AIK global Kerr identification under the full GR/SBS/PK contract

Let $(M,g)$ be a smooth connected oriented time-oriented paracompact four-dimensional vacuum spacetime with no manifold boundary, and let $T$ be a smooth Killing field. Impose all of the following independent-source hypotheses, without replacing them by the article's shorter regularity wording. Use Countable Choice for the smooth tangent-bundle conventions of the geometric suppliers.

GR. There is an embedded partial Cauchy hypersurface ${\mathrm{\Sigma }}_0$ and a diffeomorphism ${\mathrm{\Phi }}_0:\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>1\mathrm{/}2\}\to {\mathrm{\Sigma }}_0$. Here partial Cauchy means achronal and edgeless; the edge of an achronal set consists of points in its closure for which every sufficiently small neighborhood contains a timelike curve from its local past to its local future avoiding the set. For large $R_0$, this chart extends to an end diffeomorphism $\mathbb{R}\times \{\mathrm{\mid }x\mathrm{\mid }>R_0\}\to M^{\mathrm{e}\mathrm{n}\mathrm{d}}$ with $T={\mathrm{\partial }}_t$. The source end bounds are $g_{00}=-1+2M_0\mathrm{/}r+O_6(r^{-2})$, $g_{ij}={\delta }_{ij}+O_6(r^{-1})$ and $g_{0i}=-2{ϵ}_{ijk}{S}^j{x}^k\mathrm{/}{r}^3+O_6(r^{-3})$, where $M_0>0$ and $J=\mathrm{\mid }S\mathrm{\mid }<M_0^2$. The symbol $O_6$ bounds all coordinate derivatives up to order six with the corresponding extra inverse powers of $r$; stationary time derivatives vanish. Set $E=I^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})\cap I^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$ using ambient timelike curves. Require that $E$, with the restricted metric and time orientation, is globally hyperbolic, the two separate equalities ${\mathrm{\Sigma }}_0\cap I^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})={\mathrm{\Sigma }}_0\cap I^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})={\mathrm{\Phi }}_0(\{\mathrm{\mid }x\mathrm{\mid }>1\})$, nonvanishing of $T$ throughout $E$, and completeness of every $T$-orbit in $E$ with every such orbit meeting ${\mathrm{\Sigma }}_0$.

SBS. The sphere $S_0={\mathrm{\Phi }}_0(\{\mathrm{\mid }x\mathrm{\mid }=1\})$ is the bifurcation sphere. In a neighborhood of it, $H^{+}=\mathrm{\partial }{I}^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$ and $H^{-}=\mathrm{\partial }{I}^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$ are smooth null nonexpanding hypersurfaces intersecting transversely in $S_0$. For a local smooth future null generator $L$ of either branch, its positive screen is ${\mathcal{S}}_p=L_p^{\perp }\mathrm{/}\mathbb{R}{L}_p$, with $\bar{q}([X],[Y])=g(X,Y)$. A local spacelike two-dimensional section $Q$ transverse to $L$ identifies $T_{p}Q$ isometrically with this screen. With $q=g{\mathrm{\mid }}_{TQ}$, define ${\chi }_L(X,Y)=g({\mathrm{\nabla }}_{X}L,Y)$ for $X,Y\in TQ$ and ${\theta }_L={\mathrm{tr}}_q{\chi }_L=q^{AB}{\chi }_{L,AB}$, using the metric-selected connection and its pullback to $Q$. Nonexpanding means ${\theta }_L=0$ on every such local section. Under $L\mapsto e^{f}L$ both ${\chi }_L$ and ${\theta }_L$ scale by $e^f$, so this zero condition is independent of generator normalization, including the source null-pair convention $g(L,\underset{‾}{L})=-1$ and the suppliers' convention $-2$. The stationary field is tangent to both hypersurfaces and is not identically zero on $S_0$.

PK. Use the source-normalized Ernst potential $\sigma \to 1$ at infinity and the Mars–Simon tensor $\mathcal{S}$ of the declared interface, defined in an open neighborhood of ${\mathrm{\Sigma }}_0\cap E$. For the future unit normal $E_0$ to ${\mathrm{\Sigma }}_0$ and a smooth orthonormal frame $E_0,E_1,E_2,E_3$, require $\mathrm{\mid }(1-\sigma )\mathcal{S}(T,E_{\alpha },E_{\beta },E_{\gamma })\mathrm{\mid }\le \epsilon$ on ${\mathrm{\Sigma }}_0\cap E$ for a sufficiently small positive $\epsilon$ depending on the finite regularity constant $A$ constructed in source equations (2.1)–(2.5). If this tensor is zero, the PK bound is satisfied for every positive threshold, but all GR and SBS conditions remain required.

Under this complete GR/SBS/PK contract, $E$ is globally isometric to the domain of outer communication of the Kerr spacetime with mass $M_0$ and angular momentum magnitude $J$. Global isometry means one metric-preserving diffeomorphism onto that full Kerr domain. This source result does not assert that the original article's weaker displayed end and section assumptions automatically imply GR.

Concretely, the target uses the subextremal Kerr tensor on ${\mathcal{M}}_{\mathrm{K}}={\mathbb{R}}_t\times (r_{+},\mathrm{\infty })\times S^2$ from the declared Kerr interface, with its mass parameter $M=M_0$, specific angular momentum $a=J\mathrm{/}{M}_0$ and $r_{+}=M_0+\sqrt{M_0^2-a^2}$. The GR bound gives $0\le a<M_0$. This is the full exterior for one selected asymptotically flat end, including the rotation axis via the smooth sphere tensor; the horizon $r=r_{+}$ is excluded. The standard identification of this exterior with that end's Kerr DOC is the imported geometric fact in Dafermos–Rodnianski, §5.1; it applies also when $J=a=0$.

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

Killing fields of a Lorentzian metric

Metric compatibility, torsion freedom and the metric-selected connection class

Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

Chronology, causality, strong causality, global hyperbolicity and time functions

Local and global flows generated by a vector field

Ernst sublevels and the Mars–Simon tensor interface

Adopted Einstein field equation in the direct-equation presentation

Subextremal Kerr exterior in length coordinates

Null normal pairs and screen geometry

Local null normal pairs and the positive screen quotient

The Axiom of Countable Choice (AC_ω)

Imported established result: Alexakis–Ionescu–Klainerman, arXiv:0904.0982v1, §1.1 Main Theorem, PDF p.5. Its hypotheses are the preceding GR, SBS and PK clauses, including both separate chronological-section equalities, its angular-momentum end coefficient and subextremal parameter range. Its conclusion is expressly a global DOC isometry. The source's analytic continuation and final uniqueness argument are not reconstructed here. This import records a correctly scoped external theorem; application to a different hypothesis package remains a separate task.

The target identification uses Dafermos–Rodnianski, Lectures on black holes and linear waves, arXiv:0811.0354v1, §5.1, PDF pp.56–57: the standard subextremal Kerr DOC is the exterior represented by the displayed Boyer–Lindquist metric, with angular momentum $Ma$. Its metric agrees with the declared global Kerr tensor on polar charts and uses the latter's smooth sphere-axis convention. Only this target identification is imported from that source.