# How did Einstein derive general relativity?

How did Einstein derive general relativity (GR)?

Did he use: the equivalence principle? The principle of least action? Anything else?

Note, I'm not looking for a full mathematical derivation of GR! But rather, I'd like to know what Einstein's starting points were.

Note: I found a similar question that has been asked already, but I couldn't find an answer to my question in the answers.

His starting point was to realize that Newton's gravity didn't satisfy his principles of the (special) theory of relativity because it wasn't Lorentz-invariant and it included action at a distance, faster-than-light effects of gravity that could spread immediately.

So he was looking for a better theory that would be compatible with the principles of relativity. It took him a decade after special relativity was found to find and complete general relativity. Let me completely skip dead ends he had tried although these stories are interesting and one could learn something from them, too. At some point in 1911, in Prague's Viničná Street (see some letters Einstein wrote about Prague), he realized that the equivalence principle was a very special property of gravity – known already to Galileo but not appreciated as an important principle – and it led his final years.

Eventually he realized that the spacetime had to be curved, by arguments based on the equivalence principle, and it must be described by the Riemannian geometry. He was looking for the right equations that could relate the curvature of spacetime and the density of matter in the spacetime and finally in 1915, he found his Einstein's equations.

I think that he found the equations in their explicit form and the Einstein-Hilbert action from which the equations may be derived via the principle of least action were found later – also independently by Hilbert. We may say that the principle of least action wasn't necessary to discover GR; the equivalence principle was essential but Einstein needed (and one needs) more insights than just this principle.

• This is not the best way of explaining Einstein's work on GR. I remember telling my uncle that Einstein discovered GR to satisfy Lorentz invariance in gravity. He said that had little or nothing to do with it, and that it is really easy to create a relativistic extension of Newtonian gravity(a non GR extension). What Einstein was really motivated by was extending the principle of relativity to include non-inertial frames, and indeed Einstein was writing as early as 1907 that there was no good reason to believe it couldn't be done. Sep 26, 2012 at 14:37
• Right, that's what the equivalence principle is all about. The equivalence principle is a principle that allows you to discuss non-inertial frames to be equivalent to inertial frames with extra gravity. I just wrote about these issues at the end of motls.blogspot.cz/2012/09/… - deriving the red shift just like he did it in Prague of 1911-12. Your criticism may just boil down to your misunderstanding what the equivalence principle means. Sep 26, 2012 at 14:56
• Yes, I think we would have GR without Einstein much earlier than in 2012. For example, it would have been derived from string theory, a theory of the strong force, in 1974. Sep 27, 2012 at 5:26
• String theory doesn't imply that a "flat background" is any special relatively to other solutions. It implies that the configurations-without-excitations are exactly those Ricci-flat or similar backgrounds that follow from GR. At long distances, string theory is exactly equivalent to GR coupled to a few matter fields including all conceptual and "philosophical" points that have any sense. It is a fact in GR as well that GR may be expanded around a background, e.g. the flat background, and treated as spin-2 fields. It's an extremely smart, particle-physics-friendly treatment of GR. Oct 13, 2014 at 7:10
• I don't know how to say it more clearly and generally but all the suggestions that string theory is "missing something important" relatively to GR are completely wrong or irrational or nonsensical verbal games with words that don't mean anything or demagogic misinterpretations of ways to write something instead of their physical content. Everything that has a physical sense about physics of GR or ST at long distances is encoded in some observables and those are exactly the same, so the physics-legitimate parts of these theories are exactly equivalent in the limit. Oct 13, 2014 at 7:12

I highly recommend reading section 17.7, "A Taste of the History of Einstein's Equation", pages 431 through 434 of MTW's Gravitation

According to the general theory of relativity,...,by application of arbitrary substitutions of the Gauss variables $x_1, x_2, x_3, x_4,$ the equations must pass over into equations of the same form; for every transformation (not only the Lorentz transformation) corresponds to the transition of the on Gauss co-ordinate system into another.