What kind of systems of black holes satisfy the laws of black hole thermodynamics?

Question

I've come across black holes thermodynamics multiple times recently (both at this site and elsewhere) and some things started bugging me.

For one thing, first law bothers me a little. It is a reflection of the law of conservation of energy. This is fine when the space-time is stationary (as in Kerr solution) and is consistent with system being in a thermal equilibrium (so that thermodynamics apply at all). But what about more general systems of black holes?

What are the assumptions on the system of black holes for it to be in thermal equilibrium so that laws of BH thermodynamics can apply?

Note: the reason I am asking is that I heard that the laws should also be correct for system of multiple BH (so that their total event horizon area is increasing, for example). But I cannot wrap my head about how might the system of BH be in thermal equilibrium. I mean, they would be moving around, generating gravitational waves that carry energy away (violating the first law) all the time. Right?

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user Marek

Answer 1

The first law is not violated if stated properly:

$$ dM = \frac{\kappa}{2 \pi}dA + \Omega dJ + \Phi dQ $$

(reference: wikipedia)

where $\kappa,\Omega,J,\Phi,Q$ are the surface gravity,angular velocity, angular momentum, electric potential and charge of the black hole, respectively. Compare this to the usual expression for the first law:

$$ dE = TdS + PdV + \mu dN $$

One can (heuristically) make the identifications $ T = \frac{\kappa}{2\pi}$, $ S = A/4 $, $ \mu = \Phi $ and $ N = Q $. The first two of these are well-established. Hawking showed that the temperature of the black hole is proportional to its surface gravity ($T \propto \kappa$) and Bekenstein showed that its entropy should be proportional to its area ($S=A/4$). The third and fourth equalities ($\mu = \Phi$ and $ N = Q $) can be understood if we think of the black hole as an aggregate of N particle with unit charge. Adding another charged particle to this ensemble of $N$ particles, with total charge $Q$, will cost an amount of work given by $\Phi dQ$.

For the case of a single black hole, one can use the framework of dynamical horizons developed by Ashtekar, Badri Krishnan, Sean Hayward and others [refs 1, 2]. Turns out the laws of black hole entropy can be extended to completely dynamical black holes with a well-defined expressions for the first and second laws in terms of fluxes through the dynamical horizon.

The definition of a dynamical horizon is in terms of the expansion of the inward pointing, null normal vector field on the 2+1 d boundary of a 3+1 d region. I can't think of the detailed expressions of the top of my head, but you can find them in the above reference.

I can't give a concrete answer for the case of multiple black holes, but I would think you could extend the dynamical horizon framework to that case - probably not without some serious effort though.

In any case, there will be no violation of energy conservation. The sum of the energy and momentum carried off by gravitational waves (and detected by an observer at infinity) and the change in the black hole's energy and momentum (again w.r.t such an asymptotic observer) will remain a constant.

Hope that helps !

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Edit: Corrected proportionality constants for $T$ and $S$. Thanks @Jeff!

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user user346

Comments

Thank you, this helps a little (and I'll look at those papers). But I don't understand your last paragraph. While it's true that the sum of energy of gravitational waves and BH should be conserved, the gravitational waves are not accounted for in the first law (or are they?) and so obviously their part of energy will disappear from system.

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user Marek

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Gravitational waves carry linear and angular momentum, right? In averaging over the radiated gravitational wave flux through some boundary surface containing the black hole, we can get rid of the linear momentum. The rotational energy carried away by gravitational waves in then accounted for by the $ \Omega dJ $ term in the first law. During gravitational collapse this is the process by which black holes lose their "hair", btw !

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user user346

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Ah, right. Thanks for giving answer to my dumb question :-)

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user Marek

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No such thing as asking a "dumb" question ! Only dumb thing is remaining dumb and not asking :)

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user user346

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You might want to edit your answer so that the numerical factors are correct. $T= \kappa/2 \pi$, $S=A/4$.

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user pho

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Thanks for the corrections @Jeff. I'm on it!

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user user346

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In fact, not only can the laws be extended, but the area increase of the horizon can be explicitly calculated in terms of the fields crossing the horizon using the dynamical horizons formalism. It's really an amazing set of work that is underappreciated by a general physics audience.

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user Jerry Schirmer

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Also, the definition of an IH/DH isn't dependent on a 3+1 slicing. It can be defined intrinsically on a null/spacelike hypersurface. And heh, I know a reference: arxiv.org/abs/1009.0934

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user Jerry Schirmer

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Congratulations on finishing up @Jerry :) The slice-independence of IH/DHs is a very important fact, indeed. Thanks for pointing it out!

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user user346

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@space_cadet: thanks!

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user Jerry Schirmer

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@user346: Marek may have accepted the answer that the gravitational waves are reducing the "omega" in the BH thermodynamics, but it is completely false. The potential energy of the two black holes is increasingly negative, and this is what balances the outgoing radiation. The black hole theromodynamics is for the hole itself.

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user Ron Maimon

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@user346: -1 this answer is totally wrong! The radiated energy is mostly reducing the potential energy of the orbiting black holes, and you have not included that energy contribution, and pretended that the total energy of the multi-black-hole system is just the sum of the energies of its constituent black holes, without their gravitational potential energy.

This post imported from StackExchange Physics at 2014-04-01 16:46 (UCT), posted by SE-user Ron Maimon