Why didn't Einstein propose any metric solution to his equations?

Question

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?

Answer 1

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:

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

Answer 2

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

Answer 3

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.