The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation

This item assumes the axiom of countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$ through the tangent-bundle structure used by Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (c) (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); no further choice principle is used.

(a) Parameters and the Schwarzschild radius. Let $c>0$ and $G_N>0$ be real constants and let $m_{\ast }>0$ be a real mass parameter. Put

$s:=\frac{2G_N{m}_{\ast }}{c^2}>0.$

The number $s$ is the Schwarzschild radius of the datum. Neither $c$, $G_N$ nor $m_{\ast }$ is measured here; the physical reading of $c$ and $G_N$ as the framework's conversion and Newton constants and of $m_{\ast }$ as a mass is fixed by the physical conventions of the framework, not by this definition.

(b) The underlying manifold and its product charts. Let $S^2:=S_2(0,1)=\{y\in {\mathbb{R}}^3:\parallel y{\parallel }_2=1\}$ be the unit Euclidean sphere with the subspace topology of Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$. Specify its smooth structure as the maximal atlas generated by the following two charts (Smooth manifolds and their smooth charts). For $\epsilon \in \{+1,-1\}$ put $p_{\epsilon }=(0,0,\epsilon )$ and $U_{\epsilon }=S^2\setminus \{p_{\epsilon }\}$, and define

${\sigma }_{\epsilon }:U_{\epsilon }\to {\mathbb{R}}^2,{\sigma }_{\epsilon }(y)=\frac{(y_1,y_2)}{1-\epsilon y_3}\mathrm{.}$

These open domains cover $S^2$. Each chart is a homeomorphism onto ${\mathbb{R}}^2$, with inverse, for $u=(u_1,u_2)$ and $q=u_1^2+u_2^2$,

${\sigma }_{\epsilon }^{-1}(u)=(\frac{2u_1}{1+q},\frac{2u_2}{1+q},\epsilon \frac{q-1}{1+q})\mathrm{.}$

The two inverse identities follow by substitution; all displayed denominators are nonzero on their domains. On the overlap both transition maps are $u\mapsto u\mathrm{/}(u_1^2+u_2^2)$ on ${\mathbb{R}}^2\setminus \{0\}$, hence smooth. The subspace is Hausdorff and has a countable basis given by intersections with Euclidean balls of rational centres and positive rational radii. Thus these charts specify a smooth two-manifold without boundary. This is the standard sphere structure used for the round metric below: the inclusion has the displayed smooth local parametrizations, and their differentials are injective because composition with ${\sigma }_{\epsilon }$ is the identity. Let $M$ be the product manifold

$M:=\mathbb{R}\times (s,\mathrm{\infty })\times S^2$

with the product smooth structure (Products of smooth manifolds have a canonical product smooth structure); it is a smooth four-manifold without boundary. Write $x^0:M\to \mathbb{R}$ and $r:M\to (s,\mathrm{\infty })$ for the first and second projections and $\iota :M\to S^2$ for the third. A product chart of $M$ is a smooth chart on the product whose angular part is a smooth chart of $S^2$. In a spherical-polar product chart, $0<\vartheta <\pi$ and the components of the round metric ${\gamma }_{S^2}$ of The round metric on the sphere as an induced metric are ${\gamma }_{22}=1$, ${\gamma }_{23}={\gamma }_{32}=0$, ${\gamma }_{33}={\sin }^2\vartheta$. These polar charts omit the two poles; ordinary smooth sphere charts around the poles complete the product atlas.

(c) The metric. Let $f:(s,\mathrm{\infty })\to (0,\mathrm{\infty })$ be the smooth function

$f(r):=1-\frac{s}{r}\mathrm{.}$

Define $g$ globally by the coordinate-free expression

$g=-fd(x^0)\otimes d(x^0)+f^{-1}dr\otimes dr+r^2{\iota }^{\ast }{\gamma }_{S^2}\mathrm{.}$

In every spherical-polar product chart its component functions are

$g_{00}:=-f(r),g_{11}:=\frac{1}{f(r)},g_{22}:=r^2,g_{33}:=r^2{\sin }^2\vartheta ,$

and $g_{ab}:=0$ whenever $a\mathrm{\ne }b$, $a,b\in \{0,1,2,3\}$, with the index order $(x^0,x^1,x^2,x^3)=(x^0,r,\vartheta ,\phi )$. This is a smooth symmetric covariant $2$-tensor field on all of $M$ (Smooth covariant 2-tensor fields on a smooth manifold); the component display is local to spherical-polar charts, and $f>0$ because $r>s$. The pair $(M,g)$ is the Schwarzschild exterior metric of mass parameter $m_{\ast }$; the datum of this definition is the quadruple $(c,G_N,m_{\ast },g)$. With these parameters fixed, write $(M,g,N)$ for its associated geometric datum, with $N$ as in clause (d). The geometric datum depends on the parameters only through $s$ and does not retain $c$, $G_N$ and $m_{\ast }$ separately. That $(M,g,N)$ is a time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$ is not asserted here; it is proved in the companion metric-properties lemma of this page (its justification record in the frontmatter), which also records the equivalent coordinate-free description $g=-fd(x^0)\otimes d(x^0)+f^{-1}dr\otimes dr+r^2{\iota }^{\ast }{\gamma }_{S^2}$.

(d) Time orientation and the Schwarzschild chart. The field $N:={\mathrm{\partial }}_0$ is the smooth product vector field on $M$ (Derivations at a point and the tangent space); the chart components of $g$ are independent of $x^0$, so $N$ is the restriction of a global vector field, and $g(N,N)=-f(r)<0$ at every point. The datum carries the time orientation $N$. Spherical-polar product charts are Schwarzschild charts of the datum; at the poles the same metric is represented in the smooth sphere charts of the product atlas. We write $t$ for the coordinate $x^0\mathrm{/}c$ when the datum is read in the framework's unit conventions.

(e) Scope. The domain of the datum is the exterior region $r>s$; the definition asserts nothing about $r=s$, about $r\le 0$, about any matter, or about any equation satisfied by $g$. The horizon $r=s$ is not part of the datum, and a continuation of the metric to $r=s$ requires the separate constructions recorded on this page, the regularity lemma for the advanced chart and the Kruskal model-properties lemma. Whether the datum is a solution of the framework's field law is proved, not defined, in the Ricci-and-Einstein-tensor lemma of this page.

• Units and the physical reading. In the framework's conventions the coordinate $x^0=ct$ and the radial coordinate $r$ have dimension of length in metres, the angular coordinates are dimensionless, and the metric coefficients $g_{00}$ and $g_{11}$ are dimensionless, while $g_{22}$ and $g_{33}$ have units of square metres; $m_{\ast }$ is read in kilograms and $s$ in metres. No numerical value of any constant is asserted. The reading of $x^0$ as a clock coordinate of static observers is part of the framework's postulates and is not part of this geometric definition (see the static-congruence and clock-rate lemma and the exterior vacuum-model theorem of this page).

• The sphere factor. The function $r$ is the areal radius: on the level set $\{x^0=c_0,r=c_1\}$ the induced metric is $c_1^2{\gamma }_{S^2}$, so the area of the sphere is $4\pi c_1^2$. Nothing in this definition identifies $r$ with a Euclidean radial coordinate of any ambient space.