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

[bcrowell]

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

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
$$ds^2 = dt^2-f(x-vt)dx^2-dy^2-dz^2 \qquad ,$$
representing a longitudinal gravitational wave, then the Einstein tensor's nonvanishing components are $G_{yy}=G_{zz}=v^2/4$. 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 $f\rightarrow 0$.

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 ) hold.

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

 

[Mentz114]

You are saying (?), that if the spherically symmetric metric varies with time, there will be gravitational radiation. But such waves are not possible, so the source must be not-varying with time. Works for me.

In Schwarzschild's original paper he starts with this

$$ds^2=-\exp(B(r,t))dt^2+\exp(A(r,t))dr^2+r^2d\theta^2+r^2\sin(\theta)^2d\phi^2$$

as the most general spherically symmetric space time. He then calculates the components of the Ricci tensor and finds that two of them contain time derivatives

$$R_{00}=\exp(A-B)\left[-\frac{A''}{2}-\frac{A'^2}{4}+\frac{A'B'}{4}-\frac{A}{r}\right]+(time\ derivs)$$
$$R_{11}=\frac{A''}{2}+\frac{A'^2}{4}-\frac{A'B'}{4}-\frac{B}{r}+(time\ derivs)$$

From which he concludes that, in order to make R_ab vanish, A and B cannot depend on t. This also works for me.

But it's not rigorous because there is the possibility that $R^p_{qrs}$ contains time derivatives which cancel on contraction to R_qs. I guess this is where Birkhoff comes in.

 

[bcrowell]

Hi, Lut,

Many thanks for taking the time to make such a detailed and informative post :-)

I found the reference to the original paper,

Schwarzschild, K. (1916). Über das Gravitationsfeld eines Massenpunktes nach der Einstein'schen Theorie. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 1, 189-196

and despaired because (a) it was obviously in some foreign language that we Americans don't speak, and (b) it was in an old journal. But then I did a little googling, and found this translation online http://arxiv.org/abs/physics/9905030 , so now I have absolutely no excuse for not reading it very carefully.

Regards,

Ben

 

[Altabeh]

You are saying (?), that if the spherically symmetric metric varies with time, there will be gravitational radiation. But such waves are not possible, so the source must be not-varying with time. Works for me.

In Schwarzschild's original paper he starts with this

$$ds^2=-\exp(B(r,t))dt^2+\exp(A(r,t))dr^2+r^2d\theta^2+r^2\sin(\theta)^2d\phi^2$$

as the most general spherically symmetric space time. He then calculates the components of the Ricci tensor and finds that two of them contain time derivatives

$$R_{00}=\exp(A-B)\left[-\frac{A''}{2}-\frac{A'^2}{4}+\frac{A'B'}{4}-\frac{A}{r}\right]+(time\ derivs)$$
$$R_{11}=\frac{A''}{2}+\frac{A'^2}{4}-\frac{A'B'}{4}-\frac{B}{r}+(time\ derivs)$$

From which he concludes that, in order to make R_ab vanish, A and B cannot depend on t. This also works for me.

How did he conclude from these euqtions that the time derivatives must vanish?! I don't see it by any means. In Ohanian's book "Gravitation and Spacetime" a very rigorous proof of BT is given in which one can observe simply why such conclusion would be made by only looking at the components $$R^{0}_{1}$$ and $$R^{1}_{0}$$ of the mixed Ricci tensor $$R^{\mu}_{\nu}$$.

So in order to find out why a spherically symmetric metric is not time-dependent one has to proceed to calculate the mixed Ricci tensor of the line-element $$ds^2=-\exp(B(r,t))dt^2+\exp(A(r,t))dr^2+r^2d\theta^2+r^2\sin(\theta)^2d\phi^2$$.

... despaired because (a) it was obviously in some foreign language that we Americans don't speak, and (b) it was in an old journal. But then I did a little googling, and found this translation online http://arxiv.org/abs/physics/9905030 , so now I have absolutely no excuse for not reading it very carefully.

This article, I think, is incomplete as it does not contain the original Schwarzschild's proof or maybe the translators got worn out in the middle of their job and stopped translating the rest of paper!:)

AB

 

[Mentz114]

Altabeh:
Thanks for the input.
Having checked in the translation arXiv /physics/9905030, I can see that what I've quoted is not in that paper. I had also forgotten that the translators appear to be anti-bh crackpots.
[sorry, bcrowell, I may have mislead you. Better treat that paper with caution]

The Ricci components I quote are from Stephani (1986) and I left out the crucial one
$$R_{01}=\frac{\dot{B}}{r}$$
and this has to be zero if R_ab vanishes. By symmetry A=B and so all the time derivatives must vanish. I'm sorry I made a mess of my first post ( my excuse is that it's a long time since I last looked at my notes on this subject).

So it's another handwaving argument in favour of Birkoff.

bcrowell's gravity wave approach is a new (to me) twist that has more than a little merit, IMO.

 

[bcrowell]

Thanks, Altabeh and Lut, for the clarification and comments!

For reference, here are some sources of various versions of the Schwarzschild paper and commentaries on it:

http://www.scribd.com/doc/25310028/schwarzschild-1916 -- scan of the original article, in German

http://www.scribd.com/doc/25310624/...-Massenpunktes-nach-der-Einsteinschen-Theorie -- OCR'd text, in German

http://arxiv.org/abs/physics/9905030 -- translation by Antoci and Loinger

http://www.springerlink.com/content/y557m10453478852/ -- a commentary disagreeing with the anti-bh interpretation Antoci and Loinger gave along with their translation; (I think most of the anti-bh stuff is in the GRG version, not the arxiv version)

http://arxiv.org/abs/0709.2257 -- a commentary on the Schwarzschild paper, giving a simpler derivation

This article, I think, is incomplete as it does not contain the original Schwarzschild's proof or maybe the translators got worn out in the middle of their job and stopped translating the rest of paper!:)

I compared the original German paper and the Antoci translation side by side, and they appear to have all the same material. I don't think anything was omitted in the translation...?

 

[Altabeh]

I compared the original German paper and the Antoci translation side by side, and they appear to have all the same material. I don't think anything was omitted in the translation...?

Maybe I'm losing some point here... but in the end I think you should work on your own idea and injecting a little bit more mathematical insight into this issue would sound more challenging and interesting.

AB

 

[yoron]

Looking at it I found On the discovery of Birkhoff's theorem. Interesting.