Phys. Rev. D 111, 041503(2025)
Black hole thermodynamics from an ensemble-averaged theory
Peng Cheng1,*, Yu-Xiao Liu2, and Shao-Wen Wei2
1 Center for Joint Quantum Studies and Department of Physics, Tianjin University, 300350 Tianjin, People’s Republic of China.
2 Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, and Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou 730000, Gansu, China.
* p.cheng.nl@outlook.com
Abstract
The path integral approach to a quantum theory of gravity is widely regarded as an indispensable strategy. However, determining what additional elements, beyond black hole or anti–de Sitter (AdS) spacetime, should be incorporated into the path integral remains crucial yet perplexing. We argue that the spacetime with a conical singularity in its Euclidean counterpart should be the most important ingredient to append to the path integral. Therefore, physical quantities should be ensemble averaged over all geometries since they are described by the same Lorentzian metric. When the ensemble average is introduced, the Hawking-Page transition for the Schwarzschild-AdS black hole and the small-large black hole transition for the Reissner-Nordström-AdS black hole naturally arise as semiclassical approximations, when the size of the black hole system is much larger than the Planck length. Away from the semiclassical limit, the system is a superposition of different geometries, and the averaged quantities would deviate from the black hole thermodynamics. Expanding around the classical saddles, the subleading order of the Newton constant contributions can be derived, which are half of the Hawking temperature both for the Schwarzschild and Reissner-Nordström black holes. The result may imply a universal structure. The subsubleading terms and more intriguing physics that diverge from black hole thermodynamics are revealed. The ensemble-averaged theory provides a new way of studying subleading effects and extending the traditional AdS/CFT correspondence.
