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 [1] and [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 P V = 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 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 ∼ 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. * viaggiu@axp.mat.uniroma2.it and s.viaggiu@unimarconi.it

The physical origin of the cosmological constant: continuum limit and the analogy with the Casimir effect

“…In [13,14] we have advanced a physical mechanism, mimicking solid state physics, capable, thanks to Planckian fluctuations, to change the equation of state of a radiation field into an effective one with a γ− linear equation of state. This has been firstly applied in order to explain the logarithmic corrections [13] to the semi-classical Bekenstein-Hawking entropy [15,16] and further applied in [1] in order to depict the cosmological constant. To start with, we consider the free energy F (0) (V, N, T ) of a radiation field with N excitations in a proper volume V at the temperature T and with energy E (0) in the continuum limit, given, as well known, by:…”

The Cosmological Constant From Planckian Fluctuations and the Averaging Procedure

“…Only Planckian fluctuations can generate a cosmological constant equation of state. The physical mechanism, depicted also in [22] for a black hole and generalized in [23], is capable to transform a radiation field into one with a γ linear equation of state with p vac = γc 2 ρ vac . This can be done by considering the cosmological constant composed of massless excitations where Planckian fluctuations come into action to transform the initial radiation-like equation of state (with γ = 1/3) into a cosmological constant one (γ = −1).…”

On a physical description and origin of the cosmological constant

“…In any case, the physical origin of Λ, its smallness and the role of the vacuum energy in the dynamic of the universe are yet fundamental unsolved problems. Recentely, I have proposed [1,2,3,4] a way to depict the black hole entropy in terms of trapped gravitons [1,2] together with the logarithmic entropy corrections [3]. The treatment has been extended in [4] to any massless excitation.…”

“…The treatment has been extended in [4] to any massless excitation. In particular, in [3,4] a possible mechanism transforming a radiation field into a one with a linear equation of state is presented as a theorem. In [3] it has been shown that this mechanism naturally gives, after considering quantum gravity corrections, the well known logarithmic corrections to the black hole entropy.…”

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In this paper we use and extend the results present in [1,2,3,4] and in particular in [4] to obtain a statistical description of the cosmological constant in a cosmological de Sitter universe in terms of massless excitations with Planckian effects. First of all, we show that at a classical level, the cosmological constant Λ > 0 can be obtained only for T → 0.

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