# Relaxation time for deviations from spherical shape of a black hole's event horizon (and waves)

A different question about truly spherical objects in nature (Do spheres exist in nature?) made me think of a lecture I had been at where, as I recall, it was mentioned that the most perfectly spherical object in nature is in fact (the event horizon of) a black hole.

In the comments of the aforementioned question, I was informed that any deviations from a spherical shape of such an event horizon would be damped out within a very short time, related to the characteristic timescale of the system. So I was wondering what order of timescales exactly we're talking about here, I would appreciate some elaboration on this.

And something I thought of just now: would it be possible for some wave (periodic) phenomenon to occur in this horizon, periodically distorting the spherical shape?

On the question you mentioned, a commentator said, "astrophysicists would be very surprised to find a nonrotating black hole in nature". And the event horizon of a rotating black hole isn't actually going to be spherical.

Anyways, the relaxation to an oblate shape might be quick. Now, this is a messy business. There have been approximation and numerical methods used to analyse the merger of two black holes. These are way over my head, but Figure 2 of Binary black hole mergers in Physics Today (2011) shows the ring-down time being a hundred or so times $GM/c^3$. ($GM/c^3$ was the characteristic time mentioned in the comments to the other question, so this is in agreement with what was said there.)

For a solar mass black hole, the characteristic time is about 5 microseconds. The supermassive black hole at the centre of our galaxy is thought to be about 4 million solar masses, so the time would be about 20 seconds. So the ring-down time even for that monster would be only about 2000 seconds, or let's say half an hour.

That said, this only models how long it takes huge distortions to relax to small distortions. It's not clear to me that small distortions have as fast a relaxation time to even smaller distortions. More precisely, I don't see why the decay would be exponential. Again, it's over my head. [Maybe this should be another question.]

You also asked if there could be other periodic disturbances of the horizon. Technically, no, any disturbance would be subject to some damping, because it would have to produce gravitational radiation. If an object were orbiting the black hole, for example, that would have to distort the event horizon as it passed over it, while its orbit would decay via radiation. But the power radiated doesn't scale linearly with mass of the orbiting body. For very small disturbances, it could take a very long time, and you could have an almost periodic scenario. (In the limit, test particles have stable orbits and produce no distortion of the horizon.)

• I'm not entirely able to grasp all the details of the gravitational radiation process you describe in your final paragraph yet. (I'm set to follow a GR course next semester) But I can understand the principle and the reason for damping to occur. In any case, I was hoping for someone to come in and expand on the dependence of the relaxation time on the amplitude of the distortion. I didn't accept before to sollicit more viewers, but it seems better to ask a new question indeed. Thanks for the answer! – Wouter Jun 22 '13 at 17:56

How would you detect the rotation of a black hole , nothing escapes . If the ergosphere or a black holes force is oblate is then centrifugal /centripetal force stronger than a black holes gravity !