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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?

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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 !


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

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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. –  Marek Dec 7 '10 at 9:48
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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 ! –  user346 Dec 7 '10 at 23:06
    
Ah, right. Thanks for giving answer to my dumb question :-) –  Marek Dec 8 '10 at 18:06
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No such thing as asking a "dumb" question ! Only dumb thing is remaining dumb and not asking :) –  user346 Dec 9 '10 at 1:44
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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. –  Jerry Schirmer Jan 7 '11 at 3:52

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