Eckart heat-flux applicability in $F(\mathrm{\Phi },X)R$ theories and the existence of temperature gradients

We show that in single–scalar theories of the form $\mathcal{L}=F(\mathrm{\Phi },X)R+G(\mathrm{\Phi },X)$, a generic nonminimal coupling $F(\mathrm{\Phi },X)$ induces, in the scalar–comoving frame, an additional transverse contribution to the effective heat flux, proportional to $(F_X\mathrm{/}8\pi F)V_{\perp a}$, where $V_a\equiv h_a{}^c{\mathrm{\nabla }}_c{\mathrm{\nabla }}_{d}X{u}^d$ and $V_{\perp a}$ denotes the component orthogonal to the 4–acceleration $a_a$. This term cannot in general be written as a spatial temperature gradient, and therefore obstructs a standard Eckart interpretation of the scalar sector for arbitrary timelike scalar configurations. As a result, requiring an Eckart heat flux $q_a=-K(D_a{T}_g+T_g{a}_a)$ for all such configurations is possible if and only if $F_X(\mathrm{\Phi },X)\equiv 0$, i.e.\ $F(\mathrm{\Phi },X)=F(\mathrm{\Phi })$, resulting in a theory that is a subclass of Horndeski. Thus, only Jordan–like theories of the type $F(\mathrm{\Phi })R+G(\mathrm{\Phi },X)$ admit a global Eckart fluid picture of the scalar sector, while models with $F_X\mathrm{\ne }0$ can recover an Eckart–like form only on highly symmetric backgrounds where the transverse contribution vanishes or collapses to a single gradient direction. We also make a brief comment on the existence of temperature gradients $D_a{T}_g$.