# Schwarzschild solution

1. Aug 15, 2004

### blue_sky

Where can I found the rigorous Schwarzschild solution INSIDE a body?

2. Aug 15, 2004

### jcsd

3. Aug 15, 2004

### pervect

Staff Emeritus
The Scwarzschild solution works inside as well as outside a body, in the sense that it satisfies the field equations rigorously.

However, the usual symmetrical solution is probably not stable inside the event horizon. Kip Thorne talks about this in his popular book "Black Holes & Time Warps". The reference he gives is the "BKL" singularity, BKL being Belinsky, Khalatnikov, and Lif****z.

The bibliography gives BKL, 1970, "Oscillatory Approach to a Singular Point in the Relativistic Cosmology,", Advances in Physics, 19, 525 and BKL, 1982, "Solution of the Einstein Equations with a Time Singularity," Advances in Physis, 3, 639. I haven't seen any of the original papers personally, though, just what Thorne wrote in his popularization.

Other people have proposed different singularities, Thorne seems to feel that the BKL paper is the one that is most likely correct.

4. Aug 15, 2004

### jcsd

I think he's talking about the solution inside spherically symmetric objects of constant density, at least that's how I read it.

5. Aug 16, 2004

### DW

Its not called the Schwarzschild solution, but the derivation of the weak field solution for steller interiors is touched on in MTW's Gravitation. More generally the following works:
$$ds^{2} = (1 + \frac{2\Phi}{c^2})dct^{2} - \frac{dr^{2}}{1 - \frac{2r}{c^2}\frac{d\Phi}{dr}} - r^{2}d\theta ^{2} - r^{2}sin^{2}\theta d\phi ^{2}$$
Where $$\Phi$$ is the Newtonian potential for the spherically symmetric matter distribution as a function of r. To verify this as a weak field solution simply enter it into Einstein's field equations to see that to first order in the potential it yields $$T^{00} \approx \frac{\nabla ^{2}\Phi }{4\pi G}c^{2}= \rho c^{2}$$ and all other $$T^{\mu \nu} \approx 0$$. GRTensor II for Maple works well for this sort of task. Just enter the spacetime and then ask it for the Einstein tensor. MTW's is the special case of this equation for constant density.

Last edited: Aug 16, 2004
6. Aug 16, 2004

### blue_sky

Yup, you right but not costant density; I'm looking for the solution inside spherically symmetric objects with density following the rules of a perfect gas.
in particular I'm looking for p=p(r) in that case.

blue

7. Aug 16, 2004

### DW

I just gave it to you. The solution is consistent with an ideal gass. Aside from "Modern Relativity" the web site, good luck finding the the case for arbitrary density anywhere.
http://www.geocities.com/zcphysicsms/chap9.htm#BM108
Problem 9.2.5

Last edited: Aug 16, 2004