Staticity, axisymmetry and black-hole regularity hypotheses

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold, $T$ a specified Killing field and $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ a specified asymptotically flat spacelike end. Here stationary relative to the selected end means that there is a specified sufficiently large end-chart radius $R_1$ such that $T$ is future timelike on the tail $r\ge R_1$. Restrict to that tail in the stationary-flow exterior construction and denote the selected restricted end again by $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$. The decay convention is that of the supplied asymptotically flat spacelike-end interface: for some $\alpha >0$ and integer $m\ge 2$, ${\gamma }_{ij}-{\delta }_{ij}=O_m(r^{-\alpha })$ and $k_{ij}=O_{m-1}(r^{-1-\alpha })$. Completeness of $T$ means its flow is defined for every real time at every point of $M$, not only in the exterior. For complete $T$, form $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$, $D=I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})\cap I_M^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$ and ${\mathcal{E}}^{+}={\mathrm{\partial }}_{M}D\cap I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$ by the supplied stationary-flow exterior interface. All these sets use the specified tail and the ambient chronology of $M$.

The triple $(M,g,T)$ is $I^{+}$-regular in the Chruściel–Costa sense, relative to this end, when $T$ is complete, $D$ is globally hyperbolic, and there exists in $D$ a spacelike connected acausal hypersurface $S\supset S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ whose closure ${\bar{S}}^M$ is a topological manifold with boundary consisting of a compact set and finitely many asymptotic ends. Its boundary $B:={\bar{S}}^M\setminus S$ is a topological manifold, satisfies $B\subset {\mathcal{E}}^{+}$, and meets every generator of ${\mathcal{E}}^{+}$ precisely once. Use the supplied closure and boundary conventions, generator and cross-section conventions, and finite-end acausal-section data. Global hyperbolicity is computed for the Lorentzian domain $D$, with its own causal curves; acausality of $S$ is relative to the ambient $M$. No smoothness of the completed boundary or Killing generator for the event horizon is included in this definition.

On an open region where $T$ is timelike, staticity means hypersurface orthogonality: locally the hyperplanes $T^{\perp }=\{v:g(T,v)=0\}$ are tangent to spacelike hypersurfaces. Equivalently the Frobenius condition is $T^{\mathrm{♭}}\wedge d{T}^{\mathrm{♭}}=0$, where $T^{\mathrm{♭}}=g(T,\cdot )$, $d$ is the invariant exterior derivative, whose smooth-form well-definedness is supplied by the declared multilinearity lemma, and $\wedge$ is the wedge product. The equivalence is the cited external staticity criterion, not a global splitting theorem. The ergoregion is $\{p:g_p(T,T)>0\}$; points where $g(T,T)=0$ do not belong to it.

The stationary axisymmetric comparison data additionally specify a jointly smooth left action $a:S^1\times M\to M$ of the standard circle Lie group $S^1=\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$, written ${\psi }_s(p)=a([s],p)$, by isometries. Thus ${\psi }_0={\mathrm{id}}_M$, ${\psi }_{s+t}={\psi }_s\circ {\psi }_t$ and ${\psi }_{s+2\pi }={\psi }_s$; effectiveness means that ${\psi }_s={\mathrm{id}}_M$ implies $s\in 2\pi \mathbb{Z}$. Its generator is $Z_p={\frac{d}{ds}\mid }_{s=0}{\psi }_s(p)$. Require $Z$ to commute with $T$: explicitly, the flow of $T$ and ${\psi }_s$ commute wherever both compositions are defined. The axis is the fixed-point set $A=\{p:{\psi }_s(p)=p\text{ for all }s\}$, required here to be a nonempty smooth two-dimensional submanifold. To specify a regular axis, require that $g$ extend smoothly and nondegenerately across $A$, that its normal two-plane $(T_{p}A{)}^{{\perp }_g}$ be spacelike at every $p\in A$, and that the differential $D_p{\psi }_s$ on that plane be the ordinary rotation through angle $s$ up to choice of orientation. This last normalization is weight one; it excludes a multiple rotation or a conical quotient at the axis. These axis conditions are explicit additional data, not a consequence of $I^{+}$-regularity.

