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The story is usually told starting with Einstein's 1915 paper Explanation of the Perihelion Motion of Mercury from General Relativity Theory, or at least its drafts from 1913-14. It was the first triumph of general relativity. But how did Einstein zero in on Mercury in the first place?

Einstein's early work seems to be rather distant from astronomy. Special relativity grew out of classical electrodynamics, as did the photoelectric effect, the work on heat and Brownian motion is far from astronomy too. Even general relativity was driven by philosophical concerns about general covariance, etc., rather than empirical issues in astronomy. I am not sure how well the Mercury's anomaly was known at the time, but Kelvin and others did not mention it as one of the "clouds". While developing general relativity Einstein learned of tensor calculus from a geometer friend, perhaps he had an astronomer friend as well?

Who or what attracted Einstein's attention to Mercury, and when? What alerted him to the idea that Mercury's case was different from all those other cases, when a mundane explanation was involved?

Another twist is that special relativity combined with the inverse square law already causes elliptic orbits to precess. Sommerfeld knew that at least by 1916, when he refined Bohr's model of the hydrogen atom by using precessing elliptic orbits under special relativity kinematics. A calculation on Physics SE shows that this effect can account for about 7" out of 43" per century of Mercury's anomalous precession.

Was Einstein or his contemporaries aware of this effect for Mercury before 1915, and how did it reflect on special relativity if they were?

EDIT: VicAche's answer below led me to Conquering the Perihelion chapter in Kevin Brown's book Reflections on Relativity, which gives the whole story summarized below.

Einstein first mentions Mercury in a letter to Habicht in 1907:"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." He most likely got the idea from Mach's Science of Mechanics, which mentions "Paul Gerber alone, from the perihelial motion of Mercury, forty-one seconds in a century, finds the velocity of propagation of gravitation to be the same as that of light." This directly suggested that Mercury's anomaly, unlike many others, was relativistic.

Back in 1906 Seeliger gave an alternative solar corona explanation, which convinced many of his contemporaries, but not Einstein. And not Poincare, who mentioned in 1908 book Science and Method that special relativity already predicts an advance of 7" for Mercury’s perihelion. He further wrote (answering my second question):"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." This likely confirmed the relativistic nature of the anomaly in Einstein's eyes. Einstein's original version of a new gravity theory ("Entwurf") first predicted negative precession, and then 18" instead of 45", which was one of three reasons he cited for abandoning it in favor of what we now call general relativity.

P.S. Although Mercury's is the only one featured in the story orbital anomalies are plenty, as are proposals to change the laws of gravity for them. Newton could not account for all of lunar precession, and Clairaut suggested a modification to inverse square law before discovering that the issue was not retaining enough terms in the Taylor series for the solution. The anomaly of Uranus famously led to Leverier's prediction of Neptune, and Leverier, who also discovered the anomalous precession of Mercury, suggested a new planet Vulcan to explain it.

Vulcan did not materialize by 1915, but neither did Pluto until 1930, and it was blamed for orbital anomalies of Uranus and Neptune known in Einstein's time. After the discovery it turned out that little Pluto wasn't responsible after all, a misestimation of Neptune's mass was. Also in Einstein's time there was still a known lunar anomaly, which was not resolved until 1940s, the non-uniformity of Earth's rotation was the culprit. Zodiacal cloud and another modification of inverse square law were suggested instead of Vulcan for Mercury by Newcomb. A more recent Pioneer anomaly also caused a lot of speculation about new gravity physics (MOND), but was finally traced to subtle thermal pressure. Some suggested it was responsible early on.

