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On black hole thermodynamics, singularity, and gravitational entropy

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Abstract

Black holes were found to possess properties that mirror ordinary thermodynamical systems in the landmark paper by Bardeen, Carter and Hawking almost half a century ago. Since then much progress has been made, but many fundamental issues remain. For example, what are the underlying degrees of freedom of a black hole horizon that give rise to said thermodynamical properties? Furthermore, classical black holes also harbor a spacetime singularity. Although it is often believed that quantum gravity would “cure” the singularity, as emphasized by Penrose, this viewpoint requires a deeper examination. In this review, I will examine the possibility that singularities remain in quantum gravity, the roles they may play, and the possible links between singularity and black hole thermodynamics. I will also discuss how—inspired by Penrose’s Weyl curvature hypothesis—gravitational entropy for a black hole can be defined using curvature invariants, and the surprising implication that the entropy of black holes in different theories of gravity are different manifestations of spacetime curvature, i.e., their underlying microstructures could be different. Finally, I review the “Hookean law” recently established for singly rotating Myers-Perry black holes (including 4-dimensional Kerr black holes) that connect black hole fragmentation—a consequence of the second law of black hole thermodynamics—with the maximum “Hookean force”, as well as with the thermodynamic geometry of Ruppeiner. This also suggests a new way to study black hole microstructures, and hints at the possibility that some black holes are beyond the Hookean regime (and thus have different microstructures). While examining the remarkable connections between black hole thermodynamics, spacetime singularities and cosmic censorship, as well as gravitational entropy, I shall point out some subtleties, provide some new thoughts, and raise some hard but fundamental questions, including whether black hole thermodynamics is really just “ordinary thermodynamics” or something quite different.

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Notes

  1. In gauge/gravity duality, the physics of a system in the presence of gravity inside an asymptotically anti-de Sitter spacetime (“the bulk”) is equivalent to the physics of another non-gravitational field theory with one less spatial dimension at the conformal boundary. However, in most interesting applications, the field theory is not completely conformal. For example, when the field theory has a temperature dual to the Hawking radiation in the bulk, the temperature gives a length scale to the boundary. Hence the term “gauge/gravity duality” is more accurate than “Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence”. I have not much to say about this important aspect of theoretical physics, though there will be occasional related remarks.

  2. An alternative view that some sort of remnant persists is not entirely ruled out [22], though they are widely argued to be problematic. Indeed, the recently popular weak gravity conjecture [23, 24] was partly formulated to prevent such a state from ever arising.

  3. Here we recall Penrose’s statement to the contrary [25]: “It is hard to avoid the conclusion that the endpoint of the Hawking evaporation of a black hole would be a naked singularity - or at least something that one [sic] a classical scale would closely resemble a naked singularity.” In fact, Russo has argued that a complete evaporation would lead to a catastrophic event at the end of the evaporation [26]—an outbrust of Planckian curvature wave similar to a “thunderbolt” singularity [25, 27, 28].

  4. In fact, cosmic censorship is not a strictly classical phenomenon, for under some situations one needs quantum effects to maintain the censorship [29, 30] that would have otherwise failed. See also [31].

  5. We can further compare this with the ultraviolet catastrophe (Rayleigh-Jeans catastrophe) in classical thermodynamics. In that case, a divergence was predicted by the classical theory, which of course did not match observations. Once quantum mechanics is taken into account, the infinity is “cured” and one obtains the Planck spectrum. Thus quantum mechanics is required to explain the observed spectrum. In our context, however, the classical theory of GR already predicts the correct observation: namely there is a censorship bound, without which one runs into the problem with naked singularity. If QG cures away the singularity, it also removes the need for cosmic censorship. So quantum modifications in this case, seemingly introduced a new question instead of solving an existing one.

