Gravitational Waves as Thermodynamic Shear Excitations in Scalar-Tensor Gravity

In Jordan-frame scalar–tensor gravity, a timelike scalar gradient defines an effective thermodynamic medium. We show that a transverse–traceless perturbation of a homogeneous shear-free background activates the local shear-work channel ${\mathcal{W}}_{\phi }={\pi }_{ab}^{(\phi )}{\sigma }^{ab}$, with leading pure-TT contribution ${\mathcal{P}}_{\mathrm{T}\mathrm{T}}^{[1,1]}=\overline{\mathcal{K}\mathcal{T}}{\dot{\chi }}_{ij}^{\mathrm{T}\mathrm{T}}{\dot{\chi }}_{\mathrm{T}\mathrm{T}}^{ij}\mathrm{/}4$. This stress power follows from local traction mechanics, enters the scalar-sector energy balance, and is gauge invariant at second order. The same scalar–tensor coupling controls non-GR tensor damping and appears with the opposite sign in a GR-normalized amplitude balance, although its interpretation as a separate tensor-energy source is normalization dependent. Scalar–tensor gravity thus provides a controlled example in which a propagating spin-2 mode acquires an intrinsically gravitational thermodynamic interpretation.