General proof of the entropy principle for self-gravitating fluid in static spacetimes

Abstract

We show that for any perfect fluid in a static spacetime, if the Einstein constraint equation is satisfied and the temperature of the fluid obeys Tolman's law, then the other components of Einstein's equation are implied by the assumption that the total entropy of the fluid achieves an extremum for fixed total particle number and for all variations of metric with certain boundary conditions. Conversely, one can show that the extrema of the total entropy of the fluid are implied by Einstein's equation. Compared to previous works on this issue, we do not require spherical symmetry for the spacetime. Our results suggest a general and solid connection between thermodynamics and general relativity. PACS number(s): 04

Thermodynamical stability for a perfect fluid

“…Then we proved that the extrema of the total entropy δS = 0 gives the components of gravitational field equations [5]. The maximum entropy principle implies that the gravitational field equations may be replaced by a constraint equation and thermodynamic relations.…”

“…Without loss of generality, we take T χ = 1. In [5] we have shown that if the constraint equation and some boundary conditions are satisfied, and the total particle number is fixed, the variation of the total entropy is…”

“…Note that we have chosen the energy density ρ and the particle number density n as independent variables on obtaining the first variation of the total entropy [5]. But Chandrasekhar has chosen ρ and p as independent variables [14,15].…”

“…First, we briefly review the proof of the maximum entropy principle for a perfect fluid in general static spacetime [5]. We assume that all the quantities are measured by static background observers which are orthogonal to the hypersurface , and we consider the fluid over any selected region C on to satisfy the ordinary thermodynamic relations and Tolman's law, which gives T χ = const., where T and χ are the temperature of the fluid and the redshift factor, respectively.…”

“…Meanwhile, a series of papers proved that the maximum entropy principle for a perfect fluid holds in different theories. This principle shows that if the constraint equation and some thermodynamic conditions are satisfied, the gravitational field equations could be derived by the extrema of the total entropy [3][4][5][6][7][8][9]. Recently, Jacobson proposed the "causal diamond" structure and considered entanglement equilibrium to imply the Einstein equations [10].…”