  6. Interestingly, and somewhat surprisingly, there may be nontrivial connections between Lorentz symmetry and the second law of thermodynamics [47,48,49], and local Lorentz symmetry may be an emergent property of the macroscopic world with origins in a microscopic second law of causal horizon thermodynamics [50].

  7. Note that the Schwinger effect is necessarily suppressed for all values of black hole parameters [58], but it is crucial that the term be included, otherwise all Reissner-Nordström black holes will tend to extremality, and potentially violates cosmic censorship [59].

  8. It is worth emphasizing again at this point that singularities are problematic only when they cause the theory to lose predictability. If the instability of the inner horizon turns into a spacelike singularity, there is actually no problem with predictability, and as such, is a good thing. Many physicists, however, are uncomfortable with singularities, even if they are spacelike.

  9. The estimate made by Penrose is essentially based on the number of microstates corresponding to the maximum possible entropy, which is given by the Bekenstein–Hawking entropy of a black hole with the total mass of the observable universe. Surprisingly, this number is also close to the one obtained by assuming we are living inside a de Sitter spacetime (now that we know that the universe undergoes an accelerated expansion), in which the de Sitter horizon entropy is \(S\sim 1/\Lambda \sim 10^{122}\) in the Planck units. The current actual entropy inside the observable universe is about \(10^{104}\), which also includes contributions from supermassive black holes [91]. This is an example of the “holographic principle” in action, that the entropy of the cosmic horizon bounds the interior entropy.

  10. A strengthened version of the theorem can be proved even when the NEC is somewhat violated [95]. At present it is not clear whether the relaxed energy conditions that guarantee the validity of a semi-classical version of the area theorem would also be the same conditions that forbid Hawking radiation and vice versa.

  11. A lower bound on the entropy of a black hole can be deduced from a bound on the minimal (Tolman) redshift factor of gravitational waves emerging from the vicinity of its horizon [97].

  12. The Clifton-Ellis-Tavakol gravitational entropy was also shown to be increasing during structure formation in Szekeres Class I models [100]. Interestingly, in the presence of shear, the Clifton-Ellis-Tavakol gravitational entropy can increase even though the Weyl curvature decreases. Thus, the Weyl curvature hypothesis is not strictly valid in all cosmological spacetimes [101].

  13. It is debatable whether this is really a problem. For example, when singularity does form when a black string or black ring undergoes “pinch-off” under Gregory-Laflamme instability [104], the loss of classical predictability of the system is argued to be quite small. This is also the case for the naked singularity formation under black hole collision in higher dimensions. To quote [105], “even if cosmic censorship is violated, its spirit remains unchallenged.” Is a little violation acceptable?

  14. Here \(\Omega _{d-2}\) is the area of the unit \((d-2)\)-dimensional sphere, not to be confused with the angular velocity of the horizon, which I will denote \(\Omega _+\).

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Acknowledgements

I thank the organizing committee for the invitation to present my research during the SCRI21 meeting, as well as for the opportunity to contribute to the proceeding. It is an honor to participate in such a tribute to Sir Roger Penrose. Instead of focusing on a small aspect that I talked about in the presentation during the meeting, I decided to write a review to organize some of my current thoughts, which span some topics that are related to or inspired by Penrose’s own works, including the singularity theorem, cosmic censorship conjecture, and the Weyl curvature hypothesis. I would like to take this opportunity to thank Penrose himself for his remarkable contributions to mathematics, physics, and philosophy, which continue to provide rich ideas for research. I also thank the National Natural Science Foundation of China (No. 11922508) for funding support.

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  1. Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, 180 Siwangting Road, Yangzhou, Jiangsu Province, 225002, China

    Yen Chin Ong

  2. Shanghai Frontier Science Center for Gravitational Wave Detection, School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, 200240, China

    Yen Chin Ong

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Ong, Y.C. On black hole thermodynamics, singularity, and gravitational entropy. Gen Relativ Gravit 54, 132 (2022). https://doi.org/10.1007/s10714-022-03008-0

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