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

Abstract

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 [Formula: see text] for the pressure [Formula: see text] and the internal energy [Formula: see text] arises. Firstly, we study the case with [Formula: see text], where only fluids with [Formula: see text] are possible. If corrections [Formula: see text] are added to [Formula: see text], for [Formula: see text], we found no limitation on the allowed value for the areal radius of the trapped sphere [Formula: see text]. Moreover, for [Formula: see text], we have a minimum allowed value for [Formula: see text] of the order of the Planck length [Formula: see text]. Conversely, a fluid with [Formula: see text] can be obtained but with a maximum allowed value for [Formula: see text]. With the added term looking like [Formula: see text] to the BH internal energy [Formula: see text], 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.