Birkhoff's theorem can be viewed as the simplest of the no-hair theorems about black holes. It basically says that non-rotating, uncharged black holes are fully described just in terms of their mass. More formally, it says that spherically symmetric exterior solutions of the vacuum field equations must be stationary and asymptotically flat, which implies that they're the same as the Schwarzschild metric. Birkhoff published the proof in his 1923 GR textbook (which I've ordered a copy of). Another proof is given here: http://arxiv.org/abs/gr-qc/0408067(adsbygoogle = window.adsbygoogle || []).push({});

I would prefer to be able to give a more elementary argument, even if it comes at the expense of rigor. I came up with the following, which I think demonstrates the stationarity part of Birkhoff's theorem. Okay, maybe "demonstrates" is an overstatement. This is admittedly very hand-wavy. But in any case, I'd be interested in opinions as to whether this is even a believable heuristic argument or not.

If I write down a metric of the form

[tex]

ds^2 = dt^2-f(x-vt)dx^2-dy^2-dz^2 \qquad ,

[/tex]

representing a longitudinal gravitational wave, then the Einstein tensor's nonvanishing components are [itex]G_{yy}=G_{zz}=v^2/4[/itex]. Since it's not zero, this isn't a vacuum solution. (The special case of v=0 gives a flat space, i.e., it's just a coordinate wave, not a real standing wave.) So I conclude that a rectilinearly propagating, longitudinal gravitational wave can't exist. Although my justification for this result is weakened by the assumption that a longitudinal wave can be represented in a certain form in certain coordinates, I think the result is correct; although longitudinal gravitational waves can exist (they're Petrov type III), they're a near-field solution, whereas I'm just talking about rectilinearly propagating solutions. I think the coordinate-specific assumption may also be justifiable in the weak-field limit, since then the metric is approximately Minkowskian, and (t,x,y,z) have to have Minkowskian behavior in the limit of [itex]f\rightarrow 0[/itex].

As a consistency check, this kind of constraint on polarization can only occur for waves that propagate at c, since otherwise you could go into the co-moving frame, where all directions are equivalent. We know that in the low-amplitude limit gravitational waves do propagate at c, so we pass this consistency check. The fact that large-amplitude waves need not propagate at c in GR suggests that it really was necessary to invoke the approximately Minkowskian nature of the metric in the preceding paragraph.

Now suppose you have a spherically symmetric spacetime, and assume it's asymptotically flat. (Birkhoff's proof doesn't need the prior assumption of asymptotic flatness.) At large r, any nonstationary behavior of the metric has to look approximately like a low-amplitude, rectilinearly propagating gravitational wave. (This step is admittedly hand-wavy. You could imagine, e.g., that it was nonstationary because of rotation, but then it wouldn't be spherically symmetric.) Because of the spherical symmetry, this wave would have to be purely longitudinal, and that would be impossible. Therefore the metric must be stationary at large r.

If it's stationary at large r, it must also be stationary at small r, since otherwise the energy being radiated by gravitational waves in the near field would be disappearing before it got out to the far-field region. (You can't have energy being transmitted inward and outward at equal rates, because that would violate spherical symmetry.)

Since the metric is stationary everywhere, and we're assuming it's asymptotically flat, all the assumptions that uniquely determined the Schwarzschild metric (e.g., http://www.lightandmatter.com/html_books/genrel/ch06/ch06.html#Section6.2 [Broken] ) hold.

Does this work for you folks, at least as a plausibility argument?

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# Hand-wavy proof of (part of) Birkhoff's theorem

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