Bekenstein–Hawking Entropy in Classical and Quantum Gravity

Abstract

We outline the background and classical derivation of the black hole entropy formula as described in Bekenstein 1973 [Bek73]. This construction was highly physical and lacked sufficient mathematical rigour, which we use to motivate transitioning to loop quantum gravity to obtain a rigorous microscopic origin of this entropy. We recast general relativity into a form that allows us to perform deformation quantisation. Upon quantising, we describe the Hilbert space of the theory following Ashtekar et al. [ABK00]. This allows us to obtain the black hole entropy formula of quantum gravity by counting certain basis states of this Hilbert space, proving several claims made by the authors. We finally outline some directions for future research and propose an alternative direction for the derivation based on concepts from knot theory and homology.