Solving the Gravitational Field Equations

[genefalk]

I have read that:
In 1915 Einstein presents to the Prussian Academy of Sciences the General Theory of Relativity; it includes a set of Gravitational Field Equations; at this time he does not present any solution to the equations.

In 1917 he considers a greatly simplified case; presents a solution and adds the cosmological constant to give a solution which yields a static universe.

My question:
Where do the following results come from, did he have to solve the equations to conclude the following?
1) The correct prediction of the amount of precession of the orbit of Mercury.
2) The correct prediction of the amount of bending of light by matter.
3) Time passes more slowly in a gravitational field.

 

[PeterDonis]

In 1915 Einstein presents to the Prussian Academy of Sciences the General Theory of Relativity; it includes a set of Gravitational Field Equations; at this time he does not present any solution to the equations.

This isn't quite true. What Einstein presented on November 25, 1915 was the final form of the field equations; but he had already published several previous forms of them, and he had already derived predictions for your three items from those previous forms. Those predictions were unchanged for the final form of the field equations (which he kept pursuing for reasons which had nothing to do with these particular predictions). Only the first could be compared with observations at the time, and it was correct. The other two predictions had to wait for observations to catch up; there were no observations that could test them in 1915, or in 1917 for that matter.

In order to derive all of these predictions, Einstein did not need a full solution to the field equations; he only needed an approximate solution for weak fields. That is why he was able to derive those predictions without knowing any full solution (the first such was discovered by Schwarzschild soon after Einstein published the final form of the field equations). But the approximate solution he used to derive the predictions certainly counts as a partial solution.

 

[Jonathan Scott]

3) Time passes more slowly in a gravitational field.

This is inaccurate. Time runs more slowly at a lower gravitational potential, but this effect is not related to the field (which is the gradient of the potential). At the center of the earth, there is no gravitational field due to the earth, but the potential is lower and the time runs more slowly than it does higher up.

As far as I know, the time dilation effect can be predicted from the Principle of Equivalence (i.e. that gravity is locally equivalent to acceleration) and does not require Einstein's Field Equations, unlike the other two effects.

 

[PeterDonis]

As far as I know, the time dilation effect can be predicted from the Principle of Equivalence (i.e. that gravity is locally equivalent to acceleration)

Only locally. The EP cannot explain time dilation over a large enough change in height that differences in g are observable (since it can't even be applied over a patch of spacetime that large).

and does not require Einstein's Field Equations, unlike the other two effects.

This is my understanding as well.