Reversible and Irreversible Spacetime Thermodynamics for General Brans-Dicke Theories

TL;DR

This paper derives the equations of motion for Palatini F(R) gravity using thermodynamics, showing the scalar field's role in heat flux and clarifying the nature of reversible and irreversible thermodynamic contributions in Brans-Dicke theories.

Contribution

It extends thermodynamic derivations of gravitational equations to Palatini F(R) and Brans-Dicke theories, clarifying the scalar field's thermodynamic role and the nature of viscosity terms.

Findings

No bulk viscosity term in Palatini F(R) gravity.

Scalar field contributes to heat flux as a reversible process.

Shear viscosity is associated with irreversible thermodynamics.

Abstract

We derive the equations of motion for Palatini F(R) gravity by applying an entropy balance law T dS= \delta Q+\delta N to the local Rindler wedge that can be constructed at each point of spacetime. Unlike previous results for metric F(R), there is no bulk viscosity term in the irreversible flux \delta N. Both theories are equivalent to particular cases of Brans-Dicke scalar-tensor gravity. We show that the thermodynamical approach can be used ab initio also for this class of gravitational theories and it is able to provide both the metric and scalar equations of motion. In this case, the presence of an additional scalar degree of freedom and the requirement for it to be dynamical naturally imply a separate contribution from the scalar field to the heat flux \delta Q. Therefore, the gravitational flux previously associated to a bulk viscosity term in metric F(R) turns out to be actually part of the reversible thermodynamics. Hence we conjecture that only the shear viscosity associated with Hartle-Hawking dissipation should be associated with irreversible thermodynamics.