Spacetime, Phil.Spacetime

The Genesis of General Relativity

(8) Field Equations


The Genesis of General Relativity

(8) Field Equations

A number of scholars have tried to reconstruct a path, or paths, along which Einstein arrived at the final form of the field equations of 1915 (see the documents listed at the beginning of gen.GR5.html). Here are a number of facts, reasonably accepted by anyone who examined the matter: (1) When Einstein came back to Zuerich from Prague in 1912, he already knew he needed a new geometry for dealing with gravitational fields; (2) he collaborated with Grossmann, and they published a big paper (1913, Entwurf paper) which contained almost all essential ideas for the completed theory, except that its field equations are not generally covariant; (3) Einstein later devised an argument against general covariance, in order (somehow) to "justify" this defect (The Hole Argument); (4) Einstein became familiar with the "new mathematics", and he could continue his work without Grossmann's help after he moved to Berlin (1914); and (5) after a number of unsuccessful attempts, and along with a similar research by David Hilbert in the same period, the final results came out in 25 November 1915.

Now, a number of scholars (including Pais 1982) attributed some elementary mistakes or ignorance to Einstein, in order to explain this long delay for arriving at the right answer. However, it seems John Stachel's paper in 1980 (published later as Stachel 1989b) prepared the ground for balanced views on the matter. And John D. Norton has published (originally in 1984) a detailed analysis of Einstein's search for field equations during 1912-1915 (Norton, 1989b), along the line put forth by Stachel. Since the matter is highly technical, I will summarize only the crucial points for their reconstruction (see Norton 1989b, 102).

John Stachel (Photo by Roger Highfield, from The Private Lives of Albert Einstein, 1993)

(A) Einstein's understanding of the form of static gravitational fields (which must be obtained as a limiting case from the final theory) in 1913 was inconsistent with the final theory; thus he discarded all equations and conceptions incompatible with this. And this understanding or "misunderstanding" persisted until November 1915, until he realized he was wrong by way of a solution for the perihelion of Mercury (18 November).
(B) Einstein's attempt at the Hole argument and its solution decisively constributed to his understanding of spacetime in his new theory; he overcame his preconception about the relationship between the spacetime manifold and the gravitational fields; this also took place around the same time.

Neither of these two is a trivial matter. Thus, Norton's view is that, as long as we keep these two points (essentially due to Stachel) in mind, Einstein's various attempts in between 1912 and 1915 become perfectly understandable, because all mathematical tools and conceptions were available to Einstein; what prevented Einstein from arriving at a correct answer in the Entwurf paper were, essentially, these two misconceptions. (Needless to say, these misconceptions were by no means easy to overcome, because Einstein was almost alone in his struggle for creating a new profound theory.) Thus in November 1915, he regained general covariance, "rediscovered" as it were, the correct form of the final equations envisaged before, with rich byproducts of Mercury's perihelion and the deflection of light around the sun. Although Einstein and Hilbert had a close contact during the crucial period, Norton thinks Einstein's work was sufficiently self-contained.

Einstein's equations in modern notation are:

Rmn - (1/2)gmnR = -Tmn

where R is a scalar curvature obtained from the Ricci tensor. He added the second term of the left-hand side in 25 November 1915, but fortunately, this term had nothing to do with his previous prediction of Mercury's perihelion (18 November).


Howard, D. and Stachel, J., eds. (1989) Einstein and the History of General Relativity, Birkhaeuser, 1989.

Norton, J. D. (1989a) "What was Einstein's Principle of Equivalence?", in Howard & Stachel 1989, 5-47.

Norton, J. D. (1989b) "How Einstein found his Field Equations, 1912-1915", in Howard & Stachel 1989, 101-159.

Pais, A. (1982) 'Subtle is the Lord ...', Oxford University Press, 1982.

Stachel, John (1989a) "The Rigidly Rotating Disc as the 'Missing Link' in the history of General Relativity", in Howard & Stachel 1989, 48-62.

Stachel, John (1989b) "Einstein's Search for General Covariance, 1912-1915", in Howard & Stachel 1989, 63-100.


Last modified March 28, 2003. (c) Soshichi Uchii