Some technical questions about Birkhoff's theorem

[bcrowell]

Here are two statements of Birkhoff's theorem:

"Any spherically symmetric solution of the vacuum field equations must be stationary and asymptotically flat." -- http://en.wikipedia.org/wiki/Birkhoff's_theorem_(relativity)

"Any C² solution of Einstein's empty space equations which is spherically symmetric in an open set V, is locally equivalent to part of the maximally extended Schwarzschild solution in V." -- Hawking and Ellis

Hawking and Ellis also remark that the C² requirement can be relaxed to continuity plus piecewise C¹.

Question #1: Why do H&E need "locally?" Can anyone give an example where this becomes relevant, preferably one that would constitute a counterexample if "locally" were omitted?

Question #2: Presumably smoothness is needed because otherwise you could stitch together counterexamples out of a patchwork quilt of random stuff; but wouldn't the non-smooth joins violate the vacuum field equations? Or is it possible to have "kinks" in a spacetime without violating the vacuum field equations? Presumably one would have to discuss this in terms of some kind of limiting process, since the field equations involve second derivatives of the metric, which won't even be well defined if the metric isn't a C² function of the coordinates.

Question #3: I'm tempted to conclude from Birkhoff's theorem that naked singularities in GR can't be spherically symmetric, and this seems to be consistent with the fact that the examples in the WP article on naked singularities all seem to be rotating solutions that clearly aren't spherically symmetric. But I'm not clear on how the smoothness condition applies here. Maybe you could have a spherically symmetric naked singularity with a metric that wasn't C², and it wouldn't violate Birkhoff's theorem?

Question #4: To me, the essential point of Birkhoff's theorem is that there's no such thing as gravitational monopole radiation. Am I right in thinking that, say, a pointlike source of gravitational quadrupole radiation would be considered asymptotically flat (because the curvature falls off faster than some power of r) but not stationary? The technical definition of asymptotic flatness (e.g., ch. 11 of Wald) is very complicated. Is there some rule of thumb about how fast some measure of curvature would typically have to fall off as a function of r if the spacetime was to be considered asymptotically flat?

 

[Ich]

I suck at math, so take my answers with a grain of salt or treat them as questions if appropriate.

Question #1: Why do H&E need "locally?" Can anyone give an example where this becomes relevant, preferably one that would constitute a counterexample if "locally" were omitted?

Toroidal empty space. I think the locally means something like "up to topology".

Question #2: Presumably smoothness is needed because otherwise you could stitch together counterexamples out of a patchwork quilt of random stuff; but wouldn't the non-smooth joins violate the vacuum field equations?

I think they would. I don't know if there are possible solutions with a kink in the Weyl curvature, however. Doesn't seem reasonable.

Question #3: I'm tempted to conclude from Birkhoff's theorem that naked singularities in GR can't be spherically symmetric

What about shell singularities? Like crossing shells in a LT dust.

Am I right in thinking that, say, a pointlike source of gravitational quadrupole radiation would be considered asymptotically flat (because the curvature falls off faster than some power of r) but not stationary?

Not static, but stationary, I think.

Is there some rule of thumb about how fast some measure of curvature would typically have to fall off as a function of r if the spacetime was to be considered asymptotically flat?

I think it's enough if it converges to zero, no matter how fast.

 

[bcrowell]

Thanks, Ich!

What about shell singularities? Like crossing shells in a LT dust.

Are those really singularities in the sense that scalar observables blow up there?

 

[Ich]

I think that the density blows up, but that could be a harmless kink.

 

[atyy]

Question #3: I'm tempted to conclude from Birkhoff's theorem that naked singularities in GR can't be spherically symmetric, and this seems to be consistent with the fact that the examples in the WP article on naked singularities all seem to be rotating solutions that clearly aren't spherically symmetric. But I'm not clear on how the smoothness condition applies here. Maybe you could have a spherically symmetric naked singularity with a metric that wasn't C², and it wouldn't violate Birkhoff's theorem?

Maybe if the singularity is hidden in matter, so that we don't have a vacuum solution near it, although the vacuum solution may be used in the exterior?

http://arxiv.org/abs/gr-qc/0608136