Hawking horizon generator and analytic global rigidity

PDF p. 2: §1.1, Hawking rigidity discussion

Adopt the four-dimensional stationary vacuum model of Stationary vacuum model and asymptotic normalization. For the smooth local conclusion, assume the regular bifurcate non-expanding horizon pair $(S,H_{+},H_{-})$ of Regular bifurcate null horizon data before rigidity in a connected oriented neighborhood $O$, with $T$ tangent to both branches. A smooth Hawking Killing field $K_0$ exists on a possibly smaller open neighborhood $U\subseteq O$ of $S$. For every local conclusion below, replace $H_{\pm }$ by $H_{\pm }\cap U$ and retain the same notation. It may be chosen with affine horizon profiles $K_0=s{L}_{+}$ on $H_{+}$ and $K_0=-t{L}_{-}$ on $H_{-}$, where the generators are nonzero and affine, $s=t=0$ at $S$, $L_{+}(s)=L_{-}(t)=1$, and the zero sets of the branch parameters are exactly $S$. For any constant $c>0$, the field $K=c{K}_0$ is nonzero on $(H_{+}\cup H_{-})\setminus S$ and zero on $S$. It is Killing to every order along both branches. Its signed surface gravities are $c$ and $-c$ on the two nonzero branch parts, with the common nonnegative magnitude $\kappa =c$; the future branch has coefficient $c$ under this labeling. A numerical value therefore requires the chosen scale. These statements do not assign a coefficient by the acceleration equation at a zero of $K$.

If additionally $T{\mathrm{\mid }}_S\equiv ̸0$ (the local rotating symmetry hypothesis), after possibly shrinking $U$ again and restricting both branches and $K_0$ accordingly, the source construction yields $[T,K_0]=0$ and a nontrivial periodic rotational Killing field $Z=T+{\lambda }_0{K}_0$ near $S$, commuting with $T$ and $K_0$. The condition means not identically zero on the sphere, not nowhere zero there.

For global analytic rigidity, use the precise comparison data of Staticity, axisymmetry and black-hole regularity hypotheses and impose all hypotheses in the following paragraph. Let $M$ be analytic, asymptotically flat, strongly causal and vacuum, with complete stationary $T$, and let its connected exterior $D$ be globally hyperbolic. A connected future-horizon component has a compact cross-section $A\subset I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$. Assume either nonzero mean surface gravity or periodic induced $T$-flow on the generator space. The mean is the source's pre-rigidity quantity: in Gaussian null coordinates $g=r\phi d{v}^2+2dvdr+2r{h}_{a}d{x}^{a}dv+h_{ab}d{x}^{a}d{x}^b$, it is $⟨\kappa {⟩}_A=-\mathrm{\mid }A{\mathrm{\mid }}^{-1}{\int }_A\phi d{\mu }_h$, with $d{\mu }_h$ the volume measure of the section metric and $\mathrm{\mid }A\mathrm{\mid }$ its volume. Also assume either strong causality of $\bar{D}\cap I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$, or an asymptotically flat spacelike hypersurface in $D$, achronal in $D$, whose completed boundary is $A\subset {\mathcal{E}}^{+}$. Require that $T$ is not tangent to the generators of that component. These are the full alternatives of Chruściel–Costa Theorem 4.14, not consequences asserted from the local bifurcate data. In four-dimensional $I^{+}$-regular applications its Remark 4.15 supplies the periodic-generator alternative.

Under these analytic global hypotheses, the restriction of the chosen local $K$ to any connected nonempty exterior collar extends to a Killing field on all $D$. In addition the exterior is stationary axisymmetric, with a global complete periodic rotational Killing field commuting with $T$. The continuation is of the chosen local field; its scale is retained. No identification of separately normalized source generators is assumed. Smooth local rigidity by itself asserts no global extension.

Stationary vacuum model and asymptotic normalization

Regular bifurcate null horizon data before rigidity

Hawking and rotational fields near a bifurcate sphere

Bifurcate Killing horizons, rotation and normalization

Rotation, surface gravity and horizon degeneracy

Staticity, axisymmetry and black-hole regularity hypotheses

Periodic isometries and invariant normal averaging

Affine null-generator profiles vanish exactly on the bifurcation section

Affine generator profiles give constant signed acceleration

For a Killing field the acceleration and surface-gravity equations agree

An exact local Killing field is Killing to every order on a horizon

Constant combinations preserve horizon symmetries and normalize periods

AIK Theorem 1.1, Proposition 3.1 and its equation (3.1), with the affine setup of §2 and extension in §4, supply a full-neighborhood Killing field having the stated affine branch profiles. Their Theorem 1.2 and §5 supply the commuting periodic combination under the additional branch-tangent, nontrivial-on-the-sphere symmetry hypothesis. These are independent existence and extension inputs, not a local reconstruction of the characteristic PDE argument.

Chruściel–Costa Corollary 2.4 supplies simple connectedness of the globally hyperbolic asymptotically flat vacuum exterior, since vacuum satisfies the null energy condition. Their Theorem 4.14, Remark 4.15 and §7.1 supply global complete periodic rotational symmetry under all global hypotheses written in the Statement; no inference of those hypotheses from an arbitrary smooth null horizon is made.

Chruściel–Costa–Heusler §3.4.2 supplies analytic continuation of the near-horizon Killing field throughout the simply connected analytic exterior by Nomizu's theorem. This is a source-supported continuation input for the very same local field on its exterior collar; the original Nomizu theorem is not claimed to have been read or reproved here.