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

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.