On black hole thermodynamics, singularity, and gravitational entropy

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