation, the equation's motion. The starting point for Einstein was a geodesic equation given as equation 27 on the next His first move was to calculate the gammas, the Christoffel symbols of the second kind, to what he calls first approximation. It is easily checked that Einstein's approximation is given by the exact values of the Christoffel with the result that the first approximation is given by equations 28A and 28B. Now, inserting those approximations into the equation of motion and making a slow motion approximation gives us 29 as equations of motion, which are just the Newton equations of motion if one takes this approximation, S equals T, and alpha equals 2gm over C squared. From 29, then, it follows. The 30a and 30b follow, constants of motion,

37:30 where phi of n and b squared are given by the expression of the last line of transparency. action. On the next slide, we go on with Einstein's procedure to evaluate the geodesic equations to the next approximation, which he did by means of what one might call ingenious bootstrapping procedures. In the first place, from I equals 4 in the original equation of way and also use the first approximation or gammas he found correct first-order expression 32 using few equation 33 and inserting the first approximation values for for gamma on the right-hand side solving for gamma sigma map or four, we find expression 34. So this is found from 33. And this, in its turn, is inserted into the original equation of motion together with equations 32 and 33. Now, all 35, the second approximation for the equation of motion. Now from this equation, as I can know, this follows the exact validity of the area law. Using the area law in the Newtonian relations 30A and 31A, which are the two, well, let me just go back to the previous one. This is 3A, and 318 is essentially this equation. using those we find eventually finally these are the six year this is all of our what we want except for your factor to a

40:00 this factor that factor can be going to be suppressed by rescaling s and also redefining C, capital C. And this then leads to the next slide, equation 37, or alternatively equation 38, where the right-hand side of the equation of the motion is written as the gradient of potential. And this is particularly the letter expression 38 shows that Einstein's general theory revives in a sense the hypothesis. The sense is a rather attenuated one, however, since the capital C in this equation is not a universal constant, but depends on the orbit. Equations 37 and 38 imply the relativistic energy relation 39, given here and now we can eliminate dS from 39 in favor of d5 by means of the Aria law and that then finally gives expression 40 where s before u equals 1 over r now it is from equation 40 that Einstein derives and he does that in the following way next slide Einstein's method starts from the first order equation from the equation 40, which upon integration yields equation 44, we find the angle between the radius vectors from the sun to the planet, at the time superior helium and alpha 1 and alpha 2 are the roots of the cubic equation written down here This cubic equation corresponds to the classical quadratic equation that's written down here. The upshot is an enhanced per orbit of this amount. The second expression is derived from the first one by using Kepler's third law. now the next slide in the course of the derivation of the equation of motion Einstein rescaled the proper time and redefined the constant C as we have seen

42:30 this maneuver it's summarized here again, this is the original equation in which the capital A occurs, and by rescaling and redefining C we end up with equation 37 This maneuver was cited by Charles Lane Poore in 1930 at the opening wedge of his attack on Einstein's derivation. Poore, who was a professor of astronomy at Columbia University and former head of the astronomy department at Johns Hopkins, argued that if the unit of time is changed, then there must be a corresponding change in the unit of mass, and that as a result of the latter change, the constant alpha becomes, in what he called relativity units, a constant which you might call alpha 1, and which equals alpha times this expression between brackets here. So, according to Poore's viewpoint, Einstein's equation of motion reduces to a Newtonian equation, just to retain just this first term, with alpha replaced by alpha 1. As S. Coore says, this is an equation, I quote, whose gravitational solution is identically as that of Newton, a fixed ellipse, end of quote. Poore concluded that, and I quote again, the so-called relativity rotation of planetary orbits is only a mathematical illusion. The Newtonian law has not been abolished. There is no Einsteinian law of gravitation, end of quote. That the astronomical Nagysten should publish just an article is a sad commentary on the politics of German science in 1930. Although Poore's critique was worthless, it did focus on the tender point in Einstein's derivation. Einstein's equation 36 and the rescale 37, although those are equations of motion, both imply that the first order equation of motion is not equation 40. Maybe we should go back to 40 for a second. It's not equation 40, this one, but equation 40 prime, oh it's only here, I'm sorry, the form of the Piri-Helian shift entailed by 40 prime is the same as for 40, but those roots of 40 prime that correspond to Piri-Helian and F-Helian are not the same as those of 40, and the Newtonian equation This problem can be resolved in the process of clearing up another one.

