# Time dependent exact solution of E.eqs

1. Mar 28, 2014

### Vrbic

Hello,
Im looking for some time dependent exact solution of Einsteins eqs. If Im right (if not please correct me) the easiest one is Robertson - Walker cosmological solution for homogeneous and isotropic universe (this use Oppenheimer and Snyder for collapse). I cant find another in common literature.
Is it some "easyone", not necessary physicaly relevant, solution? Or some list or something like that? Im really curious how such solution in metric form (lenght element) looks like.

Thank you for replies.

2. Mar 28, 2014

### Bill_K

3. Mar 28, 2014

### Staff: Mentor

This is probably just a dumb answer, but how about an object in free fall radially with the Schwarzschild metric?

Chet

4. Mar 28, 2014

### Vrbic

If you mean, free falling frame is time dependant "solution". My opinion is that if you are in spacetime described by Schwarzschild metric, you can transform to any frame but metric coefficient doesnt change. You can observe a test particle in free falling frame, it would have time-dependant coordinates (at least r), but measuring in this frame have to carry out by same metric.
If somebody has a comment to my point of view, please dont worry.

5. Mar 28, 2014

### Staff: Mentor

That wouldn't be a time-dependent solution to the EFE, because the metric is not time-dependent. More precisely, if you take a family of observers free-falling radially, all of them see the *same* metric at any given value of the radial coordinate $r$, so all of their worldlines are exactly the same. The $r$ coordinates of each observer are time-dependent, but the observers' worldlines aren't solutions of the EFE, they're just solutions of the geodesic equation given a fixed metric. The metric is the solution of the EFE.

6. Mar 28, 2014

### PAllen

If you actually talked about a non-zero mass falling toward another much larger mass, that would be time dependent but there is no such exact solution (last I checked). Such a solution would (if exact) involve gravitational radiation.

7. Mar 28, 2014

### Staff: Mentor

As I said, it was probably a naive answer. There is still a lot I would like to learn.

Chet