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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.

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user12345
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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.

Luboš Motl
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    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. – user7348 Sep 26 '12 at 14:37
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    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 http://motls.blogspot.cz/2012/09/albert-einstein-1911-12-1922-23.html?m=1 - 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. – Luboš Motl Sep 26 '12 at 14:56
  • @Motl Okay, that's a pretty good entry, I'll admit. Do you think in the year 2012, we'd have General Relativity without Einstein? – user7348 Sep 26 '12 at 19:14
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    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. – Luboš Motl Sep 27 '12 at 05:26
  • Mach's principle, the equivalence principle and the Ehrenfest paradox all played a part. – Physiks lover Sep 27 '12 at 20:25
  • "it would have been derived from string theory": Hi Luboš, coming from a non-string theorist's standpoint: I understand ST (and there's not much of ST I understand) yields GTR in some kind of limit, but doesn't it conceive of the Einstein field equations as a propagation of a spin-2 field on a flat background? Would the geometry on a (pseudo-) Riemannian manifold interpretation of the EFE in this context been apparent to a string theorist in or sometime after 1974? Or is my understanding incorrect? – Selene Routley Oct 13 '14 at 04:51
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    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. – Luboš Motl Oct 13 '14 at 07:10
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    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. – Luboš Motl Oct 13 '14 at 07:12
  • @LubošMotl No, I'm not saying ST is missing anything: I do understand (second hand) that it completely reproduces GTR: I'm just curious as to whether you think the early string theorists after 1974 would have recognised the geometric interpretation of this limit that astonomers and other GTR users use to think about it if Einstein wouldn't have come along. The limiting theory, however you interpret it, is the same: the equations of GTR would be a bit daunting and tangled without some way (like Riemannian geometry) to organise them into structures comprehensible to normal mortals. – Selene Routley Oct 13 '14 at 11:36
  • Yes, of course that they did recognize and would (soon or later) recognize the geometric interpretation. It's really the point of this Witten's thought experiment about the world where string theory is discovered first, when GR is unknown. One can see that adding a vertex operator of the graviton to the world sheet action is equivalent to curving the metric and this point is manifest. – Luboš Motl Oct 13 '14 at 14:46
  • Doubts about this self-evident observation are usually just a misinterpretation of someone's dislike for the treatment of the gravitational field as a quantum field theory for spin-two quanta. But this is a totally legitimate description, one that is almost mandatory if one wants to use the language of the QFT, and of course that string theory naturally produces this description/decomposition, too. QFT theorists often use it. But all other valid ways to interpret or visualize the curved spacetime are there in string theory, too. – Luboš Motl Oct 13 '14 at 14:48
  • Thanks heaps Luboš, I wasn't aware of Witten's thought experiment. – Selene Routley Oct 14 '14 at 11:43
  • It wasn't a big deal. Just in some semi-popular context, he said that string theory predicted general relativity in the sense that GR on Earth was only discovered before ST due to historical coincidences. It's possible that there are other civilizations where ST was discovered first and GR with all of its usual features derived as the string theory's consequence. – Luboš Motl Oct 15 '14 at 06:19
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I highly recommend reading section 17.7, "A Taste of the History of Einstein's Equation", pages 431 through 434 of MTW's Gravitation

(Click the link to read at Google books).

Alfred Centauri
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Einstein published a book in 1916 called Relativity which he updated in 1952 just a few years before his death. In chapter 25 he discusses Gaussian coordinates and in chapter 28 he gets to the heart of the matter and states:

The Gauss co-ordinate system has to take the place of the body of reference. The following statement corresponds to the fundamental idea of the general principle of relativity: "All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature."

Later then elaborates further:

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.

Which essentially gets to his point. Whatever the variables of spacetime are, any relationship between those variables must be respected even when we arbitrarily change variables. Or in other words, our choice of coordinates is arbitrary as long as we include enough variables to describe the underlying spacetime.

It should be noted in the context of discovery that there is a long standing dispute of priority. However, despite uneducated debate, most scholars agree that Einstein developed special and general relativity largely independently.

Freedom
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