Kerr metric, parameters, exterior and extremality

PDF p. 1: §1.1, Boyer–Lindquist display; PDF p. 2: §1.1, Boyer–Lindquist display

Use geometric units $G=c=1$. Let $M>0$ and $a\in \mathbb{R}$ satisfy $\mathrm{\mid }a\mathrm{\mid }\le M$, and put $J=Ma$, ${\rho }^2=r^2+a^2{\cos }^2\theta$ and $\mathrm{\Delta }=r^2-2Mr+a^2$. The Kerr exterior is the smooth Lorentzian tensor on ${\mathbb{R}}_t\times (r_{+},\mathrm{\infty })\times S^2$ whose Boyer–Lindquist expression away from the rotation axis is $g_{\mathrm{K}\mathrm{e}\mathrm{r}\mathrm{r}}=-\mathrm{\Delta }{\rho }^{-2}(dt-a{\sin }^2\theta d\varphi {)}^2+{\sin }^2\theta {\rho }^{-2}((r^2+a^2)d\varphi -adt{)}^2+{\rho }^2{\mathrm{\Delta }}^{-1}d{r}^2+{\rho }^{2}d{\theta }^2,$ where $r_{+}=M+\sqrt{M^2-a^2}$, the azimuth has period $2\pi$, and the tensor extends smoothly over the axis by the round-sphere construction. Subextremal means $\mathrm{\mid }a\mathrm{\mid }<M$ and extremal means $\mathrm{\mid }a\mathrm{\mid }=M$. The $a=0$ member is Schwarzschild.

Here $M$ is the ADM rest mass of the specified asymptotically flat constant $t$ end and $J$ its angular momentum, using the axial orientation ${\mathrm{\partial }}_{\varphi }$. Its asymptotic stationary field is $T={\mathrm{\partial }}_t$, normalized by $g(T,T)\to -1$. In the specified future ingoing Kerr extension and relative to the future null infinity attached to this exterior, $r=r_{+}$ is the future event horizon, the boundary of the black hole region. The Boyer–Lindquist exterior itself contains no horizon points; the horizon statement uses that extension. No assertion of a bifurcation sphere is made for extremal Kerr.

Kerr exterior tensor including the extremal parameter range

A causal barrier identifies the future Kerr event horizon

The Komar angular-momentum flux of the Kerr end

The Kerr end is an ADM rest frame

A specified future ingoing Kerr extension and its time orientation

Outgoing Kerr null curves and the selected future null infinity

Asymptotic mass and axial angular-momentum conventions

The Boyer–Lindquist spatial end and differentiated expansion

The asymptotic metric flux of Kerr equals its mass parameter

Given: The exterior tensor, horizon conclusion and end-charge conclusions of [F1], the external angular-momentum identification in [F2], the extension and selected infinity in [F3], the charge conventions and spacelike end in [F4], and the energy calculation in [F5], and the explicit ADM axial-flux definition and finite-sphere identity in [F6].

Goal: Justify the definition's metric, horizon and charge interpretations throughout its parameter range.

1.1 The exterior supplier establishes smoothness, Lorentzian signature, the coordinate Killing fields, the axial period and the Schwarzschild specialization. The ingoing extension in [F3] supplies the exterior embedding and smooth future time orientation, and its outgoing-completion supplier constructs the selected conformal infinity. The causal-horizon supplier in [F1], with its stated Countable Choice and Dependent Choice assumptions, proves that precisely $r>r_{+}$ can send a future signal to that infinity. Its complement has boundary $r=r_{+}$, which is the claimed future event horizon. These constructions and the causal conclusion include $\mathrm{\mid }a\mathrm{\mid }=M$ and do not rely on two distinct roots. [given, F1, F3]

2.1 The charge conventions and spacelike asymptotically flat end are those of [F4]. The energy calculation [F5] gives $E_{\mathrm{A}\mathrm{D}\mathrm{M}}=M$, while the rest-frame supplier in [F1] gives zero linear momentum and rest mass $M$. The independently calculated axial Komar integral in [F1] exists and is $Ma$, with the future/outward orientation of [F4]. The exterior tensor gives $g(T,T)=-1+2Mr\mathrm{/}{\rho }^2\to -1$. [given, F1, F4, F5, step 1.1]

3.1 In the Cartesian chart $x=r(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$ of [F4], the axial field of [F1] is $\eta =-x^2{\mathrm{\partial }}_{x^1}+x^1{\mathrm{\partial }}_{x^2}$, with the specified positive $2\pi$ period. It is a smooth Killing field tangent to the constant $t$ slices and their spheres, including at its axis zeros. Since $\eta (r)=0$, the normal formula in [F4] gives $h({\nu }_h,\eta )=dr(\eta )\mathrm{/}\mathrm{\mid }{\mathrm{grad}}_{h}r{\mathrm{\mid }}_h=0$. Thus the trace term in [F6] vanishes identically on every sphere. The finite-sphere identity in [F6], with $K^{\mathrm{G}}=-K$ and the same future/outward orientations, identifies each ADM defining surface integral with $J_{\mathrm{K}}(S_r)$. Step 2.1 supplies its convergent limit $Ma$, proving existence of the ADM axial charge and $J_{\mathrm{A}\mathrm{D}\mathrm{M}}[\eta ]=Ma$ in precisely the convention used by [F2]. This verifies the imported identification on this end, including negative $a$, $a=0$ and $\mathrm{\mid }a\mathrm{\mid }=M$, without requiring an unproved convergence theorem for arbitrary ends. These justifications fix the intended geometric meaning of the parameters without using the original article as proof. [given, F1, F2, F4, F6, step 2.1] ∎

Reall §9.4 following (9.20) supplies the ADM–Komar identification. Gourgoulhon (2.43), (7.63) and (7.104) supply its explicit sign convention, ADM definition and finite-sphere geometric identity. The Kerr end and Komar limit are local supplier results; their application in step 3.1 is a local argument, not a literature-derived proof of the whole item.

Original claim attribution: The uniqueness of smooth stationary black holes, arXiv:2611.16853v1, §1.1, PDF pp.1–2. The original displayed metric, $M>0$, $J=Ma$, subextremal/extremal terminology, horizon location and Schwarzschild specialization are retained. This attribution supplies no prerequisite proof. Reall §§7.2–7.4, printed pp.93–97, and Visser §5, printed pp.14–15, were consulted for coordinate and asymptotic conventions; Visser §9 explicitly restricts its horizon discussion to subextremal Kerr, and is not used for the extremal assertion. The proof above uses the specified ingoing patch, not an assumed maximal-extension diagram.