A specified future ingoing Kerr extension and its time orientation

Fix $M>0$, $\mathrm{\mid }a\mathrm{\mid }\le M$ and $r_{\pm }=M\pm \sqrt{M^2-a^2}$ as in the exterior tensor. The future ingoing Kerr extension used here is ${\mathcal{K}}^{\mathrm{i}\mathrm{n}}={\mathbb{R}}_v\times (0,\mathrm{\infty }{)}_r\times S^2$, with $\mathrm{\Sigma }=r^2+a^2{\cos }^2\theta$, $\mathrm{\Delta }=r^2-2Mr+a^2$ and metric $g=-(1-2Mr\mathrm{/}\mathrm{\Sigma })d{v}^2+2dvdr-4Mar{\sin }^2\theta dvd\psi \mathrm{/}\mathrm{\Sigma }-2a{\sin }^2\theta drd\psi +\mathrm{\Sigma }d{\theta }^2+(r^2+a^2+2M{a}^{2}r{\sin }^2\theta \mathrm{/}\mathrm{\Sigma }){\sin }^2\theta d{\psi }^2\mathrm{.}$ Here $\psi$ has period $2\pi$. Precisely, take $S^2=\{(x,y,z)\in {\mathbb{R}}^3:x^2+y^2+z^2=1\}$ with its standard smooth atlas and inclusion $\iota :S^2\to {\mathbb{R}}^3$. Define the round metric $\gamma ={\iota }^{\ast }(d{x}^2+d{y}^2+d{z}^2)$, the tangent rotation field $\xi (x,y,z)=(-y,x,0)$, and its metric-dual one-form $\alpha =\gamma (\xi ,\cdot )={\iota }^{\ast }(xdy-ydx)$. On a polar chart $(x,y,z)=(\sin \theta \cos \psi ,\sin \theta \sin \psi ,\cos \theta )$, these are $\gamma =d{\theta }^2+{\sin }^2\theta d{\psi }^2$ and $\alpha ={\sin }^2\theta d\psi$. Globally, the displayed metric means the following tensor, with the sphere tensors pulled back to the product and $\mathrm{\Sigma }=r^2+a^2{z}^2$: $g=-(1-2Mr\mathrm{/}\mathrm{\Sigma })d{v}^2+2dvdr-(4Mar\mathrm{/}\mathrm{\Sigma })dv\alpha -2adr\alpha +\mathrm{\Sigma }\gamma +a^2(1+2Mr\mathrm{/}\mathrm{\Sigma }){\alpha }^2\mathrm{.}$ Products in these line elements denote symmetric tensor products, with $\beta \omega =(\beta \otimes \omega +\omega \otimes \beta )\mathrm{/}2$. This sphere construction does not restrict $a$ to the subextremal range.

Put $\tau =v-r$ and choose $-\mathrm{\nabla }\tau$ to be future timelike, using the inverse-metric and time-orientation conventions for stationary Lorentzian data. The exterior embeds by $v=t+F(r)$ and $\psi =\varphi +G(r)$ with $F^{\mathrm{\prime }}=(r^2+a^2)\mathrm{/}\mathrm{\Delta }$ and $G^{\mathrm{\prime }}=a\mathrm{/}\mathrm{\Delta }$; additive constants merely choose origins. On the sphere this angular map is the rotation $R_{G(r)}$, where $R_{\lambda }(x,y,z)=(x\cos \lambda -y\sin \lambda ,x\sin \lambda +y\cos \lambda ,z)$. The exterior coordinate tensor used for the substitution is the uncharged case $q=0$, $x^0=t$ of the Boyer–Lindquist formula, restricted to $r>r_{+}$ and $0<\theta <\pi$; its global exterior smoothness and signature are those of the exterior tensor. This specifies a global future extension with one selected exterior. It does not assert that this patch is the maximal analytic extension or contains a bifurcation sphere.

