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I've read about general relativity (GR) recently and something stroke me: Einstein came up with his equations in 1915, linking the metric of spacetime to the distribution of energy (more exactly, to the stress-energy tensor), without providing any specific solution to those equations.

It was only a few months later that Schwarzschild came up with the famous solution for a non-rotating spherically symmetric mass distribution and that particular solution provided strong arguments for Einstein's equations as it simplified to Newton's law under classical physics assumptions.

Years later in 1922, another solution (under homogeneity/isotropy assumptions about the whole universe, see FLRW metric) was found and explained some empirical observations (e.g., cosmological redshift).

I'm wondering:

  • Why didn't Einstein provide any specific solution to his equations? I mean, wouldn't it be the first thing to look for?
  • How were people convinced GR was an acceptable theory when he came without a specific solution to demonstrate consistency with Newtonian physics or other empirical observations?
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  • $\begingroup$ We don't have a solution to the standard model either. $\endgroup$
    – Connor Behan
    Sep 2 at 15:27
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    $\begingroup$ The geodesic equation can be solved using perturbation theory (without the full Schwartzschild solution) in the weak gravity limit, and the precession of the (Newtonian) elliptical orbit can be perturbatively evaluated (and the rate of precession of the perihelion of Mercury matches the experimental results very well) without finding the metric explicitly. This paper discusses some of the historical events arxiv.org/abs/1411.7370 $\endgroup$ Sep 2 at 15:48
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    $\begingroup$ Related: physics.stackexchange.com/q/160380/2451 $\endgroup$
    – Qmechanic
    Sep 2 at 15:49
  • $\begingroup$ Einstein did provide a solution -- the flat one, i.e. empty spacetime, i.e the lorentz vector space. $\endgroup$ Sep 26 at 15:34

3 Answers 3

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Einstein actually used the consistency with Newtonian theory to derive the equations. He used the assumptions that in nonrelativistic velocities and weak static gravitational fields, the equations should give you back the Newtonian potential equations.

People where convinced due to the correct calculation of the perihelion precession of the orbit of Mercury, predicted by the Schwarzschild solution.

The proof, here incomplete, goes like this:

In the nonrelativistic static and weak limit, the geodesic equation gives:

$$\frac{d^2 \vec{x}}{d\tau^2} = \frac{1}{2}\vec{\nabla}h_{00}$$

Since the Newton equation is:

$$\frac{d^2 \vec{x}}{d\tau^2} = - \vec{\nabla}\phi$$

Einstein proposed that $g_{00} = -(1+2\phi/c^2)$

Applying the Laplace operator on both sides:

$$\nabla^2 g_{00} = -\frac{8 \pi G}{c^2}\rho$$

Which is in fact

$$G_{00}= \frac{8\pi G}{c^4} T_{00}$$

Hence:

$$G_{\mu \nu} = \frac{8\pi G}{c^4} T_{\mu \nu}$$

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  • $\begingroup$ "predicted by the Schwartzschild solution" <= I'm not sure I understand. Was Schwartzschild metric already needed to explain the result or was it derived from Einstein equations without the whole Schwartzschild metric? $\endgroup$
    – Weier
    Sep 2 at 15:36
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    $\begingroup$ The Schwartzschild solution explained the perihelion, the Einsten equations reduced to the Newtonian limit alone $\endgroup$
    – LolloBoldo
    Sep 2 at 15:40
  • $\begingroup$ No, what Einsteind did was solving in 1919 the geodesic equation in Schwartzschild geometry (1915) only maintaining first order terms in the angular expression $\endgroup$
    – LolloBoldo
    Sep 2 at 16:55
  • $\begingroup$ No, what Einsteind did was solving in 1919 the geodesic equation in Schwartzschild geometry (1915) only maintaining first order terms in the angular expression $\endgroup$
    – LolloBoldo
    Sep 2 at 16:55
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The other answer by LolloBoldo is correct. But just to give some evidence, in the original 1915 paper Einstein explicitly states:

The field equations (16a) then take, as a first approximation, the form $$ \frac{1}{2} \sum_{\alpha} \frac{\partial^2g_{\mu \nu}}{\partial x_{\alpha}^{2}} = \kappa T_{\mu \nu} \tag{16b} $$ from which one sees immediately that it contains Newton’s law as an approximation.

Source: Volume 6: The Berlin Years: Writings, 1914-1917 (English translation supplement) Page 106

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    $\begingroup$ I edit my answer to give a more detailed proof of the procedure $\endgroup$
    – LolloBoldo
    Sep 2 at 15:40
  • $\begingroup$ @LolloBoldo Sure, but I thought it useful to have an original source to show Einstein did indeed show this in his original paper, contrary to the OP's question. $\endgroup$
    – Eletie
    Sep 2 at 15:44
  • $\begingroup$ Yes of course :) $\endgroup$
    – LolloBoldo
    Sep 2 at 16:55
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The question, as asked, implicitly assumes an overly advanced state of the art for physics and astronomy in 1915

GR was an elaboration of special relativity (SR), but SR had very little direct evidence in its favor in 1915. The negative result of the Michelson-Morley experiment was not clear at the time. Morley's collaborator Dayton Miller went on to do more experiments of the kind, searching for aether entrainment. The outcome was controversial as late as 1930. Other experiments supporting SR were still far in the future: Ives-Stilwell (1938), Rossi-Hall (1941). Really all the solid experimental evidence for SR was from after World War II.

It's anachronistic to expect that Einstein could have posited an expanding-universe solution to the GR field equations and said, "Aha, that matches what we see!" In 1915, it wasn't even established whether the "spiral nebulae" were outside the Milky Way. The Shapley-Curtis debate was in 1920. Cosmology wasn't even a recognized field of study.

As other answers have noted, the field equations of GR were structured from the start so as to be consistent with Newtonian gravity in the nonrelativistic limit, so it wasn't necessary to have the Schwarzschild solution in order to demonstrate that. In fact, Einstein didn't like the Schwarzschild solution, because it offended his Machian intuition.

And finally, the Einstein field equations are nonlinear, and therefore it is very, very difficult to find closed-form solutions to them. The difficulty of finding closed-form solutions has nothing to do with the existence of solutions, which is another matter.

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  • $\begingroup$ If "all" Einstein had to offer in 1915 was consistency with Newtonian physics, what convinced people of GR's validity? $\endgroup$
    – Weier
    Sep 8 at 17:45

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