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  • $\begingroup$ Another twist is that special relativity combined with the inverse square law already causes elliptic orbits to precess. This is at best a vast oversimplification. You can't simply take SR and plug in a certain force law for instantaneous action at a distance; the result is a theory that lacks self-consistency. GR is essentially the minimal thing you need to do in order to combine SR with gravity. In fact, GR is the unique such theory if you also require the equivalence principle. $\endgroup$ – Ben Crowell May 12 '15 at 16:01
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    $\begingroup$ @Ben Crowell And yet Sommefeld, and Poincare, did plug inverse square law into special relativity. Planck was inconsistent with his radiation formula, Bohr was with his atom, and Dirac with his "negative sea" in a single particle equation. Self-consistency is often sacrificed to agreement with experiments, especially at times when new theories are made, but not only. SM+GR is inconsistent now. When Sommerfeld reproduced fine structure with precessing ellipses Einstein said "Bohr's theory must be right" despite its self-inconsistency. $\endgroup$ – Conifold May 12 '15 at 23:52
  • $\begingroup$ You mention Clairaut determined an error from not keeping enough terms in a Taylor series. Can you tell me where this is discussed? $\endgroup$ – KCd Sep 10 '15 at 2:00
  • $\begingroup$ @KCd See here sites.apam.columbia.edu/courses/ap1601y/… He wasn't the first to modify the law, Émilie de Breteuil already suggested it in 1740, but for reasons unrelated to anomalies. $\endgroup$ – Conifold Sep 12 '15 at 18:48
  • $\begingroup$ The paper you pointed me to says some high-order terms turned out to be important without saying what order such terms were. The fourth page of springer.com/cda/content/document/cda_downloaddocument/… says the previously neglected second-order terms were enough to account for observations (their coefficients were unusually large). I was hoping for an example where terms of third-order or more would be needed. $\endgroup$ – KCd Sep 24 '15 at 5:29
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Who or what attracted Einstein's attention to Mercury, and when? What alerted him to the idea that Mercury's case was different from all those other cases, when a mundane explanation was involved?

I know for sure that Henri Poincaré was aware of the problem and of its singularity - had he been in Kelvin's place, he would have added it to the list, and Einstein was most certainly aware of the work of Poincaré (Étienne Klein reported La science et l'hypothèse was one of the subject of discussion in Olympia Academy).

Poincaré studied the stability of the solar system in extenso during the 1880s and 1890s. I know no work mentioning Mercury before his adress of the problem in 1910 (see page 424 of Poincaré's biography by Jeremy Gray), but despite my deception in failing to find explicit adress prior 1905, this is still a valid answer.

Note that the study of Mercury and the solar system by Poincaré led to another whole new theory, the Chaos Theory, which makes this ridiculous advance in the perihelion of mercury on of truly productive ground for XXth century science. Why? Because Einstein didn't use this example to advert his theory until 1915 .

Was Einstein or his contemporaries aware of this effect for Mercury before 1915, and how did it reflect on special relativity if they were?

Langevin says that Poincaré proposed several propagation equations for gravitational waves at the speed of light, that all had in common "that they reducted the gap between the laws of physics and the facts, in the move of the perihelia of Mercury for example". This proves that, when working on similar laws, the French academy of Science was not unused to test them on Mercury (Poincaré's failure to find special relativity can be explained by his obsession to keep his laws Lorentz-compliant. Poincaré's work led to the Relativity priority dispute . Poincaré attributes special relativity to Lorentz, Lorentz to Poincaré and Einstein, and Einstein to himself realizing that Lorentz's local time was in fact just time. It is sure the three of them looked after each other, and Poincaré was the astrophysicist of the lot.

