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 $(F_X/8\pi F)V_{\perp a}$, where $V_a\equiv {h_a}^c{\nabla }_c{\nabla }_{d}X{u}^d$ and $V_{\perp a}$ denotes the component orthogonal to the four-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 $F_X\ne 0$ 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 $D_a{T}_g$.

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• Alternative gravity theories
• Fluids & classical fields in curved spacetime
• Gravitation
• Nonequilibrium & irreversible thermodynamics