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Hawking and Ellis state the second law of black hole thermodynamics as their Proposition 9.2.7 (p. 318), in language that is a little opaque to me. Below the formal statement of the theorem, they give an interpretation, which talks about black holes growing and black holes merging in such a way that the total area increases.

In ordinary thermodynamics, there are processes in which one object's entropy goes down, but another object's entropy increases by a greater amount. Is this possible for black holes? In other words, is it possible for two black holes to interact without merging, in such a way that the total area increases? If so, then what would this interaction process look like? If not, then why not?

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    $\begingroup$ Area transfer could easily be accomplished if black holes are coupled to quantum fields. If there are mechanisms that would prevent the merging and contain the radiation, then, yes, they would be exchanging entropy and moving along toward an equilibrium configuration. $\endgroup$
    – A.V.S.
    Apr 4 '19 at 9:44
  • $\begingroup$ @A.V.S. Good comment. However, if quantum processes like Hawking radiation are within the scope of the question, then the classical area theorem can be violated, because even an isolated black hole will eventually evaporate if no radiation-containing mechanism is present. I don't have a copy of Hawking and Ellis in front of me, but since it's called a "Proposition" (as in theorem), I assume they're talking about classical GR without quantum effects. $\endgroup$ Apr 5 '19 at 2:25
  • $\begingroup$ @ChiralAnomaly: Hawking & Ellis is indeed about pure classical situation, but mine was more of a comment on the use of intuition from thermodynamics with black holes. To be applicable, it requires relaxation mechanism, which would be the Hawking radiation. $\endgroup$
    – A.V.S.
    Apr 5 '19 at 3:50
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This should really just be a comment, because I have not worked through the proof and can't offer any intuition, but the following excerpts from reputable sources seem to answer the question. The first source outlines the proof, and the second source (which cites the first) states the conclusion more clearly.

As time increases, black holes may merge together and new black holes may be created by further bodies collapsing but a black hole can never bifurcate.

In thermodynamics one can transfer entropy from one system to another, and it is required only that the total entropy does not decrease. However one cannot transfer area from one black hole to another since black holes cannot bifurcate ([1, 2, 3]). Thus the second law of black hole mechanics requires that the area of each individual black hole should not decrease.

Of course, these statements assume classical general relativity. The weak energy condition also seems to be assumed.

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  • $\begingroup$ Thanks, that's helpful. However, I think they're only claiming that it's impossible for black holes to have a transfer of area involving a merger followed by bifurcation. (I could be wrong, but that's my reading of their logic.) My question is specifically about a transfer of area without a merger. $\endgroup$
    – user4552
    Apr 4 '19 at 15:30
  • $\begingroup$ In Bardeen 1973, reference 1 is to Hawking 1972, the first one you quoted from; reference 2 seems to be to a chapter Hawking contributed to a book, which we probably won't be able to access easily; and reference 3 is to Hawking and Ellis. H&E has two index entries for bifurcation of black holes. One is to prop. 9.2.5, pp. 315-316, which is the proof that bifurcation is impossible. $\endgroup$
    – user4552
    Apr 4 '19 at 15:35
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    $\begingroup$ In higher dimensional GR bifurcation is possible (for example as developement of Gregory–Laflamme instability). Does this mean that area transfer is also possible? $\endgroup$
    – A.V.S.
    Apr 5 '19 at 3:54
  • $\begingroup$ I don't think this helps, because the are of the event horizon cannot decrease even locally, thus each black hole can only increase. But the question is whether the geometry can change, so that each black hole increases in area without mergers. $\endgroup$
    – MBN
    Apr 5 '19 at 14:28
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    $\begingroup$ @BenCrowell Theorem 12.2.6 (black hole area theorem) in Wald's book General Relativity outlines a proof that the area of the horizon cannot decrease even locally. The proof uses the expansion $\theta$ of the null geodesic generators of the event horizon, not just a generic null congruence. Related: arxiv.org/abs/1504.07627, arxiv.org/abs/1711.06480, and references in arxiv.org/abs/1804.10610 $\endgroup$ Apr 7 '19 at 15:02

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