# Do black holes lose hair in finite time?

• A
OK, so it's time to start a new thread.

I heard many times that there exists only one black hole solution for a given mass and angular momentum, but I know already that this is not true.

We all know that if we throw something into an existing black hole, its event horizon starts to ripple. So there are many solutions for a given mass and angular momentum, which can be distinguished by the shape of the event horizon.

The correct true statements about black holes are:
1. There exists only one stationary solution for a given mass and angular momentum.
2. There are no periodic (oscillating) solutions for any mass or angular momentum.

Now we know that any disturbance of event horizon will radiate gravitational waves, losing energy. That means, if we start from a disturbed solution, it will start approaching the only stationary solution.

Now my question is: will a disturbed black hole reach the stationary solution in finite time or will it radiate energy forever?

Quote from "Black Holes in Higher Dimensions", Gary Horowits, p. 17:
The Schwarzshild and Reisner-Nordstroem solutions have been shown to be stable to linearized perturbations. Black holes do not have ordinary normal modes since energy can be lost via radiation out to infinity or via waves falling through the horizon. If one considers perturbations with a harmonic time dependence exp(-i omega t) and imposes boundary conditions that the modes are purely outgoing at infinity and ingoing at the horizon then one finds solutions for a discrete set of complex omega. These are called quasinormal modes. The imaginary part of omega is negative, so these modes oscillate with exponentially decaying amplitude. A typical perturbation of the black hole decays exponentially for a while but falls off like power law at later times. This is a result of backscattering of the perturbation off the curvature around the black hole.

What I understand is that holes can be oscillating, but with decaying amplitude. Nevertheless, in absence of some quantum effects the amplitude never goes to zero. That means classical black holes do have hair. Once the hole got hair it never goes bald.

PAllen
2019 Award
Perhaps you are right in a formal sense. However, if after a fairly short time the GW emitted by the 'hair' of a massive BH are infinitesimal compared to those emitted by snapping your fingers, does it not make sense physically to say the hair has be radiated away?

Buzz Bloom
In consideration of entropy, information paradox and Hawking radiation, yes. Snapping my fingers would probably have bigger impact on the LIGO aperture than gravitational waves from two colliding black holes, but that's not the reason to not to try to detect them.

But most important, classical black holes are not information sinks and they do have entropy.

PAllen
2019 Award
All derivations of BH entropy I've seen involve quantum arguments. It would be interesting if you could point to any computation of BH entropy on a strictly classical basis, especially one due to decaying hair.

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PeterDonis
Mentor
2019 Award
there are many solutions for a given mass and angular momentum, which can be distinguished by the shape of the event horizon.

You have to be careful here, because "mass" and "angular momentum" are properties of the global spacetime geometry, not just of the "black hole" portion of it. For example, suppose I throw some mass with zero angular momentum into a Schwarzschild black hole. We normally say the hole's "mass" increases; but strictly speaking, the mass of the spacetime as a whole stays constant, because the mass I threw in was at a finite radius to start with, so an observer at infinity would see no change at all. There are ways to be more precise about these issues, but it requires care.

PAllen
2019 Award
Bekenstein's original paper on black hole entropy is here:

http://www.itp.uni-hannover.de/~giulini/papers/BlackHoleSeminar/Bekenstein_PRD7_1973.pdf

It provides a good basis for discussion of the relevant issues.
What is interesting about the OP argument is that all statistical mechanics explanations of BH entropy (i.e. counting microstates for a macrostate) I've seen in the literature are explicitly quantum (string theoretic, LQG, the 'universality' explored by Carlip). The OP suggests a possible classical microstate derivation in terms of decayed 'hair'. This seems like a really interesting idea if it could be quantified.

PeterDonis
Mentor
2019 Award
The OP suggests a possible classical microstate derivation in terms of decayed 'hair'.

The OP is suggesting that the microstate is "the shape of the event horizon". However, because of the issues raised in my earlier post, I'm not sure how to pick out the macrostates, i.e., what the thermodynamic variables are. Mass and angular momentum, as I said in that earlier post, aren't really properties of the black hole itself; they are global properties of the spacetime as a whole. (Similarly for electric charge.)

