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An Anomaly In Translation |
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The agreement between general relativity and the precession of Mercury’s orbit was, and remains, one of the strongest confirmations of Einstein’s theory because, of all the classical tests, it alone is sensitive to the second-order in m/r. The equivalence principle by itself strongly suggests that gravity can be modeled by a metrical theory of spacetime (in which particles follow extremal paths), but it does not necessarily single out Einstein’s field equations as the laws governing the metric. In general we can only say that the diagonalized coefficients of the metric in the vicinity of a spherically symmetrical gravitating body should be expressible (at least in the weak field) as power series in m/r. In terms of the usual Schwarzschild coordinates r and t the conventional way of writing this generalized metric is |
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where |
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where a, b and g are constants. The constant a is directly measurable in terms of the gravitational acceleration experienced by a static object in the weak field limit, and since we define the mass of an object on this basis, we effectively define a = 1 in any theory that satisfied the equivalence principle. The other two parameters, b and g, are dependent on the field equations of whatever theory of gravity we choose. Einstein’s field equations of general relativity give these constants the values b = g = 1, but in other metrical theories of gravity these constants have different values. |
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Of the three classical tests of general relativity, the gravitational redshift depends only on gtt and can only be evaluated up to the first order, so it really verifies only the fact that a = 1, which is to say, it verifies only the equivalence principle. Needless to say, this is an important verification, but it doesn’t single out Einstein’s field equations as opposed to the field equations of other possible metric theories. The second classical test was the deflection of starlight grazing the Sun during a solar eclipse, and it can be shown that this test verifies not only a = 1, it also depends on grr up to the first order, so it verifies g = 1 as well. Likewise the “Shapiro test” based on the time delay of radar echoes from the inner planets allows us to evaluate a and g, but none of these tests enable us to evaluate the gtt coefficient to the second order, i.e., they do not constrain the value of b. |
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However, according to the general metrical theory, the precession per revolution of an orbit with semi-latus rectum L is 6pm/L times the factor (2 – b + 2g)/3. Since the precession of Mercury’s orbit is in close agreement with the value 6pm/L, and since we can determine g = 1 by other means, we can conclude that b = 1, consistent with Einstein’s field equations. Thus the perihelion observations are by far the strongest confirmation we have of the correctness of general relativity. |
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Because of the importance of this observational verification, it is of interest to know whether, or how much, Einstein was influenced or guided by it in the formulation of general relativity. Throughout the years from 1908 to 1914 while he was working on the theory, he often assessed the redshift and the starlight deflection predictions of his current theory, but he never mentioned (in print) the precession of Mercury. But when he presented his completed theory late in 1915 the agreement with Mercury’s precession was cited prominently as the crowning glory. |
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Actually we know Einstein was quite conscious of the Mercury precession anomaly throughout the years when he was developing general relativity. For one thing, we know he was an avid reader of the works of Henri Poincare, who wrote about the anomaly in his 1908 book Science and Method. In the section entitled “The New Mechanics and Astronomy” he notes that for a theory of gravity with a velocity dependent potential (along the lines suggested by Weber for electromagnetism) |
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there would result, in the perihelion of Mercury, a secular variation of 14” [seconds of arc per century], in the same direction as that which has been observed and not explained, but smaller, since the latter is 38”. |
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He goes on to say that Lorentz’s theory of relativity (and therefore also Einstein’s special relativity, although Poincare never mentioned Einstein in connection with relativity) predicts an advance of 7” seconds of arc per century for Mercury’s perihelion. He concludes that |
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This cannot be regarded as an argument in favor of the new dynamics, since we still have to seek another explanation of the greater part of the anomaly connected with Mercury, but still less can it be regarded as an argument against it. |
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This is interesting because it clearly shows that, at least in Poincare’s mind, there was an anomaly connected with Mercury, and moreover that this anomaly was roughly 38”. According to some accounts (e.g., Roseveare) there was no pressing anomaly perceived at this time, because Seeliger’s hypothesis (1906) of a solar corona was thought to be adequate to account for the extra precession of Mercury’s orbit. Roseveare says |
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I think that the reason for Nordstrom’s attitude and the general neglect of the perihelion of Mercury as an anomaly to be explained by any new gravitational theory [in 1913] was that Seeliger’s hypothesis was being taken very seriously… Since it was felt by both Einstein and Nordstrom that no empirical argument existed beyond the light deflection predictions, one can only assume that the perihelion motion of Mercury was not considered to be anomalous and that the prevailing hypothesis explaining it, Seelinger’s hypothesis, was valid. |
