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 [S. Viaggiu, Physica A 473 (2017) 412; 488 (2017) 72.] 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 = γU for the pressure P and the internal energy U arises. Firstly, we study the case with U ∼ R, where only fluids with γ > −1 3 are possible. If corrections ∼ 1/R are added to U, for γ ∈ [0, 1 3], we found no limitation on the allowed value for the areal radius of the trapped sphere R. Moreover, for γ > 1 3, we have a minimum allowed value for R of the order of the Planck length LP. Conversely, a fluid with P < 0 can be obtained but with a maximum allowed value for R. With the added term looking like ∼ 1/R to the BH internal energy U, the well-known logarithmic corrections to the BH entropy na...