45:00 One can wonder under what conditions Einstein's energy relation is compatible to first order with the exact relations 14 to 16 for his line element. So, here we have, this is what is called the energy relation, it's one of those constants of motion again. And the question is, under what conditions is this relation compatible with the exact here, 14 to 60 for the line element. Now, the answer is written down here is where minus one must be equal to two A. Now, since in effect, Einstein takes E equal to one, A must be zero. Whatever cost one might have about the dazzling series of approximations and bootstrap maneuvers used in Einstein's derivation, were shortly to be swept away with Schwarzschild producing the exact solution to the field equations and produced it in that it was the most general solution satisfying Einstein's condition C, the four conditions that I missed on the previous transparency. Droste was CST2. Just a second. An unproblematic derivation of the perinealian shift was now possible, which Schwarzschild duly supplied. Droste's identical solution appeared in print after Schwarzschild, but since Droste proceeded independently, the solution should rightly be called the Schwarzschild-Droste solution. Now, the next section deals with reactions to the reaction of Gerber's theory, which was republished in 1917 in the Annale de Physique. Outside Germany there was no coordinated anti-relativity campaign, but Charles Lane Boer waged his own campaign, charging that Einstein's derivation of the pyrihelion advance was incoherent, and that the Solian gravitation, together with the form of Zeyniger's hypothesis, would suffice. Second, challenges to the data. Grossman, 1921, argued that the re-analysis of the astronomical data used by NuCom believes the residual advance for Mercury's pyrihelion of between 29 seconds and 38 seconds per century.

47:30 Wichert in 1921 said the anomalous advance of 34 seconds per century. Von Bleich in 1923 took the discrepancy between Einstein's prediction and the figures of Brosnan and Wichert to raise doubts about the political general theory of relativity. Third category, misunderstandings. As already mentioned, Anderson in 1920 claims that no theory-helienship is predicted by Einstein's theory. His error was quickly discovered, and Anderson himself published a correction. 21 complained that while the perihelion had found bien été obtenu à propos de la théorie de la relativité, mais qu'il n'en est pas une conséquence et ne constitue même pas un adouement en sa faveur. This complaint, which sounds better in his native French, not in my non-native French, but in English, is too uninteresting to discuss here. A misunderstanding with more serious consequences was spotted by Alvar Gullstam, professor of at Uppsala, and the recipient of the 1911 Nobel Prize in Physiology and Medicine. Gullstrand claimed that the Schwarzschild metric was not the unique static solution of Einstein's field equations for a central mass, and that the general solution contained, in addition to alpha, another parameter beta, which affects the value of the perihelion shift. As a result, Gullstrand thought that Einstein's perihelion accumulation was merely an artifact of the coordinates system of Einstein and Freud. Under pressure from Kretschmann, Gullstrand was forced to retreat, in 1921 report to the Nobel Committee for Physics, giving a negative opinion about the general theory of relativity. This is undoubtedly part of the reason why Einstein's Nobel Award does not mention his work on gravitation, and was given, and I quote, for his services to theoretical physics, and especially his discovery of the law of the possible electric effect. The next category is a category of alternative theories. This category can be subdivided the uninteresting and the relatively more interesting. In the former belongs Wichelt's 1916 attempt to retain the ether, and to explain the perihelion motion on the basis of an electromagnetic theory of gravitation. In the latter category belongs two theories of Silberstein. The first was presented in 1916 when Einstein's redshift prediction was in doubt, and before the British Eclipse expeditions verified the light-bedding prediction. In these circumstances, Silverstein thought it worthwhile to investigate how much of the anomalous advance could be accounted for on the basis of Einstein's old theory of relativity.

50:00 Silverstein proceeded to rediscover a version of the Citus 1911 result, namely that by introducing a factor of psi to the power n, where psi equals square root of 1 minus 3 squared over c squared, into the special relativistic force law, the Piri-Helian shift formula becomes this expression here. No, the first one. If the entire excess of mercury spherohelium is to be attributed to the sum, then n equals Sierwestein confessed that he did not know why the value of n is just 6, but he added in self-defense, and I quote, as little do we know why the exponent of r in Newton's law is just or exceedingly near 2 minus 2, and I quote, and besides, quote, such a naturalistic method of improving the Newton's law of gravitation seems a great deal safer than those based on fantastic constructions or RAS generalizations. In 1918, Silberstein published his own theory of gravitation, which he called, I quote, general relativity without the equivalence hypothesis, I quote. By this he meant that the G4-4 component, which controls the redshift, is a constant. Silberstein wrote that even the one, I quote, conspicuous and fascinating success, end of the deduction of the 43 seconds, is stated by the fact that the secular motion of the Pyregelian is, I quote, most vitally conditioned by G44, which, to everybody's regret, has thus discredited itself at the Mount Wilson Observatory. Silverstein's all-alternative theory, based on the postulate that space-time has a fixed constant curvature, gives a secular a perihelian motion of this expression, given with this expression here, that is, a retrograde motion of one-sixth of Einstein's value. This feature was not regarded by Silverstein as a defectivist theory, since he was able to refer to the forthcoming work by Jeffries, which attempted to show that a version of Zaliga's hypothesis would bring Silverstein's predictions into harmony with the observed secular motions of the inner planet. In the following year, however, Jeffy has abandoned his daily arising. Now we get to the fifth category of objections.

