# Birkhoff's Theorem

1. Oct 20, 2008

### La Guinee

Hi. Birkhoff's Theorem says that the Schwarzschild metric is the unique spherically symmetric vacuum solution. Isn't the Robertson-Walker metric spherically symmetric?

2. Oct 20, 2008

### atyy

The Schwarzschild solution is a vacuum solution. The Robertson-Walker metric may include matter.

3. Oct 20, 2008

### La Guinee

Why can't the Robertson Walker metric be a vacuum solution? The Ricci tensor for the Robertson Walker metric can be easily calculated. Choosing a linear scale factor with appropriate coefficients satisfies R_mu nu = 0 (or am I doing something wrong)?

4. Oct 20, 2008

### MeJennifer

May? The FRW solution contains only matter there is no vacuum in this solution.

Interesting things to consider in the light of this fact:
- Why does the majority of cosmologists consider the FRW solution a rough aproximation of our universe if the FRW has no vacuum.
- How can they possibly make statements about the distance between objects if the FRW solution has no distance between objects.

Beware, it might be similar to asking a catholic priest too many questions about hell, he may answer it is the place whe you go if you keep asking those questions.

Last edited: Oct 20, 2008
5. Oct 20, 2008

### La Guinee

Sorry, I still don't understand. You have the Robertson Walker metric. From this you can construct the Ricci tensor. Now set the components of the Ricci tensor equal to 0. This gives you a condition on the scale factor, namely that it's linear in time. What's wrong with this?

6. Oct 20, 2008

### atyy

So if I keep asking the "majority of cosmologists" too many questions, where do I end up - the singularity?

7. Oct 21, 2008

### atyy

Is there any way to change coordinates to get it into the Schwarzschild form?

8. Oct 21, 2008

### Jonathan Scott

I may have missed something here, but as far as I know:

The RW metric describes a whole universe with the same space-time curvature and energy density everywhere, such as a hypersphere.

The Schwarzschild solution describes the shape of space-time around a single central mass where mass is assumed to be distributed in a spherically symmetrical way about the central point and space-time is assumed to be flat at sufficient distance from that mass.

I find it difficult to see anything in common between these two cases.

9. Oct 21, 2008

### Ich

La Guinee,

you have an interesting point here.
FRW and Schwarzschild are (in the empty case) in fact the same metric, expressed in different coordinates.
My interpretation: You can arrange receding test particles in flat spacetime such, that the overall picture becomes isotropic if you associate an appropriate boost with each space translation, fitting your velocity to that of the test particle at the new location.
FRW space is a slice through Minkowski space that makes the distribution of test particles additionally homogeneous, as necessary for cosmologigal models.

Last edited by a moderator: May 3, 2017
10. Oct 21, 2008

### George Jones

Staff Emeritus
Have you calculated the Riemann tensor in this case? I think

is relevant.

11. Oct 21, 2008

### La Guinee

Thank you for the replies. I don't see how the FRW can reduce to the schwarzschild metric with a suitable coordinate change because the FRW is time dependent.

12. Oct 21, 2008

### George Jones

Staff Emeritus
Careful. Massless FRW reduces to a patch of flat Minkowski spacetime in unusual coordinates. Minkowski spacetime is isotropic.

13. Oct 21, 2008

### La Guinee

Consider the following FRW metric:

ds2 = -dt2 + 9t2 [ dr2 / (1+9r2) + r2 dOmega2 ]

This satisfies einsteins equations in vacuum. So are you're saying under suitable coordinate change this reduces to Minkowski? This seems weird because one is time dependent and the other isn't.

14. Oct 21, 2008

### George Jones

Staff Emeritus
Look at post #30 in the thread to which I previously gave a link.

15. Oct 21, 2008

### La Guinee

I see. Thank you.

16. Oct 21, 2008

### atyy

Birkhoff and Minkowski
http://scienceworld.wolfram.com/physics/BirkhoffsTheorem.html

Burko et al
http://arxiv.org/abs/gr-qc/0008065
"The spacetime is Schwarzschild outside the shell and Minkowski inside the shell. Note that gtt → −1 as r → ∞, but gtt != −1 inside the shell, although spacetime is (locally) Minkowski."

Edit: Heuristically, it seems the Minkowski metric is obtained by setting both the Schwarzschild radial coordinate and mass parameter to zero. I have no idea if this can be rigorously justified.

Edit: I can just set the mass parameter to zero without touching the radial coordinate.

Last edited: Oct 21, 2008