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.