Supplied thermodynamic chart and gravitational entropy variation model

Assume $\mathrm{Z}\mathrm{F}+{\mathrm{A}\mathrm{C}}_{\omega }$ exactly as in Proper spherical entropy and particle constrained functionals. Adopt a separate static, spherically symmetric, isotropic $\mathrm{\Lambda }=0$ Einstein-fluid preparation using the geometry and $(-,+,+,+)$, $x^0=ct$ conventions of Smooth static spherical geometry and state domain, without a separate barotropic relation $\epsilon =\mathcal{E}(P)$. The adopted field equation is $G^a{}_b=(8\pi G\mathrm{/}{c}^4)T^a{}_b$ with $T^a{}_b=\mathrm{diag}(-\epsilon (r),P(r),P(r),P(r))$ and the signed curvature convention in Smooth static spherical geometry and state domain. Fix a radius $R>0$, and supply a $C^2$ open state chart $\mathcal{U}\subset (0,\mathrm{\infty }{)}^2$ with coordinates energy density $\epsilon$ in $\mathrm{J}{\mathrm{m}}^{-3}$ and fixed-composition marker number density $n$ in ${\mathrm{m}}^{-3}$. Supply $s\in C^2(\mathcal{U},\mathbb{R})$ in $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$ with $s_{\epsilon }>0$, and define $\mathrm{\Theta }:=1\mathrm{/}{s}_{\epsilon },\mu :=-\mathrm{\Theta }{s}_n,P:=\mathrm{\Theta }s+\mu n-\epsilon \mathrm{.}$ Assume this chart is compatible with the adopted material energy-per-marker function $e(v,{\sigma }_p)$ of Restricted local equilibrium parcel with adopted material state function: $v=1\mathrm{/}n$, ${\sigma }_p=s\mathrm{/}n$, $\epsilon =ne(1\mathrm{/}n,s\mathrm{/}n)$, and $\mathrm{\Theta }=e_{{\sigma }_p}>0$ on the supplied chart. The symbols $\mathrm{\Theta },\mu ,P$ respectively have units kelvins, joules, and $\mathrm{J}{\mathrm{m}}^{-3}$. The local identities $d\epsilon =\mathrm{\Theta }ds+\mu dn$ and $dP=sd\mathrm{\Theta }+nd\mu$ are those of the exact mathematical chart definition Proper spherical entropy and particle constrained functionals.

For positive even $C^2$ profiles $(\epsilon (r),n(r))$ with compact range in $\mathcal{U}$, define $M,b,S,N_b,E$ exactly by Proper spherical entropy and particle constrained functionals and require $b(r)\ge b_{\ast }>0$ on $[0,R]$. The adopted proper entropy and particle number are $S=4\pi {\int }_0^{R}s{r}^2{b}^{-1\mathrm{/}2}dr$ and $N_b=4\pi {\int }_0^{R}n{r}^2{b}^{-1\mathrm{/}2}dr$, while the constrained mass-energy is $E=c^{2}M(R)=4\pi {\int }_0^R\epsilon r^{2}dr$. Restrict variations to the supplied positive-$b$ chart at fixed $R$ and fixed $E,N_b$, and choose the boundary time normalization $e^{\varphi (R)}=\sqrt{b(R)}$.

This model supplies local material thermodynamics and a gravitational mass constraint as assumptions. Its variation problem is not an independent derivation of all Einstein equations. A stationary profile is only a first-variation critical point; no entropy maximum, existence of a maximizing profile, boundary vacuum matching, or physical observation is assumed.