The Genesis of General Relativity
(5) Perihelion of Mercury, a highlight in Einstein's 1915 Papers
[(1)//(2)//(3)//(4)//(5)//(6)//(7)//(8)]
Einstein's 1915 papers:
(a) Zur allgemeinen Relativitaetstheorie, Preussische Akademie der Wissenschaften, Sizungsberichte (1915), 778-786.
(b) Zur allgemeinen Relativitaetstheorie (Nachtrag), Preussische Akademie der Wissenschaften, Sizungsberichte (1915), 799-801.
(c) Erklaerung der Perihelbewegung des Merkur aus der allgemeinen Relativitaetstheorie, Preussische Akademie der Wissenschaften, Sizungsberichte (1915), 831-839. 邦訳 水星の近日点の移動に対する一般相対性理論による説明（内山龍雄訳）、『アインシュタイン選集』2、115-124、共立、1970。
(d) Die Feldgleichung der Gravitation, Preussische Akademie der Wissenschaften, Sizungsberichte (1915), 844-847.
It is quite remarkable that Einstein's energetic work toward the completion of general relativity took place in the midst of his personal catastrophe, the breakup of the first marriage (with Mileva). Einstein moved to Berlin in the spring of 1914; but his wife Mileva did not like Einstein's relatives, and moreover Einstein was having an affair with his cousin Elsa (who was to become the second wife). His wife returned to Zuerich with two sons in the summer of 1914, and never came back. Highfield and Carter tell us: "Einstein wept after he had seen off Mileva's train; [Fritz] Harbor had to support him as he walked home from the railway station" (Highfield, R. and Carter, P., The Private Lives of Albert Einstein, St. Martin's Press, 1993,168).
But according to Highfield and Carter,
It remains remarkable how diligently Einstein strove to keep contact with his sons during 1915, for this was the year in which his scientific labours reached their fiercest intensity. As he struggled to complete the extension of relativity, he increasingly cut himself off from the outside world. Letters were far less likely to be answered than to be impaled on a large meat hook and later burnt. The work was most intense in mid to late November, when Einstein wrote to almost no one. The sole exceptions--aside from David Hilbert, a German mathematician whose work on gravity ran remarkably close to his own-- appear to have been Mileva and Hans Albert [elder son]. (op. cit.,173.)
The final step for completing general relativity took place in November 1915; the published documents are the four papers listed above, and a chronological account by Pais (1982, 14c) is quite useful and the reader is referred to that for details. Briefly, (a) comes close to the final form but not yet; (c) derives two predictions from the new theory, and (d) presents the final equations. Here, I wish to focus on (c) which contains two of the three predictions (i.e., perihelion, deflection of light, and red-shift; the first two appears in this paper with "correct values") Einstein made from his new theory. We will postpone our discussion of the field equations until we come to Einstein's 1916 paper (the most definitive review paper), although we are going to make some references to special cases for them.
A detailed analysis of this paper (c), which has only two sections besides an introduciton, and of its background, is already made by Earman and Janssen (1993), so that my account in the following heavily depends on this; but I will try to check the German original text at crucial steps. The reason for forcusing on this paper is that it is quite conducive to our understanding of general relativity, because it clearly shows how the theory is used for solving specific problems, how one proceeds in order to obtain a "solution" for a field equation. In the case of perihelion, Einstein arrived at his prediction by an "iterative approximation", not by a strict solution (which was obtained by Schwarzscild and Droste, in 1916 and 1917, respectively); still you can have a good view of what it is like to "solve a field equation".
According to the modern notation, the field equation Einstein used in this paper is expressed as follows:
(1) Rmn = -κTmn
where Rmn is the Ricci tensor, Tmn is the stress-energy tensor, and κis a constant (of gravitation). The reader does not have to worry about the mathematical detail of these symbols; what is important here is to keep track of the line of Einstein's reasoning. In view of this, it suffices to say that the Ricci tensor can be derived from the Riemann tensor, which in turn is determined by the metric tensor gmn. But the equation becomes much simpler if there is no "matter" (as distinguished from the gravitational field); then the stress-enrgy tensor vanishes, and we have, if we wisely choose a coordinate system (because of the general covariance, we can do this)
(2) Rmn = 0
and
(3) √- g = 1
where g is the determinant of gmn (never mind, because it is obtained from gmn; the minus sign appears because g is negative, according to Einstein's symbolism).