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According to this account, the paper written by Einstein’s friend Freundlich in February 1915 arguing against Seeliger’s hypothesis was motivated by a desire to restore the precession of Mercury’s orbit to the status of an anomaly so that it could be used as a test of gravitational theories. This seems like rather odd reasoning, and Roseveare himself admits that his argument is undermined by Einstein’s later comment (in 1916) that Freundlich’s attack on Seeliger’s hypothesis was “forcing an open door”, which clearly implies that Einstein (like Poincare) did not take Seeliger’s hypothesis seriously. |
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Admittedly in 1906 (the year Seeliger published his hypothesis), Poincare stated that although the extra precession in Mercury’s orbit was at that time the most grave discordance known for Newton’s laws, he recognized that it could be explained by a ring of matter around the Sun. However, he may just have been acknowledging the latest hypothesis. In his review of astronomy and the new mechanics just two years later (quoted above) he made no mention of Seeliger’s hypothesis or a circumsolar ring as a possible explanation for the anomaly, and he clearly treated the anomaly as a fact that could be used to discriminate between theories of gravity. (Incidentally, it’s odd that Poincare was apparently familiar with the current astronomical literature regarding Seeliger’s hypothesis in 1906, and yet in 1908 he was still using for Mercury’s perihelion advance the figure 38”, which was Leverrier’s original value, but which had been raised to about 43” by Newcombe in 1882.) |
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Similarly, in a review of gravitation theories, Walter Ritz wrote in 1909 |
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Astronomical observations carried out over many centuries have revealed some deviations between observation and calculation, which cannot be explained by Newton’s law up to now, and which a new theory will have to explain. Of these anomalies by far the largest is of the planet Mercury, whose ellipse precesses slowly, under the effect of the remaining planets; but the observed precession is larger by approximately 42 arc-seconds per century than the computed. The difference is small, but nevertheless unquestionable and unexplained. |
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Again this clearly indicates not only that the precession of Mercury’s orbit was considered anomalous, but that it was widely suspected that its resolution would come from a new theory of gravity. Of course, Einstein was very familiar with Ritz’s work, having engaged him in a public debate in 1909 on the subject of the advanced solutions of Maxwell’s equations. |
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In any case, we do have one definite piece of evidence for the fact that Einstein (like Poincare) regarded Mercury’s precession as anomalous, and as something to be explained by a new theory of gravity – even before Poincare’s 1908 book. In December 1907, just as Einstein was beginning to work seriously on his ideas about a relativistic theory of gravity (and just after having “the happiest thought of my life”, i.e., the equivalence principle), he wrote in a letter to his friend Conrad Habicht |
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At the moment I am working on a relativistic analysis of the law of gravitation by means of which I hope to explain the still unexplained secular changes in the perihelion of Mercury. |
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We also know that around 1913 Einstein and his friend Michele Besso worked out the perihelion advance implied by the so-called Entwurf theory of gravity that Einstein had developed with the help of Marcel Grossmann. They were disappointed to find that the theory actually predicted a negative value for the precession, making the anomaly even worse. They decided not to publish the derivation. |
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Despite the failure to account for Mercury’s precession, Einstein was initially enthusiastic about the Entwurf theory, but gradually he began to lose confidence in it, and resumed the search for a satisfactory theory. Finally in November of 1915 Einstein arrived at the final generally covariant field equations, which, to his delight, not only reduced to Newton’s theory in the first-order approximation, but gave in the second-order approximation the correct value for Mercury’s anomalous precession. In a letter to Sommerfeld on 28 November Einstein described how he had progressed from the Entwurf theory to general relativity: |
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In the last month I had one of the most stimulating, exhausting times of my life, and also one of the most successful…. For I realized that my existing gravitational field equations [the Entwurf theory of 1913] were entirely untenable! The following indications led to this: |
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1) I proved that the gravitational field on a uniformly rotating system does not satisfy the field equations. |
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2) The motion of Mercury’s perihelion came to 18” rather than 45” per century. |
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3) The covariance considerations in my paper of last year do not yield the Hamiltonian function H. When it is properly generalized, it permits an arbitrary H. From this it was demonstrated that covariance with respect to “adapted” coordinate systems was a flop. |
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Once every last bit of confidence in the result and the method of the earlier theories had given way, I saw clearly that it was only through a link with general covariance theory, i.e., with Riemann’s covariant, that a satisfactory solution could be found. Unfortunately, I have immortalized the final errors in this struggle in the Academy contributions… The final result is as follows: The gravitational field equations are generally covariant. |
