Subextremal Kerr exterior in length coordinates

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the tangent-bundle smooth structure of the Lorentzian conventions.

Let $M>0$ and $a\in \mathbb{R}$ satisfy $\mathrm{\mid }a\mathrm{\mid }<M$, and put $r_{\pm }:=M\pm \sqrt{M^2-a^2}$.

Let ${\mathcal{M}}_{\mathrm{K}}:=\mathbb{R}\times (r_{+},\mathrm{\infty })\times S^2$ carry the product smooth structure, where $S^2=\{(x,y,z)\in {\mathbb{R}}^3:x^2+y^2+z^2=1\}$ has its usual smooth structure and round metric ${\gamma }_{S^2}$. Let $\pi :{\mathcal{M}}_{\mathrm{K}}\to S^2$ be the third projection, let $\xi$ be the smooth rotation field on $S^2$ generated by $(x,y,z)\mapsto (-y,x,0)$, and put $\gamma :={\pi }^{\ast }{\gamma }_{S^2}$ and ${\xi }^{\mathrm{♭}}:={\pi }^{\ast }({\gamma }_{S^2}(\xi ,\cdot ))$. Let $t,r$ be the first and second coordinate functions on ${\mathcal{M}}_{\mathrm{K}}$, and let $\zeta :=z\circ \pi$ be the third Euclidean coordinate on the sphere factor. Define the global smooth functions

$\mathrm{\Sigma }:=r^2+a^2{\zeta }^2,\mathrm{\Delta }:=r^2-2Mr+a^2\mathrm{.}$

On the axis-avoiding polar coordinate domain, let $0<\theta <\pi$ and regard $\varphi$ as the circle-valued azimuth in $\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$. Each choice of an open real interval of length less than $2\pi$ lifting $\varphi$ gives an ordinary local chart. In any such chart, $\zeta =\cos \theta$, $\mathrm{\Sigma }=r^2+a^2{\cos }^2\theta$, $\gamma =d{\theta }^2+{\sin }^2\theta d{\varphi }^2$, and ${\xi }^{\mathrm{♭}}={\sin }^2\theta d\varphi$.

Define the symmetric covariant $2$-tensor $g$ globally on ${\mathcal{M}}_{\mathrm{K}}$ by

$g=-(1-\frac{2Mr}{\mathrm{\Sigma }})dt\otimes dt+\frac{\mathrm{\Sigma }}{\mathrm{\Delta }}dr\otimes dr+\mathrm{\Sigma }\gamma +\frac{a^2(\mathrm{\Sigma }+2Mr)}{\mathrm{\Sigma }}{\xi }^{\mathrm{♭}}\otimes {\xi }^{\mathrm{♭}}-\frac{2aMr}{\mathrm{\Sigma }}(dt\otimes {\xi }^{\mathrm{♭}}+{\xi }^{\mathrm{♭}}\otimes dt)\mathrm{.}$

On any such local polar chart, its Boyer-Lindquist line element is

$d{s}^2=-(1-\frac{2Mr}{\mathrm{\Sigma }})d{t}^2-\frac{4Mar{\sin }^2\theta }{\mathrm{\Sigma }}dtd\varphi +\frac{\mathrm{\Sigma }}{\mathrm{\Delta }}d{r}^2+\mathrm{\Sigma }d{\theta }^2+\frac{(r^2+a^2{)}^2-a^2\mathrm{\Delta }{\sin }^2\theta }{\mathrm{\Sigma }}{\sin }^2\theta d{\varphi }^2\mathrm{.}$

The displayed cross term is $2g_{t\varphi }dtd\varphi$. This definition makes no nondegeneracy or signature claim; Lorentzian signature and a time orientation are established separately.

Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$

The round metric on the sphere as an induced metric

Products of smooth manifolds have a canonical product smooth structure

Smooth covariant 2-tensor fields on a smooth manifold

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

A smooth vector field is a smooth section of the tangent bundle

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The Boyer-Lindquist chart excludes the rotation axis, while the global tensor definition uses the smooth round metric and the smooth rotation field on $S^2$. The domain has $r>r_{+}$; the hypersurface $r=r_{+}$ is not part of this definition and is included only after the regular ingoing extension is constructed.

The Lorentzian signature and time-orientation properties are proved by Lorentzian signature and time orientation of the Kerr exterior datum ↗, not assumed by this tensor definition.

The smooth tangent-bundle convention assumes only countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$; no stronger choice principle is used.

The parameters correspond to $M=G_N{m}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{c}^2$ and $a=J_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(m_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}c)$ when a physical interpretation is adopted. No claim that this datum describes a measured rotating object is made.