Further, the Ricci tensor can be split into two parts, say A and B, and given (3), B also vanishes. Thus (2) takes the following form:
The equations (3) and (4) are nothing but the field equations where there is no matter (Einstein 1916, sect. 14), and the exterior of the sun can be covered by these equations.
Now Einstein stipulates that the sun is located at the origin of this (Cartesian 4 dimensional) coordinate system; and since he does not have a strict solution for (4) yet (Schwalzschild was to find one in 1916), he applies a method of "iterative approximation" (but we have to be careful, because such methods may well depend on the choice of a coordinate system, and solutions may become invalid if applied in another coordinate system), starting from the 0th approximation which is the Minkowski metric according to the special relativity:
-1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 +1
Notice that Einstein now scales the time axis (the fourth coordinate) differently from those papers we have examined. How to proceed to 1st and 2nd approximation (with the assumption that the true metric differs only slightly--the difference is small enough compared to 1--from the Minkowski metric), we will see shortly.
Einstein is aware that the equations may not give a unique solution for gmn (recall the "hole" argument), but he ignores this problem by stating his "conviction" that all such solutions can be reduced to one another (by coordinate transformations), differing only formally, not physically. Since this remark has an important bearing on the problem of the "hole" argument, let me quote from the German original:
Es ist indessen wohl zu bedenken, dass die gmn [Greek subscripts in the original] bei gegebener Sonnenmasse durch die Gleichungen (1) [our (4)] und (3) mathematisch noch nicht vollstaendig bestimmt sind. Es folgt dies daraus, dass diese Gleichungen bezueglich beliebiger Transformationen mit der Determinante 1 kovariant sind. Es duerfte indessed berechtigt sein, vorauszusetzen, dass alle diese Loesungen durch solche Transformationen aufeinander reduziert werden koennen, dass sie sich also (bei gegebenen Grenzbedingungen) nur formell, nicht aber physikalish voneinander unterscheiden. Dieser Ueberzeugung folgend begnuege ich mich vorerst damit, hier eine Loesung abzuleiten, ohne mich auf die Frage einzulassen, ob es die einzig moegliche sei. (832)
This remark clearly shows that Einstein already had a distinction between formal (or mathematical) and physical difference (and hence identity also) between two solutions, but we cannot judge, from this remark alone, whether he had, at this point, a definite solution for the "hole" argument. But since the final form of gravitational field equations appeared only a week later (25 Nov.), we may safely presume that he was already convinced that the general covariance can be maintained; and it is hard to imagine that Einstein came to this conviction without strong reasons.
Next, Einstein states the following four conditions ("boundary conditions") for solving the equations:
(i) All components of gmn are independent from the time (fourth) axis (in other words, the metric is stationary).
(ii) The metric gmn is symmetric with respect to the spacial origin (in other words, spherically symmetric).
(iii) The metric has the following form (in other words, time orthogonal):
g11 g12 g13 0
g21 g22 g23 0
g31 g32 g33 0
0 0 0 g44
(iv) The metric approaches the Minkowski metric at infinity (in other words, asymptotically Minkowskian).
You can now see what sort of conditions we need in order to solve the field equations. Just putting the specific values of the mass of the sun, mercury etc. (initial conditions), will not do, and we need several "boundary conditions" (some are obvious, given the problem, but the equations themselves do not imply them). Thus, in order to solve the field equations, e.g. for the interior of the star, we need different boundary conditions. We should keep in mind that "solving the field equations" has this problem-dependent or context-dependent character. In the current terminology of the "semantic" view of theories, given a problem, we look for a "model which satisfies" the field equations, under the assumption of suitable "boudary conditions".
Earman and Jenssen (1993, 136) points out that Einstein, with his friend Besso, tried to apply the same method of approximation in 1913 to the same problem, but with a wrong theory of Einstein & Grossmann 1913. Another difference is that while Einstein and Besso tried to obtain values for gmn, Einstein now represents the gravitational field by Γ's in (4) so that he needs values for gmn in so far as necessary for determining these Γ's. Actually, he computes these values only up to 2nd order (starting from 0th order) of approximation (section 1), and Earman and Jenssen points out that for this there is no need for computing gmn up to 2nd order. In any case, Einstein computed Γ's up to 2nd order, and obtained solutions for the gravitational field (for details of approximation procedure, see Earman and Jenssen 1993).
