Asymmetry in black hole formation, and a possible hand-wavy no-hair theorem

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bcrowell

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Suppose that we have an ellipsoidal shell of particles, all initially at rest in some frame, which are going to collapse to form a black hole. Since the cloud has a nonvanishing mass quadrupole moment Q, and Q is varying with time, we should get gravitational radiation.

First let's consider the weak-field limit, i.e., the early rather than late stages of the collapse.

Based on very general arguments about quadrupole radiation, the radiated power should scale as [itex]Q^2\omega^6[/itex], where [itex]1/\omega[/itex] is a measure of the time-scale of Q's time-variation. In the weak field limit, I'm pretty sure [itex]P \propto Q^2\omega^6[/itex] is right, since it gives the right result for gravitational radiation from the Hulse-Taylor binary pulsar, up to a dimensionless constant of order unity ( http://www.lightandmatter.com/html_books/genrel/ch09/ch09.html#Section9.2 [Broken] , subsection 9.2.5).

The next question is how to estimate [itex]\omega[/itex]. Still assuming the weak-field limit, I think the appropriate estimate is [itex]\omega\sim\dot{r}/r[/itex], where it doesn't really matter what time coordinate the dot is talking about differentiation with respect to, because time dilation effects are small. By conservation of energy, we get [itex]\dot{r}\sim r^{-1/2}[/itex], so [itex]\omega\sim r^{-3/2}[/itex].

As a crude approximation, let's imagine that Q scales like r2, i.e., that the whole cloud just shrinks uniformly without changing the proportions of its axes. Say the cloud is a spheroid, with two equal axes, and the ratio of the short to long axes is [itex]1+\epsilon[/itex]. Then we're basically assuming that [itex]\epsilon[/itex] stays constant.

Then the resulting estimate of the power radiated in gravitational waves is [itex]P\sim Q^2\omega^6\sim (r^2)^2(r^{-3/2})^6\sim r^{-5}[/itex]. Except for the possibly flaky assumption about uniform shrinking of the cloud, I'm pretty confident that this is a valid weak-field estimate. This estimate blows up so badly for small r that if you integrate it, you get infinite radiated power, which is clearly wrong -- but there was no reason to expect it to be right in the limit of small r, because it was derived using weak-field approximations.

So what needs to be changed to get any hope of a reasonable estimate in the strong-field case? Well, for one thing we can't have [itex]\dot{r}\sim r^{-1/2}[/itex], since this would exceed the speed of light for small enough r. Suppose we just take [itex]\dot{r}= 1[/itex] (the speed of light). With this modification, I get [itex]P\sim Q^2\omega^6\sim (r^2)^2(r^{-1})^6\sim r^{-2}[/itex]. This expression blows up much less badly at small r than the weak-field one, but integrating it still produces a result that diverges, so that's still unphysical.

I can see two possible ways of interpreting this:

(1) Maybe my method of tinkering with the weak-field result in order to go over to the strong field, simply by taking [itex]\dot{r}= 1[/itex], was overly simplistic. Maybe all kinds of other modifications have to be made, even if all we want is to get something as crude as the right exponent in [itex]P\propto r^m[/itex].

(2) Maybe everything is okay *except* for the assumption that the cloud maintains its shape. Then the interpretation is as follows. By assuming that [itex]\epsilon[/itex] would remain constant, i.e., [itex]\epsilon\propto r^0[/itex], we got infinite radiated power. This is unphysical. Therefore we conclude that [itex]\epsilon[/itex] must get smaller as r gets smaller. To keep the radiated power from integrating to infinity, we need [itex]P\propto r^{m}[/itex], where [itex]m>-1[/itex]. This means [itex]\epsilon\propto r^n[/itex], where n>1/2. In other words, the cloud has to become more spherical as it collapses, and we can put a bound on how fast it has to lose its deformation.

If #2 were right, it would be kind of sweet. It would be a very simple and direct way of proving the simplest no-hair theorem, the one for the case of zero angular momentum and zero charge (i.e., the case that normally requires Birkhoff's theorem).
 
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If #2 were right, it would be kind of sweet.
I think #2 has to be right for the idealized case you describe, which by hypothesis has exactly zero angular momentum. The gravitational wave emission would have to be such that the net angular momentum of all emitted waves was zero; then the final black hole would be a Schwarzschild hole and would be perfectly spherical.

A more interesting case, I think, would be that of a rotating ellipsoidal cloud, which would open up the possibility of the final hole being a Kerr hole.
 

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