Proper spherical entropy and particle constrained functionals

Assume $\mathrm{Z}\mathrm{F}+{\mathrm{A}\mathrm{C}}_{\omega }$ as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Fix $R>0$, an open state chart $\mathcal{U}\subset (0,\mathrm{\infty }{)}^2$, and a function $s\in C^2(\mathcal{U},\mathbb{R})$ with $s_{ϵ}>0$. The coordinates on $\mathcal{U}$ are energy density $ϵ$ in $\mathrm{J}{\mathrm{m}}^{-3}$ and particle number density $n$ in ${\mathrm{m}}^{-3}$; the entropy density has units $[s]=\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$. Define the local temperature, chemical potential, and pressure functions by $\mathrm{\Theta }(ϵ,n):=\frac{1}{s_{ϵ}(ϵ,n)},\mu (ϵ,n):=-\mathrm{\Theta }(ϵ,n)s_n(ϵ,n),p(ϵ,n):=\mathrm{\Theta }s+\mu n-ϵ\mathrm{.}$ These are $C^1$ functions on $\mathcal{U}$; their formal SI dimensions are $\mathrm{K}$, $\mathrm{J}$, and $\mathrm{J}{\mathrm{m}}^{-3}$, respectively. By definition $ϵ+p=\mathrm{\Theta }s+\mu n$, and differentiation gives $dϵ=\mathrm{\Theta }ds+\mu dn$ and $dp=sd\mathrm{\Theta }+nd\mu$ on the chart.

Let $ϵ,n$ be restrictions to $[0,R]$ of even $C^2$ functions on an open interval containing $[-R,R]$, positive on $[0,R]$, with image in a compact subset of $\mathcal{U}$. This evenness is the center regularity convention. With $K:=4\pi G\mathrm{/}{c}^4$, define $M(r):=\frac{4\pi }{c^2}{\int }_0^rϵ(t)t^{2}dt,b(r):=1-\frac{2GM(r)}{c^{2}r}(r>0),b(0):=1.$ Equivalently, $b(r)=1-r^{2}q(r)$ on $[-R,R]$, where $q(r):=2K{\int }_0^1{s}^2ϵ(rs)ds\mathrm{.}$ The compact-parameter differentiation theorem Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral makes $q$ even $C^2$. Therefore Even radial coefficients give a C² Cartesian center gives the corresponding $C^2$ Cartesian spatial metric coefficients. Assume $b(r)\ge b_{\ast }>0$ on $[0,R]$. The displayed integrals are Riemann integrals in the Darboux convention of The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$; their continuous integrands are integrable by A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, and $b^{-1\mathrm{/}2}$ is continuous on this positive range by Continuity and derivatives of positive-base real powers. Use the spatial metric $h=b(r{)}^{-1}d{r}^2+r^2\gamma$ on $(0,R)\times S^2$, where $\gamma$ is the unit round sphere metric used in Smooth static spherical geometry and state domain. Here $h$ has $C^2$ coefficients and the Cartesian center extension just established; no globally smooth static state is required. Define its volume density in a spherical coordinate chart by $\sqrt{\det (h_{ij})}drd\theta d\phi$. Since $\det (h_{ij})=b^{-1}{r}^4{\sin }^2\theta$, integration over the unit sphere gives the radial density $4\pi r^{2}b(r{)}^{-1\mathrm{/}2}dr$. This uses the same spatial metric formula as the smooth static domain, at the present $C^2$ regularity. Define the proper entropy, proper particle number, and spherical mass-energy functionals $S[ϵ,n]:=4\pi {\int }_0^{R}s(ϵ(r),n(r))r^{2}b(r{)}^{-1\mathrm{/}2}dr,N_b[ϵ,n]:=4\pi {\int }_0^{R}n(r)r^{2}b(r{)}^{-1\mathrm{/}2}dr,E[ϵ,n]:=c^{2}M(R)=4\pi {\int }_0^Rϵ(r)r^{2}dr\mathrm{.}$ Their formal dimensions are $[S]=\mathrm{J}{\mathrm{K}}^{-1}$, $[N_b]=1$, and $[E]=\mathrm{J}$. These are mathematical functionals on the supplied state chart and positive-$b$ profile domain; they assert no entropy law, equilibrium, maximum, dynamics, or empirical fact.

The energy functional is the spherical mass-energy constraint $c^{2}M(R)$, not a generic additive gravitational energy on arbitrary spacetimes. No relation between this state chart and any particular material equation of state is assumed beyond the displayed definitions.