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

TL;DR

The paper investigates when the scalar sector of metric single-scalar theories with action S = (1/16π)∫√{-g}[F(Φ,X)R+G(Φ,X)]+S_m admits an Eckart-like heat flux in the scalar-comoving frame. It derives a general expression for the scalar-fluid heat flux that includes a new transverse term proportional to (F_X/8πF) V_{⊥a}, which cannot generally be recast as a temperature gradient, thus obstructing a universal Eckart interpretation unless F_X ≡ 0. Consequently, a global Eckart description for all timelike scalar configurations is possible only for F(Φ,X)=F(Φ), i.e. for Jordan-like theories, while models with F_X ≠ 0 may still exhibit Eckart behavior on highly symmetric backgrounds. The results provide a simple, purely kinematical selection criterion that favors Jordan-like nonminimal couplings and connects to Horndeski constraints, with notable exceptions in symmetric spacetimes where the obstruction can disappear. The work advances the understanding of gravitational thermodynamics in scalar-tensor theories by clarifying when a single-temperature fluid picture for the scalar sector is tenable.

Abstract

We show that in single--scalar theories of the form $\mathcal{L}=F(Φ,X)R+G(Φ,X)$, a generic nonminimal coupling $F(Φ,X)$ induces, in the scalar--comoving frame, an additional transverse contribution to the effective heat flux, proportional to $(F_X/8πF)V_{\perp a}$, 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 $F_X(Φ,X)\equiv 0$, i.e.\ $F(Φ,X)=F(Φ)$, resulting in a theory that is a subclass of Horndeski. Thus, only Jordan--like theories of the type $F(Φ)R+G(Φ,X)$ 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 $D_aT_g$.