Abstract
We attempt to provide a mesoscopic treatment of the origin of black hole entropy in (3 + 1)-dimensional spacetimes. We ascribe this entropy to the non-trivial topology of the space-like sections Σ of the horizon. This is not forbidden by topological censorship, since all the known energy inequalities needed to prove the spherical topology of Σ are violated in quantum theory. We choose the systoles of Σ to encode its complexity, which gives rise to the black hole entropy. We present hand-waving reasons why the entropy of the black hole can be considered as a function of the volume entropy of Σ . We focus on the limiting case of Σ having a large genus.
Highlights
The statistical origin of the black hole entropy [1,2,3,4,5] has been a perplexing problem since the earliest works on black hole thermodynamics more than forty years ago [6,7,8]
A space-like section Σ of the horizon whose area A(Σ) is a multiple of the entropy must have spherical topology: this is the content of a theorem due to Hawking [14,15] and of the closely related “topological censorship”
In a very closely related work [19], we explored the possibility of assuming the preservation of the spherical topology of Σ, where we ascribed the thermodynamic entropy of Σ to a different measure of its complexity rather than its genus, which in the case of a spherical topology is trivial
Summary
The statistical origin of the black hole entropy [1,2,3,4,5] has been a perplexing problem since the earliest works on black hole thermodynamics more than forty years ago [6,7,8]. All classical energy conditions such as the ones in [15] are known to be violated, point-wise at least, at the quantum level This allows for the possibility of the sections Σ of the horizon of such stationary asymptotically flat spacetimes even in (3 + 1) dimensions to have any topology, at least at such a mesoscopic level. Axioms 2020, 9, 30 the black hole entropy We push this fact to a logical limit, in the present work, by assuming that this “wrinkling” is a topology change of the horizon, which is not precluded because of the violation due to the quantum effects of the classical energy conditions.
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