More generally, the issue is that, ordinarily, we define thermodynamic variables based on stationary states, and we look at how thermodynamic variables change when the system transitions from one stationary state to another. But the OP is trying to look at the details of what happens during the transition, which is a different question and might not even be amenable to the same type of thermodynamic analysis.

martinbn
I suppose there are non-stationary solutions, but is there a heuristic argument why a solution will not be stationary after all the matter has fallen in the black hole and nothing else will ever fall in? What I don't find obvious is why the solution shouldn't be stationary.

PAllen
2019 Award
I suppose there are non-stationary solutions, but is there a heuristic argument why a solution will not be stationary after all the matter has fallen in the black hole and nothing else will ever fall in? What I don't find obvious is why the solution shouldn't be stationary.
The OP quotes a respectable reference that the deviations from SC or Kerr fall off exponentially; classically, you never 100% reach the stationary SC or Kerr, though there is no conceivable measurement that can detect the exponentially decayed deviation.

PAllen
2019 Award
The OP is suggesting that the microstate is "the shape of the event horizon". However, because of the issues raised in my earlier post, I'm not sure how to pick out the macrostates, i.e., what the thermodynamic variables are. Mass and angular momentum, as I said in that earlier post, aren't really properties of the black hole itself; they are global properties of the spacetime as a whole. (Similarly for electric charge.)

More generally, the issue is that, ordinarily, we define thermodynamic variables based on stationary states, and we look at how thermodynamic variables change when the system transitions from one stationary state to another. But the OP is trying to look at the details of what happens during the transition, which is a different question and might not even be amenable to the same type of thermodynamic analysis.
I don't quite agree with this. The mass, angular momentum, and charge for a BH are not just the global properties measured at infinity - else they would never change over the whole dynamical process. What is really being discussed is that the spacetime becomes indistinguishable from Kerr or SC, over a 'large spatial region' after a relatively short time. Then, the interesting thermodynamic question is whether one can relate possible decayed horizon oscillation states to the Bekenstein entropy (which is derived without reference to microstates). In the literature, all derivations of Bekenstein entropy in terms of microstates are based on quantum gravity arguments. The hypothesis here is whether there could conceivably be appropriate classical microstates. This seems like a genuinely interesting question, perhaps not addressed in the literature (or I am simply not aware of where it has been).

PeterDonis
Mentor
2019 Award
The mass, angular momentum, and charge for a BH are not just the global properties measured at infinity - else they would never change over the whole dynamical process.

And according to an observer at infinity, they don't. That's the point. To see a "change" in these things at all, you have to come up with some other definition for them besides the standard ones.

PAllen
2019 Award
And according to an observer at infinity, they don't. That's the point. To see a "change" in these things at all, you have to come up with some other definition for them besides the standard ones.

Agreed, you would need to get into the various (imperfect) quasilocal definitions, or the more adhoc approach of fitting a quasilocal region to an asymptotically flat extension of 'appropriate type', and computing e.g. Bondi quantities (to get change due to radiation).

martinbn
The OP quotes a respectable reference that the deviations from SC or Kerr fall off exponentially; classically, you never 100% reach the stationary SC or Kerr, though there is no conceivable measurement that can detect the exponentially decayed deviation.

In that reference there isn't anything more from the quote in the first post. I was just asking for a heuristic explanation, of course a full explanation or a reference would be better.

I don't suppose anyone will see this; but, in the 60s, Wheeler was working with a concept that gravity waves created matter in a manner analogous to ocean waves making droplets when crashing into a sea shore. Much more recently, I have toyed with the possibility that the hairs on a black hole produce elementary particles, both matter and antimatter, thus providing the positron flux from the poles of the black holes.

PeterDonis
Mentor
2019 Award
Wheeler was working with a concept that gravity waves created matter in a manner analogous to ocean waves making droplets when crashing into a sea shore.

"Analogous" only in a very, very loose sense. What Wheeler was talking about was the idea that a sufficiently energetic pair of gravitons (the quantum particle corresponding to gravitational waves) could create a matter particle-antiparticle pair (for example, an electron-positron pair), just as a sufficiently energetic pair of photons can. This should be possible in principle, but we would not expect to observe it in practice because gravity is so weak compared to the other interactions.

Much more recently, I have toyed with the possibility that the hairs on a black hole produce elementary particles, both matter and antimatter, thus providing the positron flux from the poles of the black holes.

What positron flux from the poles of black holes are you talking about?

Also, please bear in mind the PF rules on personal theories.