Exact Inner Metric and Microscopic State of AdS$_3$-Schwarzschld BHs

Abstract

Through full solvability of 2+1 dimensional general relativity we derive out exact dynamic inner metric of collapsing stars with inhomogeneous initial mass distribution but joining with outside Anti-deSitt-Schwarzschild black holes smoothly. We prove analytically by standard quantum mechanics that the log-number of such solutions, or microscopic states of the system is proportional to the perimeter of the outside black holes. Key formulas for generalizing to 3+1D Schwarzschild black holes are also presented. Our result provides a bulk space viewpoint to questions on what the microscopic degrees of freedom are and who their carriers are in various holographic and/or asymptotic symmetry methods to black hole entropies. It may also shed light for singularity theorem and cosmic censorship related researches.