# Relation between gravitational waves and black holes

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This is something I've been curious for some time. I've heard that there is a relation between gravitational waves and black holes. Moreover, this year the quite important paper "Observation of Gravitational Waves from a Binary Black Hole Merger" was published.

Now, I'm starting to study General Relativity and I want to understand better the relation between gravitational waves and black holes from a more rigorous standpoint. In truth I would like to get to some mathematical derivation of that relation. Something that starting from black hole mathematics we end up showing there is a gravitational wave generation.

Is there some paper about that out there? I've been searching for this in the last few days but didn't find anything. I belive this might be in the context of the Kerr black holes as it is the one related to rotation as far as I know. Any paper is appreciated!

## Answers and Replies

PeterDonis
Mentor
2020 Award
Heuristically, the relationship between gravitational waves and black holes is that, any time something happens to a black hole that perturbs the symmetry of its horizon, the hole will radiate gravitational waves until the horizon's symmetry is restored. Note that this applies to any black hole, not just a rotating (Kerr) one. The black hole merger that was recently detected by LIGO is one example of something that perturbs the symmetry of a hole's horizon--the merger creates a larger hole whose horizon is asymmetrical, because the two holes that merged to form it were not identical and were not moving in perfectly symmetrical orbits around their common center of mass before they merged. So gravitational waves are produced by the merger.

The general mathematical result underlying the above is called Price's Theorem; it is described briefly on the Wikipedia page for its discoverer, Richard Price:

https://en.wikipedia.org/wiki/Richard_H._Price

The more general property of black holes that describes what "symmetrical" means is the "no hair theorem"; Price's Theorem shows us that gravitational waves are a way that a black hole that has "hair" because of something that just happened to it radiates away the "hair". But different kinds of "hair" correspond to different kinds of radiation; for example, a hole that has a magnetic field because of something that just happened to it will radiate it away as electromagnetic waves, not gravitational waves. There has been plenty of study of the general "no hair" theorem, and you should be able to find papers on that topic; but I don't know that many of them will talk about gravitational waves specifically.

pervect
Staff Emeritus
This is something I've been curious for some time. I've heard that there is a relation between gravitational waves and black holes. Moreover, this year the quite important paper "Observation of Gravitational Waves from a Binary Black Hole Merger" was published.

Now, I'm starting to study General Relativity and I want to understand better the relation between gravitational waves and black holes from a more rigorous standpoint. In truth I would like to get to some mathematical derivation of that relation. Something that starting from black hole mathematics we end up showing there is a gravitational wave generation.

Is there some paper about that out there? I've been searching for this in the last few days but didn't find anything. I belive this might be in the context of the Kerr black holes as it is the one related to rotation as far as I know. Any paper is appreciated!

I'd suggest reading the LIGO paper https://dcc.ligo.org/public/0122/P150914/014/LIGO-P150914_Detection_of_GW150914.pdf

In particular, the section of the paper that talks about why it is felt that the gravity wave came from the inspiral of a pair of black holes, rather than some other source (such as a neutron star and a black hole) seems relevant to your question.

The basic features of GW150914 point to it being produced by the coalescence of two black holes.
...
A pair of neutron stars, while compact, would not have the required mass, while a black hole neutron star binary with the deduced chirp mass would have a very large total mass,and would thus merge at much lower frequency.

Price's Theorem shows us that gravitational waves are a way that a black hole that has "hair" because of something that just happened to it radiates away the "hair".
I never understood the argument really. Assuming pure (not quantum) GR, if the hole is ratiating hair at exponential rate, it will never radiate them completely! So this is rather the argument that once the hole got hair, it will have them forever (shrinking exponentially though, but never reaching zero anyway).

Is quantum gravity also a part of this argument? Don't physicists add something like "fluctuations hit the Planck size at some point and then they vanish"?

Let me ask again: does pure non-quantum general relativity say that a hairy hole loses all hair completely in finite time?

PeterDonis
Mentor
2020 Award
Assuming pure (not quantum) GR, if the hole is ratiating hair at exponential rate, it will never radiate them completely!

Do you mean "radiate at an exponentially decreasing rate"? AFAIK that's not what GR predicts. But you would probably have to dig into the literature to see the detailed math; I don't know that this subject is treated in detail in textbooks. (Price's Theorem hadn't been discovered yet when MTW was published, IIRC, and I don't remember Wald discussing it in any detail.)

Is quantum gravity also a part of this argument?

No. Quantum "hair" is a separate issue--I believe there are some quantum analogues of no-hair theorems, but I don't think it's as clear cut as it is in pure classical GR.

does pure non-quantum general relativity say that a hairy hole loses all hair completely in finite time?

As best I understand it, yes.