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    $\begingroup$ Great find! When you mentioned Poincare I was able to find this mathpages.com/rr/s8-10/8-10.htm. He wrote in 1908 book Science and Method that based on special relativity "there would result, in the perihelion of Mercury, a secular variation of 14”, in the same direction as that which has been observed and not explained, but smaller, since the latter is 38”... 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", which answers the second question. $\endgroup$ – Conifold May 11 '15 at 23:53
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    $\begingroup$ It turned out to be quite interesting: Einstein knew of it since 1907, and they speculate that it was from Mach's book Science of Mechanics. And there was a controversy as to the significance of this anomaly at the time, Seeliger conjectured that it was because of solar corona. Maybe you could add some of it to the answer. $\endgroup$ – Conifold May 11 '15 at 23:58
  • $\begingroup$ I'm sure Poincaré was aware of it. The only problem is, I did my research in French and "Mercure de France" is the editor of some of Poincaré's work, which kind of spoiled the results.. $\endgroup$ – VicAche May 11 '15 at 23:58
  • $\begingroup$ The find is not mine, but Claude Aslangul's. Only he taught that orally, without any references, and does not mention it in the written talk... $\endgroup$ – VicAche May 11 '15 at 23:59
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    $\begingroup$ @Conifold, For posterity, hjere's a mirror of mathpages.com/rr/s8-10/8-10.htm at archive.is/VYBZ7. It's part of a book, found mathpages.com/rr/rrtoc.htm and Amazon: Reflections on Relativity, Kevin Brown, 2016: "Reflections on Relativity" is a comprehensive presentation of relativity, including in-depth historical perspectives. I have no connection wtih the author and have not read the book. Just thought it would be nice to reference. $\endgroup$ – CoolHandLouis Jul 15 '16 at 9:32
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I think the answer is rather anticlimactic, tl;dr is: The anomalous precession of Mercury was in fact a well-known problem amongst astronomers at that time. Correctly predicting the anomalous precession was a big and well-known test for any theory of gravitation which was not Newtons $1/r^2$ law.

In the 1840's, Le Varrier, based on his calculations on the orbit of the back-then known planet Uranus, proposed the existence of another planet we today know as Neptune which was indeed discovered just shortly after. Around the same time, he also calculated the orbit of Mercury, noticing, that the theoretically predicted result was largely differing from the real one. In a letter to Hervé Faye, he considered this to be

[...] a difficult problem, worthy of the attention of all astronomers.

Being one of the leading astronomers of his time, I am pretty sure that this encouraged many people to search for solutions. Varrier himself douted the existence of a undiscovered intra-Mercury planet (Vulcan). The only solution within the known effects was a at least 10%-increase of the Venus mass, which seemed pretty absurd.

As you already mentioned, there were two major approaches to solve this:

  • Taking into account unknown celestial bodies or altering the properties of the known one. Known examples are the existence of the planet Vulcan, or the suggestion to flatter the sun. These examples sticked to the Newtonian law of gravitation. However, none of these phenomenons were ever observed.

  • As you already mentioned, also many theorists tried to explain this by modifying Newton's holy $1/r^2$-law. Two major approaches were popular here: Avoiding a completely static law, or correcting the Newtonian law by a velocity-dependent term (e.g. the Lorentzian theory of gravitation). All of these approaches were either unsuccessful or contained arbitrary parameters.

Einsteins approach is obviously one of the theoretical kind, but, apart from the fact that it actually worked, it had major advantages over all the other approaches - in particular, unlike many other approaches, it was not specifically designed to solve this problem and Einstein did not take into account any additional hypotheses apart from his laws of gravitation. Therefore, it was a very successful test for Einsteins theory of gravitation.

See also IV.14c in Pais' Einstein Biography.

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  • $\begingroup$ could you add some references? $\endgroup$ – hjhjhj57 May 12 '15 at 3:24
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    $\begingroup$ No, turns out it was much more complicated than that. In 1906 Seeliger proposed a solar corona explanation, which many found convincing, although not Einstein and Poincare. Einstein first mentions Mercury's precession in 1907, long before any draft of new gravity theory. His original approach (1913) did not produce correct precession, and he modified it largely because of that. $\endgroup$ – Conifold May 13 '15 at 0:06
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    $\begingroup$ And it seems he focused on Mercury not because of Le Verrier's fame, but because of an obscure comment in Mach's book: "Paul Gerber alone from the perihelial motion of Mercury, forty-one seconds in a century, finds the velocity of propagation of gravitation to be the same as that of light." This suggested that of all orbital anomalies Mercury's was the one that might actually be related to relativity. See Conquering the Perihilion chapter in Kevin Brown's Reflections on Relativity. mathpages.com/rr/s8-10/8-10.htm $\endgroup$ – Conifold May 13 '15 at 0:11

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