Jordan-like invariant representations of scalar-tensor gravity and the return of imperfect-fluid thermodynamics

It has been shown that the thermodynamics of frame-invariant scalar-tensor gravity admits an Einstein-frame-like invariant representation, in which the scalar sector is minimally coupled and behaves as a perfect fluid with vanishing temperature. We show that this is not the only invariant thermodynamic organization available. Using instead the invariant matter metric, we construct a Jordan-frame-like invariant representation in which matter is minimally coupled and the gravitational action takes the Brans-Dicke-Bergmann-Wagoner form with invariant fields $Ψ=\mathcal I_1^{-1}$, $\mathcal U(Ψ)$, and $ω_{\mathrm{inv}}(Ψ)$. In this representation the metric field equations retain the Hessian sector characteristic of Jordan-frame scalar-tensor gravity. A $1+3$ decomposition of the corresponding effective stress tensor yields nonvanishing heat flux and anisotropic stress in a generic congruence. In the $Ψ$-comoving frame, the usual first-order thermodynamic identifications are recovered in invariant form, including $K_{\mathrm J}T_{\mathrm J}=-\dotΨ/(κ^2Ψ)$ and $η_{\mathrm J}=\dotΨ/(2κ^2Ψ)$, with the bulk channel obtained in homogeneous and isotropic sectors. Thus, frame-invariant scalar-tensor thermodynamics is not intrinsically restricted to an Einstein-frame-like perfect-fluid description: it also admits a Jordan-frame-like invariant formulation in which the effective imperfect-fluid interpretation is manifest.