Eckart heat-flux applicability in F(Φ,X)R theories and the existence of temperature gradients

We show that in single--scalar theories of the form $\mathcal{L}=F(\Phi,X)R+G(\Phi,X)$, a generic nonminimal coupling $F(\Phi,X)$ induces, in the scalar--comoving frame, an additional transverse contribution to the effective heat flux, proportional to $(F_X/8\pi F)V_{\perp a}$, where $V_a \equiv h_a{}^c\nabla_c\nabla_d X\,u^d$ 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 $q_a = -K\bigl(D_a T_g + T_g\, a_a\bigr)$ for all such configurations is possible if and only if $F_X(\Phi,X)\equiv 0$, i.e.\ $F(\Phi,X)=F(\Phi)$, resulting in a theory that is a subclass of Horndeski. Thus, only Jordan--like theories of the type $F(\Phi)R+G(\Phi,X)$ admit a global Eckart fluid picture of the scalar sector, while models with $F_X\neq 0$ 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 $D_aT_g$.