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Thus, one of the three indications leading to his loss of confidence in the Entwurf theory was the failure of that theory to correctly account for the anomalous precession of Mercury’s orbit, and one of the main pieces of evidence he could cite in support of the generally covariant field equations was that they give the correct precession. Of course, his reasons for believing that he had finally arrived at the correct theory were mainly related to the logical coherence of it (“The sense of the thing is too evident”), and there really is very little arbitrariness in the correct derivation of the field equations from Einstein’s basic conceptual premises. The Entwurf theory and the other mis-steps along the way were indeed “immortalized errors”, and it was probably inevitable that sooner or later those errors would have been corrected, and the generally covariant field equations would be discovered. Indeed, Hilbert arrived at those same equations rather quickly once he began to work on the problem as Einstein had outlined it in a seminar at Gottingen the preceding summer. Nevertheless, the quantitative agreement with Mercury’s precession seems to have been psychologically very powerful for Einstein, and certainly contributed to his impression that the theory was correct. Pais says |
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This discovery was, I believe, by far the strongest emotional experience of Einstein’s scientific life, perhaps in all his life. Nature had spoken to him. He had to be right. ‘For a few days I was beside myself with joyous excitement’. Later he told Fokker that his discovery had given him palpitations of the heart. What he told de Haas is even more profoundly significant: when he saw that his calculations agreed with the unexplained astronomical observations, he had the feeling that something actually snapped in him… |
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Oddly enough Pais’ translation of the 1907 letter to Habicht differs from the translations of Anna Beck (Einstein’s Collected Papers) and Roseveare. Pais translated the passage as |
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At this time I am [again] busy with considerations on relativity theory in connection with the law of gravitation… I hope to clear up the so-far unexplained secular changes of the perihelion length of Mercury… [but] so far it does not seem to work. |
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This is strange for several reasons. First, why did Pais split the quote with ellipses? The other translations give no indication that anything is missing. (In Anna Beck’s translation of this letter the reference to Mercury is just the single sentence quoted previously.) Second, did Einstein really refer to the length of the perihelion? A perihelion doesn’t have a length. (Perhaps he meant the length of the period between perihelia?) Third, and most puzzling, where did Pais get the final phrase “so far it does not seem to work”? This phrase doesn’t appear in any of the other translations. It is such a meaningful phrase that it’s hard to imagine a translator leaving it out, but it’s equally hard to imagine it being casually inserted without justification. What does the original German text say? |
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Incidentally, the popular book “The Evolution of Physics” (1938) by Einstein and Infeld says |
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The deviation of the motion of the planet Mercury from the ellipse was known before the general relativity theory was formulated, and no explanation could be found. On the other hand, general relativity developed without any attention to this special problem [my emphasis]. Only later was the conclusion about the rotation of the ellipse in the motion of a planet around the sun drawn from the new gravitational equations. |
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Considering that Einstein began his search for a new gravitational theory in 1907 with the expressed purpose (as stated in his letter to Habicht) of explaining the anomalous precession of Mercury, and kept this objective in view throughout the intermediate development (including the Entwurf of 1913), and considering that Einstein listed the failure of the Entwurf theory to give the correct perihelion of Mercury as one of the three reasons that led him to lose faith in that theory, which then led him to the fully covariant theory of general relativity, it seems hard to justify the claim that general relativity was developed without any attention to this problem. This claim is somewhat similar to Einstein’s assertions that special relativity was developed without any attention to the Michelson and Morley experiment – despite the fact that at other times (notably his 1922 talk in Japan on how he developed the theory of relativity) he acknowledged that this experiment had been an important factor in his thinking. Of course, in both cases it’s perfectly correct to say that the theories follow logically and almost without ambiguity from very broad and fundamental principles, so they were certainly not ad hoc explanations of the respective experimental facts. Nevertheless it is historically inaccurate to claim that general relativity was developed “without any attention” to Mercury’s anomalous precession. |
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We may be able to account for the clearly erroneous claim in “Evolution of Physics” by the fact that Einstein had very little to do with the actual writing of the book. He apparently agreed to lend his name to the project in order to help his friend Infeld raise money so that Infeld might be allowed to remain in the United States after his grant expired. The prospect of being exported back to Poland in 1938 was not very appealing to Infeld. Jefferson Weaver says Einstein’s actual contribution to the book was “negligible”, and Pais reports that Einstein was “not enthusiastic” about the book, and then concludes his brief discussion of it by quoting Einstein’s cryptic comment in reference to the project: |
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One should not undertake anything which endangers the tenuous bridge of confidence between people. |
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