Building a linear equation of state for trapped gravitons from finite size effects and the Schwarzschild black hole case

TL;DR

This paper investigates how finite size effects modify the spectrum of trapped gravitons, leading to a linear equation of state and potential implications for black hole entropy and dark energy formation.

Contribution

It introduces a model linking finite size effects to a linear graviton equation of state and explores its implications for black hole physics and dark energy.

Findings

Finite size effects can produce a linear equation of state for trapped gravitons.

Logarithmic corrections to black hole entropy naturally emerge from the model.

Constraints on the areal radius R depend on the equation of state parameter γ.

Abstract

In this paper we continue the investigations present in \cite{1} and \cite{2} concerning the spectrum of trapped gravitons in a spherical box, and in particular inside a Schwarzschild black hole (BH). We explore the possibility that, due to finite size effects, the frequency of the radiation made of trapped gravitons can be modified in such a way that a linear equation of state $PV=\gamma U$ for the pressure $P$ and the internal energy $U$ arises. Firstly, we study the case with $U\sim R$, where only fluids with $\gamma >-\frac{1}{3}$ are possible. If corrections $\sim 1/R$ are added to $U$, for $\gamma\in[0,\frac{1}{3}]$ we found no limitation on the allowed value for the areal radius of the trapped sphere $R$. Moreover, for $\gamma>\frac{1}{3}$ we have a minimum allowed value for $R$ of the order of the Planck length $L_P$. Conversely, a fluid with $P<0$ can be obtained but with a maximum allowed value for $R$. With the added term looking like $\sim 1/R$ to the BH internal energy $U$, the well known logarithmic corrections to the BH entropy naturally emerge for any linear equation of state. The results of this paper suggest that finite size effects could modify the structure of graviton's radiation inside, showing a possible mechanism to transform radiation into dark energy.