The Kruskal model datum: domain, implicit radial function, metric and time orientation

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

(a) Parameters. Let $c>0$, $G_N>0$ and $m_{\ast }>0$ be the constants and mass parameter of the Schwarzschild exterior datum, and put $s:=2G_N{m}_{\ast }\mathrm{/}{c}^2>0$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation clause (a)). The Kruskal model datum of mass parameter $m_{\ast }$ is constructed from these data as follows.

(b) Domain and smooth structure. Let $\mathrm{\Omega }:=\{(U,V)\in {\mathbb{R}}^2:UV<1\}$ and let $\mathcal{K}:=\mathrm{\Omega }\times S^2$, where $S^2\subset {\mathbb{R}}^3$ is the unit Euclidean sphere, with the product smooth structure (Products of smooth manifolds have a canonical product smooth structure, Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$). It is a smooth four-manifold without boundary. Write $U,V:\mathcal{K}\to \mathbb{R}$ for the first two projections and $\iota :\mathcal{K}\to S^2$ for the third. A Kruskal product chart uses any smooth chart on $S^2$. In a spherical-polar Kruskal chart the angular coordinates satisfy $0<\vartheta <\pi$ and ${\gamma }_{S^2}=\mathrm{diag}(1,{\sin }^2\vartheta {)}_{ij}$; these charts omit the two poles, which are covered by ordinary sphere charts.

(c) The radial function. The datum carries a function $r:\mathcal{K}\to (0,\mathrm{\infty })$, the Kruskal radial function, characterized by the implicit relation

$UV=-(\frac{r}{s}-1)e^{r\mathrm{/}s}\text{at every point of }\mathcal{K}\mathrm{.}$

That for every $(U,V)\in \mathrm{\Omega }$ there is exactly one $r(U,V)>0$ satisfying this relation, and that $r$ is smooth, is not asserted here; it is proved in Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar ↗ on this page, the justification record of this definition. The value $r=0$ is not attained on $\mathcal{K}$, and $r=s$ exactly on the axes $U=0$ or $V=0$.

(d) The metric. Define the metric globally by

$g_{\mathcal{K}}=-\frac{2s^3}{r}{e}^{-r\mathrm{/}s}(dU\otimes dV+dV\otimes dU)+r^2{\iota }^{\ast }{\gamma }_{S^2}\mathrm{.}$

In every spherical-polar Kruskal chart its component functions are

$g_{UV}:=g_{VU}:=-\frac{2s^3}{r}{e}^{-r\mathrm{/}s},g_{\vartheta \vartheta }:=r^2,g_{\phi \phi }:=r^2{\sin }^2\vartheta ,g_{UU}:=g_{VV}:=g_{U\vartheta }:=g_{U\phi }:=g_{V\vartheta }:=g_{V\phi }:=0,$

where $r$ is the function of clause (c) evaluated at the point. This global expression defines a smooth symmetric covariant $2$-tensor field on $\mathcal{K}$ once the smoothness of $r$ is established by Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar ↗. In another sphere chart the angular block is $r^2({\gamma }_{ij})$. That $g_{\mathcal{K}}$ is Lorentzian is proved, not asserted, in the same lemma.

(e) Time orientation. Define the field $N_{\mathcal{K}}:={\mathrm{\partial }}_U+{\mathrm{\partial }}_V$ on $\mathcal{K}$. Its component expression is the same in every Kruskal chart; that it is a smooth timelike field and hence a time orientation of $g_{\mathcal{K}}$ is proved in Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar ↗.

(f) Scope. The datum is a geometric object: the definition asserts nothing about whether $g_{\mathcal{K}}$ satisfies any field equation, about its causal structure, about the maximality of the extension it provides, or about nature. The comparison with the exterior datum, the vacuum property and the causal structure are the content of the two following items on this page. The region $\{UV=1\}$, where the implicit relation of clause (c) has no solution with $r>0$, is not part of $\mathcal{K}$ and is not appended to it by this definition.

• Four regions. The set $\{UV\mathrm{\ne }0\}$ has four connected components (the quadrants $U<0<V$, $U>0,V>0$, $U>0>V$ and $U<0,V<0$), separated by the null axes $U=0$ and $V=0$. On components with $UV<0$ the radial function exceeds $s$, while on components with $UV>0$ it is less than $s$; both facts follow from the implicit relation once clause (c) is justified. The causal and metric meaning of the four regions is the subject of the following items.