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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 Q^2\omega^6, where 1/\omega is a measure of the time-scale of Q's time-variation. In the weak field limit, I'm pretty sure P \propto Q^2\omega^6 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 , subsection 9.2.5).
The next question is how to estimate \omega. Still assuming the weak-field limit, I think the appropriate estimate is \omega\sim\dot{r}/r, 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 \dot{r}\sim r^{-1/2}, so \omega\sim r^{-3/2}.
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 1+\epsilon. Then we're basically assuming that \epsilon stays constant.
Then the resulting estimate of the power radiated in gravitational waves is P\sim Q^2\omega^6\sim (r^2)^2(r^{-3/2})^6\sim r^{-5}. 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 \dot{r}\sim r^{-1/2}, since this would exceed the speed of light for small enough r. Suppose we just take \dot{r}= 1 (the speed of light). With this modification, I get P\sim Q^2\omega^6\sim (r^2)^2(r^{-1})^6\sim r^{-2}. 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 \dot{r}= 1, 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 P\propto r^m.
(2) Maybe everything is okay *except* for the assumption that the cloud maintains its shape. Then the interpretation is as follows. By assuming that \epsilon would remain constant, i.e., \epsilon\propto r^0, we got infinite radiated power. This is unphysical. Therefore we conclude that \epsilon must get smaller as r gets smaller. To keep the radiated power from integrating to infinity, we need P\propto r^{m}, where m>-1. This means \epsilon\propto r^n, 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 Q^2\omega^6, where 1/\omega is a measure of the time-scale of Q's time-variation. In the weak field limit, I'm pretty sure P \propto Q^2\omega^6 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 , subsection 9.2.5).
The next question is how to estimate \omega. Still assuming the weak-field limit, I think the appropriate estimate is \omega\sim\dot{r}/r, 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 \dot{r}\sim r^{-1/2}, so \omega\sim r^{-3/2}.
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 1+\epsilon. Then we're basically assuming that \epsilon stays constant.
Then the resulting estimate of the power radiated in gravitational waves is P\sim Q^2\omega^6\sim (r^2)^2(r^{-3/2})^6\sim r^{-5}. 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 \dot{r}\sim r^{-1/2}, since this would exceed the speed of light for small enough r. Suppose we just take \dot{r}= 1 (the speed of light). With this modification, I get P\sim Q^2\omega^6\sim (r^2)^2(r^{-1})^6\sim r^{-2}. 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 \dot{r}= 1, 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 P\propto r^m.
(2) Maybe everything is okay *except* for the assumption that the cloud maintains its shape. Then the interpretation is as follows. By assuming that \epsilon would remain constant, i.e., \epsilon\propto r^0, we got infinite radiated power. This is unphysical. Therefore we conclude that \epsilon must get smaller as r gets smaller. To keep the radiated power from integrating to infinity, we need P\propto r^{m}, where m>-1. This means \epsilon\propto r^n, 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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