Reversible and irreversible spacetime thermodynamics for general Brans-Dicke theories

We derive the equations of motion for Palatini F(R) gravity by applying an entropy balance law TdS = 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.