The effect of a non-minimally coupled scalar field upon classical thermodynamics

It is known that a scalar field that is non-minimally coupled to the geometry implies a varying gravitational “constant” G_eff, and hence a violation of the continuity equations, $\text{T}\text{?}{}^{\mathrm{ik}}{}_{\mathrm{;k}}\text{≠ 0}$, where $\text{T}\text{?}{}_{\mathrm{ij}}$ is the uncorrected energy-momentum tensor. This in turn upsets classical thermodynamics. 3he simplest resolution of this difficulty is to multiply all energetic quantities by $\text{G}{}_{\mathrm{<mtext>eff</mtext>}}\text{G}{}_{\mathrm{<mtext>N</mtext>}}$, where G_N is the newtonian gravitational constant. This modified thermodynamics is applied to the scalar-field version of the cosmological model of Zee, for which it is shown to cause restoration of the symmetry above some critical temperature T_c close to the Planck temperature. We also illustrate how the second law of thermodynamics is always obeyed, correcting a recent discussion by Davies.