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

## TL;DR

This paper derives the exact inner metric of collapsing stars in 2+1D AdS spacetime, linking microscopic states to black hole perimeter and offering insights into black hole entropy and singularity issues.

## Contribution

It provides an exact solution for the inner metric of inhomogeneous collapsing stars in 2+1D AdS space and relates microscopic states to black hole perimeter, with generalizations to 3+1D.

## Key findings

- Microscopic states proportional to black hole perimeter
- Exact inner metric for inhomogeneous collapse in 2+1D
- Formulas for extending results to 3+1D black holes

## 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.