Universality of entropy principle for a general diffeomorphism-covariant purely gravitational theory

Abstract

Thermodynamics plays an important role in gravitational theories. It is a principle independent of the gravitational dynamics, and there is still no rigorous proof to show that it is consistent with the dynamical principle. We consider a self-gravitating perfect fluid system in a general diffeomorphism-covariant purely gravitational theory. Based on the Noether charge method proposed by Iyer and Wald, considering static off/on-shell variational configurations which satisfy the gravitational constraint equation, we rigorously prove that the extrema of the total entropy of perfect fluid inside a compact region for fixed total particle number demands that the static configuration is an on-shell solution after we introduce some appropriate boundary conditions, i.e., it also satisfies the spatial gravitational equations. This means that the entropy principle of the fluid stores the same information as the gravitational equation in a static configuration. Our proof is universal and holds for any diffeomorphism-covariant purely gravitational theories, such as Einstein gravity, f (R) gravity, Lovelock gravity, f (Gauss-Bonnet) gravity and Einstein-Weyl gravity. Our result shows the consistency between the ordinary thermodynamics and the gravitational dynamics.

The maximum entropy principle of self-gravitating fluid system and field equations

“…In [3], for a spherically symmetric and static radiation fluid system in general relativity, it is shown that the initial value constraint equations (equivalent to the time-time component of the Einstein equation) and requiring for the total entropy to be maximum with the total mass fixed derive the Tolman-Oppenheimer-Volkoff (TOV) equation of hydrostatic equilibrium. Generalizations have been done, i.e., for arbitrary equation of state of fluids [4], non-spherically symmetric system [5], and attempts to apply other field theories, namely, the Lovelock theory [6], the f (R) theory [7], Einstein-Maxwell theory [8], and generally covariant purely gravitational theories [9]. However, since these calculations are based on variations restricted by some constraint conditions such as spherical or time translational symmetries, Tolman's law, or the time-time component of the gravitational field equation, calculations for general field theories and based on general variations have not yet done.…”