# Klein Gordon particle near R-N black hole, generalized sl(2) algebra and harmonic oscillator energy

## TL;DR

This paper analyzes Klein-Gordon particles near Reissner-Nordström black holes, revealing a connection to generalized sl(2) algebra, harmonic oscillator energy spectra, and thermodynamic properties, with implications for black hole entropy and information theory.

## Contribution

It introduces a novel approach linking black hole symmetry to generalized sl(2) algebra and harmonic oscillator energy spectra, enabling thermodynamic and informational analysis.

## Key findings

- Energy spectrum matches three-dimensional harmonic oscillator
- Derived thermodynamic properties of the black hole
- Established connection between black hole entropy and information theory

## Abstract

In this paper, we consider Klein Gordon particle near Reissner-Nordstr\"{o}m black hole. The symmetry of such background lead us to compare the corresponding Laplace equation with the generalized Heun functions. Such relation help us achieve the generalized $sl(2)$ algebra and some suitable results for describing the above mentioned symmetry. On the other hand in case of $r\rightarrow r_{+}$ which is near the proximity black hole, we obtain the energy spectrum and wave function. When we compare the equation of $R-N$ background with Laguerre differential equation, we show that the obtained energy spectrum is same as three dimensional harmonic oscillator. So, finally we take advantage of harmonic oscillator energy and make suitable partition function. Such function help us to obtain all thermodynamical properties of black hole. Also, the structure of obtained entropy lead us to have some bit and information theory in the $R-N$ black hole.