As a byproduct, Einstein realized that for the gravitational field around the sun, gmn (for m and n from 1 to 3) deviates from the Minkowski metric even at the 1st order of approximation (in other words, space is curved, not flat like a Minkowski space; he already knew that time is curved in a gravitational field); thanks to this new insight, he corrected the value of deflection of light around the sun, the correct value being a double of his previous result in Einstein 1911. Indeed, John Stachel claims that Einstein "remained convinced that that static gravitational fields have spatially flat cross sections--at least to "first order", i.e., in a linearized approximation ...--until late 1915, when a study of approximate solutions to a set of generally covariant field equations finally showed that this is not the case" (Stachel 1989, 68).
The section 2 is devoted to determining planetary motions around the sun. I will skip the detail of derivation, and point out only the following. Einstein obtained an equation for the movement of periphelion (for any planet), and found out that only one new term is added to the Newtonian equation. Thus if you compute the effect of this term on the movement of a planet (Mercury, in particular), you can obtain a relativistic effect (i.e. an additional movement, not explained by Newtonian mechanics; but this not Einstein's own word) on perihelion movement. The value Einstein thus obtained for Mercury is 43'' per a century (for a picture, see perihelion.html).
Now, some of Einstein's remarkable comments in the introductory part must be understood in view of this result. He said he will show an important confirmation of his "most radical relativity theory"; that is,
es zeigt sich naemlich, dass sie die von Leverrier entdeckte saekulare Drehung der Merkurbahn im Sinne der Bahnbewegung, welche etwa 45'' im Jahrhundert betraegt qualitative und quantitative erklaert, ohne dass irgendwelche besondere Hypothese zugrunde gelegt werden muesste. (831)
The phrase "without (support of) any special hypothesis" must be understood as meaning that the explanation directly comes from his new equation (applied to Mercury, of course), because the value 43'' is obtained as a "relativistic effect"; everything else is left as identical as the "Newtonian" attempts so far (which is justified because there is no reason to expect relativistic effects there). Further, it becomes quite understandable why Einstein mentions only the value of 43'', when he explains his results in his popular writings (e.g., Einstein 1961, 144). He did not bother to calculate the total amount of the perihelion movement, due to other planets as well as to the relativistic effect.
For a much easier derivation of the planetary motion, and the motion of the perihelion of Mercury in particular, see Taylor & Wheeler (2000), ch. 4 and Project C; orbits are obtained in terms of "effective potential" in Schwarzschild geometry, so that you do not have to wrorry about tensor calculus.
References
Earman, J. and Jenssen, M. (1993) Einstein's Explanation of the Motion of Mercury's Perihelion, The Attraction of Gravitation, ed. by J. Earman, M. Janssen, and J.E. Norton, Einstein Studies Vol.5, Birkhaeuser, 1993, 129-172.
Einstein, A. (1911)Ueber den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes, Annalen der Physik 35 (1911), 898-908.
Einstein, A. (1916) Die Grundlage der allgemeinen Relativitaetstheorie, Annalen der Physik 49 (1916), 769-822.
Einstein, A. (1961) Relativity: the special and the general theory, Three Rivers Press, 1961 (German original, 1917).
Einstein, A. and Grossmann, M. (1913) Entwurf einer verallgemeinerten Relativitaetstheorie und einer Theorie der Gravitation, Zeitschrift fuer Mathematik und Physik 62 (1913), 225-261.
Howard, D. and Stachel, J., eds. (1989) Einstein and the History of General Relativity, Einstein Studies vol. 1, Birkhaeser.
Pais, A. (1982) 'Subtle is the Lord ...', Oxford University Press, 1982.
Stachel, J. (1989) "Einstein's Search for General Covariance, 1912-1915", in Howard &Stachel 1989, pp. 63-100.
Taylor & Wheeler (2000), Exploring Black Holes, Addison Wesley Longman.
August 25, 2000; Last modified March 28, 2003. (c) Soshichi Uchii
