Time Dependent Exact Solutions of Einstein's Equations

[Vrbic]

Hello,
I am looking for some time dependent exact solution of Einsteins eqs. If I am 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 can't find another in common literature.
Is it some "easyone", not necessary physicaly relevant, solution? Or some list or something like that? I am really curious how such solution in metric form (lenght element) looks like.

Thank you for replies.

 

[Bill_K]

The first one that comes to mind is the pp wave:

https://en.wikipedia.org/wiki/Pp-wave_spacetime

 

[Chestermiller]

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

Chet

 

[Vrbic]

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

Chet

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 doesn't 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 don't worry.

 

[PeterDonis]

how about an object in free fall radially with the Schwarzschild metric?

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.

 

[PAllen]

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

Chet

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.

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.

 

[Chestermiller]

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 doesn't 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 don't worry.

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.

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.

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

Chet