# Quantum thermodynamics in the interior of a Schwarzschild B-H

## TL;DR

This paper explores the interior of a Schwarzschild black hole using relativistic quantum geometry, revealing discrete energy levels, the validity of the uncertainty principle at each level, and connections to black hole thermodynamics.

## Contribution

It introduces a novel approach to black hole interior analysis via relativistic quantum geometry, deriving discrete energy levels and linking quantum states to thermodynamic properties.

## Key findings

- Discrete energy levels for scalar fields inside the black hole.
- Validation of the uncertainty principle at each energy level.
- Derivation of temperature, entropy, and mass dependence on quantum states.

## Abstract

We study the interior of a Schwarzschild Black-Hole (B-H) using Relativistic Quantum Geometry described in \cite{rb} and \cite{rb1}. We found discrete energy levels for a scalar field from a polynomial condition for Heun Confluent functions expanded around the Schwarzschild radius. From the solutions it is obtained that the uncertainty principle is valid for each energy level of space-time, in the form: $E_n\, r_{sh,n}=\hbar/2$. Temperature, entropy and the B-H mass are dependent on the number of states in the B-H, such that the Bekenstein-Hawking (BH) results are obtained in a limit case.