Smooth static spherical geometry and state domain

Assume the countable-choice convention of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) for the canonical smooth tangent- and tensor-bundle structures in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model. Fix positive formal parameters $c>0$ and $G>0$ with SI dimensions $[c]=\mathrm{m}{\mathrm{s}}^{-1}$ and $[G]={\mathrm{m}}^3{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$. In this mathematical definition they are fixed parameters, not derived quantities, and no numerical or empirical value is asserted. Let $I_0\subseteq \mathbb{R}$ be a nonempty open interval for a length coordinate $x^0$, and define the corresponding time coordinate $t:=x^0\mathrm{/}c\in I_0\mathrm{/}c$, measured in seconds. Let $r$ be the coordinate on $(0,\mathrm{\infty })$, and let $(S^2,\gamma )$ be the unit round sphere with the induced smooth round metric of The round metric on the sphere as an induced metric. On the product smooth four-manifold $M=I_0\times (0,\mathrm{\infty })\times S^2$, using the canonical product smooth structure of Products of smooth manifolds have a canonical product smooth structure, define, for $\varphi ,b\in C^{\mathrm{\infty }}((0,\mathrm{\infty }),\mathbb{R})$ with $b(r)>0$, $g=-e^{2\varphi (r)}(d{x}^0{)}^2+b(r{)}^{-1}d{r}^2+r^2\gamma \mathrm{.}$ Set $N={\mathrm{\partial }}_{x^0}$ as time orientation. The signature convention is $(-,+,+,+)$, $x^0$ and $r$ are measured in metres, the line element has units ${\mathrm{m}}^2$, and $\varphi$ and $b$ are dimensionless. The sphere at fixed $(x^0,r)$ has induced metric $r^2\gamma$. In spherical coordinates its area element is $r^2\sin \theta d\theta d\phi$ by The round metric on the sphere as an induced metric, so ${\int }_0^{2\pi }{\int }_0^{\pi }{r}^2\sin \theta d\theta d\phi =4\pi r^2$; hence $r$ is the areal radius. The flow of $N$ is translation in $x^0$ and preserves $g$ because its coefficients are independent of $x^0$; here a Killing field means a vector field whose local flow preserves the metric. Moreover, $g(N,\cdot )=-e^{2\varphi }d{x}^0$, so $N$ is normal to the hypersurfaces $x^0=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$. In this definition, the metric is static in the precise sense that it has a timelike Killing field normal to such hypersurfaces; $N$ is timelike because $g(N,N)=-e^{2\varphi }<0$.

A static isotropic diagonal source field is a smooth tensor field of type $(1,1)$ in the sense of Tensor fields of type (r,s) on a Lorentzian four-manifold and their components, defined globally using the product splitting $TM\cong T{I}_0\oplus T(0,\mathrm{\infty })\oplus T{S}^2$ by $T(v_0,v_r,v_S)=(-ϵ(r)v_0,p(r)v_r,p(r)v_S)\mathrm{.}$ In every standard spherical chart its mixed components are $T^a{}_b=\mathrm{diag}(-ϵ(r),p(r),p(r),p(r)),$ where $ϵ,p\in C^{\mathrm{\infty }}((0,\mathrm{\infty }),\mathbb{R})$ represent energy density and isotropic pressure, respectively, under the formal physical reading, with $[ϵ]=[p]=\mathrm{J}{\mathrm{m}}^{-3}$. The associated covariant tensor has components $T_{ab}:=g_{ac}{T}^c{}_b$; it is smooth and symmetric because both component matrices are diagonal, as defined for covariant $2$-tensor fields in Smooth covariant 2-tensor fields on a smooth manifold. Define the mass function $M:(0,\mathrm{\infty })\to \mathbb{R}$ by $M(r):=\frac{c^{2}r}{2G}(1-b(r)),$ which has units $\mathrm{k}\mathrm{g}$. No Einstein equation, equation of state, central regularity, physical applicability, or empirical claim is part of this definition. The Einstein tensor and component conventions used later are those of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors and Tensor fields of type (r,s) on a Lorentzian four-manifold and their components.

The coordinate domain excludes $r=0$; a regular center is a separate conclusion and is defined by the center regularity result and the local-existence theorem on this page. Countable Choice is inherited only through the stated canonical smooth-bundle suppliers; the coordinate formulas below add no stronger choice principle.

Let $\delta >0$, let $f,h\in C^2((-\delta ,\delta ),\mathbb{R})$ be even, and let $B_{\delta }(0)\subseteq {\mathbb{R}}^3$ carry its standard Euclidean coordinates $x=(x_1,x_2,x_3)$ and radius $r=\mathrm{\mid }x\mathrm{\mid }$ from The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Then $F(x)=f(\mathrm{\mid }x\mathrm{\mid })$ is $C^2$ on $B_{\delta }(0)$, and each component $H_{ij}(x)={\delta }_{ij}+x_i{x}_{j}h(\mathrm{\mid }x\mathrm{\mid })$ is $C^2$ there. In particular, if an areal spherical spatial metric has $b(r)=1-r^{2}q(r)$ with even $q\in C^2((-\delta ,\delta ))$ and $b>0$, then its Cartesian coefficients $g_{ij}(x)={\delta }_{ij}+x_i{x}_j\frac{q(\mathrm{\mid }x\mathrm{\mid })}{b(\mathrm{\mid }x\mathrm{\mid })}$ extend $C^2$ across the center. This result is a coordinate regularity statement; it asserts no field equation or physical claim.

