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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 r

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).

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 r

^{2}, 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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