52:30 Objections to the completeness of Einstein's explanation. The Italian paradoxer Gurali Forti, in 1922 and 1923, complained that Einstein's derivation did not really explain Mercury's perihelion motion because it did not contain an account of the perturbations of the other planets, showing that these would add up to the 570-some seconds of arc per century. Einstein had shown that general relativity yields Newtonian equations of motion in the weak field slow motion approximation only for the Schwarzschild metric, whereas the relevant metric for computing, say, the perturbation of Mercury's orbit caused by Jupiter certainly will not be a Schwarzschild form. reason, Burali-Forti claimed that as a result, the Einstein value of the Jupiter's perturbation will be different from the Newtonian value, leading to a discord between the general theory of relativity and the observed 570 seconds of arcs per century. If Burali-Forti had wanted to create a paradox, he could have argued as follows. In the case of the Sparsio metric, where there is a stationary, irrotational time-like backwards view, the notion of the Pirehelian shift can be given an invariant meaning, that general relativistic explanation of the observed 570 seconds of art of shift, stationarity, and the other nice features will presumably be lost, and it thus becomes difficult to say precisely what the perihelion shift means. Positive reactions to Einstein's resolution of the perihelion anomaly were swift in coming, and they came from influential sources. The sitter declared, I quote, Zelliger's explanation of the anomalous motion of the theory of healing of mercury by the attraction of Navigot's matter in the neighborhood of the sun now becomes superfluous. This declaration was contained in a lengthy three-part review article detailing the principles and consequences of Einstein's general theory. Appearing in the monthly notices of the Royal Astronomical Society, the article served during the war years as the chief source of knowledge of Einstein's theory for scientists in England. who was to become the most effective of the early champions of general relativity. Harold Jeffries did not give up on Zellig's hypothesis until 1919, after Eddington had reported the verification of Einstein's light deflection prediction. Thus ended the series of Zellig arising in England. Now, we come to the conclusion.

55:00 In the decades following 1915, the solar redshift measurement stubbornly refused... I may omit the conclusion if we're short of time, but I think I'd like to read it anyway. In the decades following 1950, the solar redshift measurement stubbornly refused to conform to Einstein's measured by several eclipse expeditions, exceeded the theoretical value. In this context, the resolution of the Peryllium anomaly served as the main observational anchor of the general theory. But this anchor has always been a potential Achilles heel. To render superfluous Zedega's hypothesis, it was important that Einstein obtained the missing 43 seconds. But the fact that the theory does give the 43 seconds leaves no room for maneuver if additional sources of perturbation of Mercury's Peryllium are found. What was stunning about the explanation of the 43 seconds is that it was achieved without the leeway of any adjustable parameter, but it is exactly this feature of the theory that makes it vulnerable. In the 1960s, Dickey tried to pierce this Achilles heel, claiming that optical measurements revealed a solar of lateness that would account for 3 to 5 seconds of the advance, and would thus throw the general relativistic prediction into doubt. The controversy that ensued had no clean resolution, but insofar as the consensus developed, it was in favor of Einstein. Recently, however, the controversy over solar lateness has been rekindled by implications for the internal motion of the sun to offer observations of its five-minute oscillations. At the same time, the other classical tests have been perfected and stand solidly behind Einstein's theory, and new tests, such as the radar delay measurements, together with the deeper appreciation of the range of possible alternative theories have greatly strengthened the case that Einstein's theory of gravitation will prove to be as durable as Newton's. It would thus be ironic indeed if the perihelion problem were to prove to be an Achilles heel. While claiming no powers of prognostication in this matter, we must state our opinion that such an irony would be most unseen. Thank you. I think we have some time for questions, but I don't know how these people will...

