Smooth global data for the axisymmetric comparison

Work in ZFC, including choice. Use the supplied stationary Lorentzian and causal conventions. The global comparison data consist of a smooth connected time-oriented four-dimensional spacetime $(M,g,N)$ with a specified asymptotically flat spacelike end $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and a complete stationary Killing flow ${\varphi }_t$ generated by $T$, future timelike on that end, together with a smooth effective axial circle action ${\psi }_s$ of period $2\pi$ by isometries. Require these actions to commute. End decay, $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$, $D=I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})\cap I_M^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$, $I^{+}$-regularity and regular axis have exactly the predicates of the supplied black-hole comparison-hypothesis interface. Use the supplied normalized circle-action convention. Require the future event horizon to be connected.

When retaining the original item's mean-nondegeneracy assumption, supply a smooth compact horizon section $S$ without boundary and compatible Gaussian-null coordinate patches near it of the form $g=r\phi d{v}^2+2dvdr+2r{h}_{a}d{x}^{a}dv+h_{ab}d{x}^{a}d{x}^b$, with horizon $r=0$ and section $v=r=0$ in these patches. Require their section restrictions to define a single global smooth scalar ${\phi }_S:S\to \mathbb{R}$ and a smooth positive-definite metric $h$ on $S$: on overlapping section charts the coefficients $\phi {\mathrm{\mid }}_S$ agree as scalar values and $h_{ab}{\mathrm{\mid }}_S$ obey the metric tensor transition law. Let $d{\mu }_h$ be the intrinsic Riemannian density and interpret ${\int }_S{\phi }_{S}d{\mu }_h$ by the supplied compactly supported smooth-density integration interface. The relevant mean condition is ${\int }_S{\phi }_{S}d{\mu }_h\mathrm{\ne }0$ in that supplied normalization. This is a nonvanishing predicate; no numerical surface-gravity normalization or implication to a Killing horizon is asserted. It is an optional additional hypothesis for the imported smooth axisymmetric classification, whose source theorem does not need it.

For a specified comparison spacetime, supply separate stationary Lorentzian data $(\tilde{M},\tilde{g},\tilde{N},\tilde{T})$ with a selected asymptotically flat spacelike end ${\tilde{S}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and complete stationary Killing flow ${\tilde{\varphi }}_t$, future timelike on that end. Apply the same end and causal-exterior conventions to these data, setting ${\tilde{M}}_{\mathrm{e}\mathrm{x}\mathrm{t}}={\bigcup }_{t\in \mathbb{R}}{\tilde{\varphi }}_t({\tilde{S}}_{\mathrm{e}\mathrm{x}\mathrm{t}})$ and $\tilde{D}=I_{\tilde{M}}^{+}({\tilde{M}}_{\mathrm{e}\mathrm{x}\mathrm{t}})\cap I_{\tilde{M}}^{-}({\tilde{M}}_{\mathrm{e}\mathrm{x}\mathrm{t}})$. A global metric classification of the exterior by this comparison spacetime means that there exists a diffeomorphism $F:D\to \tilde{D}$ satisfying $F^{\ast }(\tilde{g}{\mathrm{\mid }}_{\tilde{D}})=g{\mathrm{\mid }}_D$. Diffeomorphism and tensor pullback use their supplied mathematical definitions. This does not mean an isometry of all extensions beyond the horizons.

Stationary Lorentzian data and tensor conventions

Staticity, axisymmetry and black-hole regularity hypotheses

Causal exterior, chronological boundaries and Cauchy data

Periodic isometries and invariant normal averaging

The Axiom of Choice

Riemannian volume of a compactly supported smooth density

Diffeomorphisms and local diffeomorphisms of manifolds

The pullback of a covariant tensor field

Given: The supplied end, global flows, causal data, optional compatible Gaussian-null section data and, when a comparison is specified, its separate spacetime data.

Goal: Verify that these define precise conditional comparison data.

1.1 Complete flow and circle-action predicates specify maps for all real parameters, while the causal interface and the comparison-hypothesis supplier determine $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$, $D$ and the full $I^{+}$-regularity predicate. Commutation, regular-axis and horizon-connectedness are separate stated conditions; none is inferred from a local Killing field. [given, F1]

1.2 For the optional mean condition, the required overlap compatibility supplies a global smooth ${\phi }_S$ and Riemannian metric $h$. The support of ${\phi }_S$ is a closed subset of compact $S$, hence compact. Thus the density-integration interface applies under ZFC and makes its scalar integral finite and the nonzero test meaningful in the supplied normalization. Multiplication by a nonzero fixed normalization factor does not change whether the integral vanishes. No existence or gauge-independence of these Gaussian-null data is deduced. [given, F1, F2, F4]

2.1 Applying the same exterior construction to the separate comparison data fixes ${\tilde{M}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and $\tilde{D}$ in the ambient $\tilde{M}$. The diffeomorphism and tensor-pullback interfaces then make the displayed condition on $F$ a definite predicate, exactly the stipulated isometry between the entire exteriors. No such map or comparison spacetime is asserted to exist by this definition. These definitions require no rigidity, nondegeneracy bridge or classification result. [given, F1, F3, step 1.1, step 1.2] ∎

Chruściel–Costa–Heusler supplies the end, stationarity, exterior and $I^{+}$-regularity definitions, axial terminology and the Gaussian-null mean criterion under the recorded hypotheses and reading limits. Theorem 3.2 omits mean-nondegeneracy. Only that omission is noted here; its classification is not proved or applied in this definition. No numerical surface-gravity identity is used.