Thermodynamics and entropy of self-gravitating matter shells and black holes in d dimensions

Abstract

The thermodynamic properties of self-gravitating spherical thin matter shells an black holes in d > 4 dimensions are studied, extending previous analysis for d = 4. The shell joins a Minkowski interior to a Tangherlini exterior, i.e., a Schwarzschild exterior in d dimensions, with d 4, The junction conditions alone together with the first law of thermodynamics enable one to establish that the entropy of the thin shell depends only on its own gravitational radius. Endowing the shell with a well-defined power-law temperature equation of state allows to establish a precise form for the entropy and to perform a thermodynamic stability analysis for the shell. A particularly interesting case is when the shell's temperature has the Hawking form, i.e., it is inversely proportional to the shell's gravitational radius. It is shown in this case that the shell's heat capacity is positive, and thus the shell is stable, for shells with radii in-between their own gravitational radius and the photonic radius, i.e., the radius of circular photon orbits, reproducing unexpectedly York's thermodynamic stability criterion for a d = 4 black hole in the canonical ensemble. Additionally, the Euler equation for the matter shell is derived, the Bekenstein and holographic entropy bounds are studied, and the large d limit is analyzed. Within this formalism the thermodynamic properties of black holes can be studied too. Putting the shell at its own gravitational radius, i.e., in the black hole situation, obliges one to choose precisely the Hawking temperature for the shell which in turn yields a black hole with the Bekenstein-Hawking entropy. The stability analysis implies that the black hole is thermodynamically stable substantiating that in this configuration our system and York's canonical ensemble black hole are indeed the same system. Also relevant is the derivation in a surprising way of the Smarr formula for black holes in d dimensions.

Thermodynamical and dynamical stability of a self-gravitating uncharged thin shell

“…The generalization and consequences of our results to the interesting cases of self-gravitating shells in d dimensions [1], and charged [15] and rotating [13] shells are left for future work.…”

“…Thin shells have been frequently employed to probe thermodynamical properties of black holes, see for instance [14][15][16][17]24] and, by taking them to their own gravitational radius, they can be transformed into quasi-black holes [18], and used to calculate black hole properties (see for instance [12]). 1 In view of these applications, it is important to determine whether relevant thin-shell configurations are stable, both thermo- 1 For a complete list of applications of the thin-sell formalism see [9]. a e-mail: sepbergliaffa@gmail.com b e-mail: chiapparini.uerj@gmail.com c e-mail: luzmarinareyes@gmail.com (corresponding author) dynamically and dynamically.…”

Compact objects in general relativity: From Buchdahl stars to quasiblack holes

“…41,42 The entropy of static shells in d-dimensions and the quasiblack hole limit has also been analyzed, see. 43…”

Analysis of Energy and Wave Functions and the Thermodynamics Properties of the 6-Dimensional Schrodinger Equation Under Double Ring-Shape Oscillator (Drso) and Manning-Rosen Potentials Using Susy Qm Method

“…The potential form that allows the variable separation in 6-dimensional system obtained from equation (25) have been given in equations (3)(4)(5)(6)(7)(8)(9). By setting up 1 ,…, −1 ( −1 = ) (̂) = ( ) ( ) ; = 1,2,3,4,5 (Eq.…”

“…Improvement to solve the Schrodinger equation in a higher dimensional system still necessary to carry out because it is important under the field of quantum physics (André, et al, 2019;Falaye and Oyewumi, 2011;Onate, et al, 2018;Wang, et al, 2002). One of the motivations that attracted the attention of scientists to continually identify D-dimensional systems is the pretentious unified theory 20th century between the Relativity and Quantum theory.…”