57:30 First of all, I want to reaffirm that Drost was 16 in print. Second, Shahziz wrote a two-volume book on la relativité générale de la mécanique celeste, and he is a celestial mechanician and he most people seem to forget about which you rightly noted that the shift really is 570 and not 43 and Shazi said in essence to put it in the American terms you should be happy if you have the same ballpark it is really meaningless to argue about a few seconds plus or minus well I don't know That's the mistake that I should talk about, but there was definitely a mistake, even in the isotropic coordinate system, you have an effect. I mean, because after all it's a coordinate independent effect, and if you do the same derivations with arbitrary parameters in the metric, which can be fixed after you fix your coordinate system and you get an expression and it turns out it's the same I don't doubt this for a moment in fact Don and I spent a considerable amount of time trying to figure out how best to present that result. I don't in fact know why it's not in there because perhaps it's because he couldn't decide which was the best way to see it. Is there any simple way of saying exactly what goes wrong. I have my own idea. Certainly, I mean, if you plug in the linearized metric not as in the first T44 as 1 plus 1 over R, M over R, but as a constant numerical factor and in the other metric component you add another numerical factor. You do the whole derivation and you add two, three numerical factors in the end result. And then, depending on your coordinate systems, these coefficients They combine it in a certain way which miraculously, not miraculously, because after all it's an observable one, not dependent on the garden system, turn off to give the same value. From all the garden systems I've tried so far.

1:00:00 And this is in the literature, by the way, I've forgotten, it's in some of the summer schools, I don't know if it was Dicky or somebody else who wrote down this formula first. Yes, I wanted to point out the same mistake as Saugune. I think it's an understandable problem, and it's amazing to understand it. But I wanted to point out that Bertotti, as far as I know, wrote the first formula in the coordinate independence of the form. In 1962, yes, in 1962. I'm done. There's no problem about this, because it's an angular. Yeah. I just wanted to mention that, just a second, this Plavansky, I don't remember when, in the late 50s or early 60s, worried about this question of what it depended to be effective. And therefore, he gave a physically-invariant definition of a very advanced... Who had Plavansky? He imagined that you take light rays that project the motion onto a plane at infinity, and this is obviously an effect that's going to be totally independent of the forces, and they showed you get the usual per-median projection effect that way. So there is a one-minute independent plane of the plane, but it must be the same result, no matter what questions you use. Could you know that that exists? They should probably include that result in their survey. I'm sure there are other references I haven't I don't have it here Well, I think John would appreciate it. I was struck by the one in the early section of the paper entitled, Einstein's neglect of the problem from 1907 to 1915. It seems like it's almost a pseudo-problem that has been created there, given we've got so little material surviving the way of correspondence in this period to argue that there's a neglect of the problem from its not being mentioned in correspondence in the period. It's really a good invention problem that isn't really there. And I think it would be helpful to rephrase the nation.

1:02:30 There are many public papers that he doesn't mention. Yeah, but again, that doesn't mean that he's not thinking about the problem, that it's not a So it's not the main preoccupation. I'm not sure if you carefully read what your office write, that's not what they mean. There's no friend that's mentioned. I don't know the exact wording, but I seem to... I have a feeling that they don't suggest that he didn't think about it. So I would have to look at the frame. Mr. Maul. There's one more. Colster, and I think tonight was the Nobel Prize. I think actually it's more serious, but that special relativity wasn't mentioned, and I believe in the introduction by the Nobel Academy, they know certain problems with prototype philosophers against the theory, and I think around that time, in 1921, there was a philosophy that was on the end of the works, and the first began I just want to mention the point which has been made by someone who could have been here, possibly. Thank you. Germans? In 1922. The Germans were using our federal cultural investments. Yes, okay, but I mean, the investments were, well, that's what I think. But from what time did you see? Yes, but they were.

1:05:00 comment at the back of the room. On this issue of the Nobel Prize, to Einstein, doesn't it need to be put into the context what for the most of the Swedish I've got to remember, Swedish scientific community and awarding these prizes and that has been brought out that there has been a very strong experimental position in the Sweden's physical community and around this time there is a certain shift of tourism. So to only look at the award of an overpriced client and for this or that in a context extraneous to Sweden was thus to read to experts on how it would also be to create a suitable problem because it doesn't take and the internal politics installed. Interesting point, you heard that one mentioned before. Is there anything else here? Well, just to tell that it's a political in the archive of Sylvester in Sweden, between Gülström, Orzen, and other people on Kretschmann about that, it's all the literature. You may start to my institute on Detroit. So, one last comment. Let's thank our pseudo-speakers. Thank you. Thank you.