Specified bulk extended entropy and relaxation law

Use the fixed-composition local-equilibrium state, smooth time-oriented Lorentzian geometry of signature $(-,+,+,+)$, smooth future unit observer congruence $u$, and length coordinates $x^0=ct$ from Restricted local equilibrium parcel with adopted material state function and Rest fluid state, frame and geometric current definitions. In particular, $n>0$, $T>0$, ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, $D=u^a{\mathrm{\nabla }}_a$, and $\theta ={\mathrm{\nabla }}_a{u}^a$ have those suppliers' meanings. In a selected bulk-only branch, adopt the stress, number current, and source-free balances ${\mathsf{T}}^{ab}=\epsilon u^a{u}^b+(p+\mathrm{\Pi }){\mathrm{\Delta }}^{ab},J^a=n{u}^a,{\mathrm{\nabla }}_a{\mathsf{T}}^{ab}=0,{\mathrm{\nabla }}_a{J}^a=0,$ where $\mathrm{\Pi }$ is a smooth scalar bulk pressure. Let $\zeta ,\tau \in C^1(O)$ be positive coefficient fields with SI units Pa s and s, respectively, and define $\beta =\tau \mathrm{/}\zeta$, with units Pa${}^{-1}$. Adopt the extended entropy current and derivative-corrected relaxation law $S^a=(s-\frac{\beta {\mathrm{\Pi }}^2}{2T})u^a,\beta D\mathrm{\Pi }+\frac{\mathrm{\Pi }}{\zeta c}=-\theta -\frac{T\mathrm{\Pi }}{2}{\mathrm{\nabla }}_a\text{\hspace{0.17em}⁣}\left(\frac{\beta u^a}{T}\right)\mathrm{.}$ Here $\mathrm{\Pi }$ has units Pa, $D$ and $\theta$ have inverse-length units, and each side of the relaxation law has units m${}^{-1}$; $\beta {\mathrm{\Pi }}^2\mathrm{/}(2T)$ has entropy-density units J K${}^{-1}$ m${}^{-3}$.

For each application, specify a nonempty open near-equilibrium regime in the pointwise state and first-derivative data around $\mathrm{\Pi }=0$, based at constant zero-gradient reference data, and use the current and law only while the fields remain in that declared regime. The state map and state function are $C^2$, $u$ is smooth, the coefficient fields are $C^1$, and $\mathrm{\Pi }$ is smooth; thus $J^a$, ${\mathsf{T}}^{ab}$, and $S^a$ are $C^1$. Require the balances and corrected law to hold pointwise. The mathematical result Corrected bulk relaxation and exact extended entropy supplies the exact matching conditional entropy identity and one homogeneous witness; its actual statement and proof are conditional on the listed mathematical inputs and assume exactly countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). It motivates this selection and is used by the downstream deductions, but it does not derive the adopted physical postulate.

This is a restricted constitutive assumption for the selected material. It does not derive the equation of state, assert universal validity, or adopt existence of a general nonlinear solution. It makes no causal or stability claim; positive relaxation time alone supplies neither. No empirical observation is supplied.