Assume $\mathrm{Z}\mathrm{F}+{\mathrm{A}\mathrm{C}}_{\omega }$ with ${\mathrm{A}\mathrm{C}}_{\omega }$ exactly as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let $J\subseteq (0,\mathrm{\infty })$ be an open interval; let the state domain, metric, mixed source tensor, mass function, units, and signature be those of Smooth static spherical geometry and state domain. Write $\kappa =8\pi G\mathrm{/}{c}^4$, and let $G^a{}_b$ be the mixed Einstein tensor defined from the signed curvature convention in The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. Suppose $\varphi ,b,ϵ,p$ are smooth on $J$ and $b>0$. As a conditional mathematical system, assume the component equations $G^a{}_b=\kappa T^a{}_b\mathrm{.}$ Then on $J$, $M^{\mathrm{\prime }}=\frac{4\pi r^2ϵ}{c^2},{\varphi }^{\mathrm{\prime }}=\frac{GM\mathrm{/}{c}^2+4\pi G{r}^{3}p\mathrm{/}{c}^4}{r^{2}b},p^{\mathrm{\prime }}=-\frac{G(ϵ+p)(M+4\pi r^{3}p\mathrm{/}{c}^2)}{c^2{r}^{2}b}\mathrm{.}$ Conversely, if these three ordinary differential equations hold with $M=c^{2}r(1-b)\mathrm{/}(2G)$, then every mixed Einstein component satisfies $G^a{}_b=\kappa T^a{}_b$. The identity ${\mathrm{\nabla }}^a{G}_{ab}=0$ used for this converse is the geometric theorem of The second Bianchi identity and the vanishing divergence of the Einstein tensor, not a field equation or empirical premise. This lemma is a pure conditional calculation and makes no claim that the model describes nature.

• The claim is conditional mathematics. Its ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption is inherited exactly from the smooth tangent/tensor-bundle and curvature suppliers; no full Axiom of Choice is invoked, and the coordinate algebra performs no selection.
• The direct component computation verifies all independent Einstein components on $r>0$. The converse uses Bianchi only after the mass, lapse, and conservation equations hold.
• The equation is confined to a smooth isotropic static spherical domain with $b>0$; it says nothing about anisotropic, rotating, nonstatic, weak, or distributional sources.

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.

Assume $\mathrm{Z}\mathrm{F}+{\mathrm{A}\mathrm{C}}_{\omega }$, where ${\mathrm{A}\mathrm{C}}_{\omega }$ is exactly The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) and is inherited from the smooth Lorentzian/tensor suppliers. Let $ϵ:I_p\to (0,\mathrm{\infty })$ be $C^{\mathrm{\infty }}$ on an open interval $I_p\subseteq (0,\mathrm{\infty })$, let $P_c\subset I_p$ be a nonempty compact interval, and choose $\delta >0$ such that the closed $\delta$-neighbourhood of $P_c$ lies in $I_p$. Fix ${\varphi }_c\in \mathbb{R}$ and use the units and signature of Smooth static spherical geometry and state domain. Put $K=4\pi G\mathrm{/}{c}^4$, with formal units $\mathrm{m}{\mathrm{J}}^{-1}$; $v$ has units ${\mathrm{m}}^{-2}$. There is $R_0>0$, common to all $p_c\in P_c$, such that for each $p_c$ there is a unique pair of even continuous functions $p,v$ on $[-R_0,R_0]$ with $\mathrm{\mid }p(r)-p_c\mathrm{\mid }\le \delta$ and satisfying $v(r)=K{\int }_0^1{s}^2ϵ(p(rs))ds,p(r)=p_c-{\int }_0^{r}t\frac{(ϵ(p(t))+p(t))(v(t)+Kp(t))}{1-2t^{2}v(t)}dt\mathrm{.}$ They are $C^{\mathrm{\infty }}$, $p(0)=p_c$, $v(0)=Kϵ(p_c)\mathrm{/}3$, $p^{\mathrm{\prime }}(0)=v^{\mathrm{\prime }}(0)=0$, and $b(r):=1-2r^{2}v(r)\ge 1\mathrm{/}2$ on $[0,R_0]$. Define $M(r)=\frac{c^2}{G}{r}^{3}v(r),\varphi (r)={\varphi }_c+{\int }_0^{r}t\frac{v(t)+Kp(t)}{1-2t^{2}v(t)}dt(\mathrm{\mid }r\mathrm{\mid }\le R_0)\mathrm{.}$ Then $(M,p,\varphi )$ satisfies the mass, pressure, and lapse equations of Complete spherical Einstein and conservation reduction for $0<r\le R_0$. The metric extends across $r=0$ as a $C^2$ Lorentzian metric in Cartesian spatial coordinates, with $b(0)=1$ and lapse $e^{{\varphi }_c}>0$; this is the regular-center conclusion in this item. On the common interval, $p,v,\varphi$ depend continuously in the uniform norm on $p_c\in P_c$. For $r>0$, the solution continues past any finite endpoint at which its graph remains in a compact subset of $\{(r,p,v):r>0,p\in I_p,1-2r^{2}v>0\}\mathrm{.}$ No assertion is made that every equation of state reaches a zero-pressure surface, avoids a horizon globally, or gives a dynamically stable solution.