Analyticity means a compatible real-analytic atlas in which the metric components are locally convergent real power series. Horizon connectedness means connectedness of the specified horizon subset (for example ${\mathcal{E}}^{+}$). For the following surface-gravity convention use the nonzero-generator portion of the supplied Killing-horizon interface: $H\subset M$ is a smooth embedded null hypersurface and $K$ is a specified smooth Killing field on a neighborhood of $H$, nonzero on $H$, tangent and normal to it, so $K_p\in T_{p}H$ and $g_p(K_p,X)=0$ for every $X\in T_{p}H$. Here null means that the metric restricted to $T_{p}H$ has a nonzero radical at every point. On this smooth Killing-horizon portion, surface gravity $\kappa$ is the unique smooth function satisfying $d(g(K,K){)}_p=-2\kappa (p)K_p^{\mathrm{♭}}$ for $p\in H$. This is an equality of ambient covectors in $T_p^{\ast }M$, not of their pullbacks to $T_{p}H$. It is degenerate if $\kappa$ vanishes identically and nondegenerate if $\kappa$ is nowhere zero on the component under consideration; if constancy is assumed or separately supplied this is just $\kappa =0$ versus $\kappa \mathrm{\ne }0$. No sign, constancy or normalization-independent numerical value is asserted here. Analyticity, connectedness, and this Killing-horizon nondegeneracy are separate hypotheses; none follows by definition from $I^{+}$-regularity. The mean-nondegeneracy used in Chruściel–Costa Theorem 1.3 is not silently identified with an arbitrary event-horizon surface gravity.

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

Killing fields of a Lorentzian metric

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

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

Asymptotically flat spacelike end and derivative decay

Complete stationary flow and its exterior region

Ambient closure and intrinsic boundary of a hypersurface

Null generators and once-only cross-sections

Acausal hypersurface with compact core and finitely many ends

The inverse metric, musical isomorphisms and index raising

The exterior derivative by the invariant vector-field formula

The wedge product of differential forms

Smooth left actions of Lie groups

The invariant exterior-derivative formula is C^∞-multilinear

Causal solutions, Killing horizons, escape and metric extensions

Given: The objects and specified data in the Definition.

Goal: Verify that the displayed conditions define unambiguous objects and predicates on these data.

1.1 The supplied cutoff and future timelikeness satisfy the exterior interface's end hypothesis. For complete $T$ its flow-saturated region and ambient causal exterior are therefore defined. The five interfaces specify the end decay, topological completion and once-only generator intersections; conjoining those predicates with global hyperbolicity of $D$ gives the stated $I^{+}$-regularity condition. The compact-core condition retains its asymptotic ends, and the boundary condition uses ${\mathcal{E}}^{+}$ with its required $I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$ factor. This uses no generator-existence or horizon-smoothness assertion. [F1, F2, F4]

1.2 Metric duality gives the smooth one-form $T^{\mathrm{♭}}$. The multilinearity lemma gives a smooth two-form $d{T}^{\mathrm{♭}}$, to which the wedge-product construction applies, yielding the well-defined three-form $T^{\mathrm{♭}}\wedge d{T}^{\mathrm{♭}}$. On the timelike region the covector is nonzero, so the cited staticity criterion applies to its orthogonal hyperplanes. The scalar $g(T,T)$ also determines the ergoregion by the strict inequality in the definition. [F2, F3, F4]

1.3 The jointly smooth circle action determines a smooth generator, fixed points and differentials at those points. Differentiating ${\psi }_{s+h}={\psi }_h\circ {\psi }_s$ in $h$ at zero shows that ${\psi }_s$ is the flow of its generator $Z$; differentiating ${\psi }_s^{\ast }g=g$ at zero then gives ${\mathcal{L}}_{Z}g=0$. At $p\in A$ its differential is the identity on $T_{p}A$ because ${\psi }_s$ fixes $A$ pointwise, and preserves $(T_{p}A{)}^{{\perp }_g}$ because ${\psi }_s$ is an isometry. Thus the stated weight-one test is a predicate on its normal representation. The regular-axis and commutation conditions are specified data; no existence of an axial action is inferred. [F2, F5, given]

2.1 On the specified Killing-horizon portion the hypotheses of [F6] hold: the embedded hypersurface is null and its Killing field is nonzero, tangent and normal. That supplier therefore gives existence, uniqueness and smoothness of $\kappa$ in the ambient-covector equation, not just uniqueness conditional on existence. Analytic charts, connectedness and vanishing or nonvanishing of that scalar test the separate predicates in the Definition. These verifications establish no rigidity or uniqueness theorem, zeroth law or existence of a horizon-generating Killing field. [F2, F3, F4, F6, step 1.2, step 1.3] ∎

Chruściel–Costa Definition 1.1, (1.1), together with §§2.1–2.2, supplies the independent regularity, end-decay and exterior-definition inputs. Their §4.1 supplies generator and cross-section terminology, and §2.3, (2.11), fixes the surface-gravity sign convention. Chruściel–Costa–Heusler §3.1 supplies the local staticity equivalence; §3.2 supplies axial-field terminology, not a derivation of the specified effective action, commutation or regular-axis conditions. The declared mathematical action and horizon suppliers furnish the missing formulation and well-definedness interfaces. These are definition/criterion inputs with the hypotheses and reading limits recorded in the three source records; no uniqueness, generator-existence, horizon-smoothness or zeroth-law theorem is imported here as a locally established result.