Kerr exterior tensor including the extremal parameter range

Stationary Lorentzian data and tensor conventions

Kerr-Newman metric and potential on the exact Boyer-Lindquist domain

The n-sphere with its standard smooth atlas

The pullback of a covariant tensor field

Given: The parameters and global tensor in the Definition, with the exterior properties [F1], tensor and orientation conventions [F2], pole-free coordinate formula [F3], and sphere and pullback interfaces [F4].

Goal: Justify smoothness, signature, embedding and time orientation.

1.1 The sphere inclusion is smooth in the stereographic atlas of [F4], as its inverse chart formulas have smooth rational ambient components. Its induced Euclidean metric $\gamma$ is smooth, and $\xi$ is tangent since $x(-y)+yx+z\cdot 0=0$. For any tangent vector $X$, $\gamma (\xi ,X)=-ydx(X)+xdy(X)$, proving the stated formula for the smooth one-form $\alpha$. The polar parametrization yields the displayed expressions for $\gamma$ and $\alpha$. Since $r>0$ gives $\mathrm{\Sigma }=r^2+a^2{z}^2\ge r^2>0$, every coefficient of the global tensor is smooth. Its angular part agrees with the polar metric because $r^2+a^2=\mathrm{\Sigma }+a^2{\sin }^2\theta$, so it is $\mathrm{\Sigma }\gamma +a^2(1+2Mr\mathrm{/}\mathrm{\Sigma }){\alpha }^2$. This establishes smoothness at both poles without dividing by $\sin \theta$ or assuming $\mathrm{\mid }a\mathrm{\mid }<M$. [given, F4, algebra]

2.1 On the exterior, [F1] gives $\mathrm{\Delta }>0$, so the smooth radial primitives $F,G$ exist on the interval $(r_{+},\mathrm{\infty })$. Substitute $dt=dv-(r^2+a^2)dr\mathrm{/}\mathrm{\Delta }$ and $d\varphi =d\psi -adr\mathrm{/}\mathrm{\Delta }$ in [F3]. Writing $A=r^2+a^2$ and $b={\sin }^2\theta$, the two squared forms become $dt-abd\varphi =dv-abd\psi -(\mathrm{\Sigma }\mathrm{/}\mathrm{\Delta })dr$ and $Ad\varphi -adt=Ad\psi -adv$. Thus the $d{r}^2$ term cancels and expansion gives exactly the polar metric in the Definition. The global coordinate map is $(t,r,p)\mapsto (t+F(r),r,R_{G(r)}p)$, with smooth inverse $(v,r,p)\mapsto (v-F(r),r,R_{-G(r)}p)$. Rotations are smooth also at the poles, so this is a diffeomorphism onto $r>r_{+}$ of the extension. Equality of its pulled-back metric with the exterior tensor holds on the dense pole-free region by substitution and hence at the poles by smoothness. [given, F1, F3, F4, step 1.1, algebra]

3.1 The determinant relative to $dvdr$ and round-sphere area is $-{\mathrm{\Sigma }}^2$. It is nonzero for every $r>0$, including $r=r_{+}$ and the poles, so the exterior Lorentzian signature extends throughout this connected product. Direct matrix inversion on the pole-free region gives $g^{vv}=a^2{\sin }^2\theta \mathrm{/}\mathrm{\Sigma }$, $g^{vr}=(r^2+a^2)\mathrm{/}\mathrm{\Sigma }$, $g^{rr}=\mathrm{\Delta }\mathrm{/}\mathrm{\Sigma }$; the resulting scalar identities extend to the poles by smoothness. Hence $g^{-1}(d\tau ,d\tau )=-1-2Mr\mathrm{/}\mathrm{\Sigma }<0$. Thus the chosen time orientation is smooth by [F2]. Near infinity, $T={\mathrm{\partial }}_t={\mathrm{\partial }}_v$ is timelike and $g(-\mathrm{\nabla }\tau ,T)=-d\tau (T)=-1<0$, so it is future-directed. None of these computations divides by $r_{+}-r_{-}$, so the extremal case is included. [given, F1, F2, step 1.1, step 2.1, algebra] ∎

The cited Kerr coordinate formulas provide background for the local calculation in this item.