Regular bifurcate null horizon data before rigidity

Use the smooth Lorentzian metric, time orientation and Levi–Civita connection conventions of the stationary Lorentzian data, but supply no stationary Killing field for the horizon construction. Work in their inherited ZFC conventions. Let $S$ be an embedded spacelike surface diffeomorphic to the unit sphere in ${\mathbb{R}}^3$. Its induced positive metric is denoted $\gamma$. Use the null-screen conventions for null normals, spacelike sections and expansions.

Curves and reachability use the causal-curve interface: curves are continuous and locally absolutely continuous in smooth charts, with nonzero future or past causal, respectively timelike, velocity almost everywhere. For a chosen open neighborhood $O$ of $S$, let $J_O^{+}(S)$ consist of $S$ and all endpoints of future causal curves from $S$ on compact parameter intervals, with the entire curve contained in $O$; let $I_O^{+}(S)$ use future timelike curves with distinct parameter endpoints and without adjoining $S$. Define $J_O^{-}(S)$ and $I_O^{-}(S)$ by past-directed curves. Thus $S$ is included in each causal set by the reflexive endpoint convention. An affinely parametrized geodesic is a smooth curve $c$ satisfying $D_{\lambda }{c}^{\mathrm{\prime }}=0$ for the supplied connection; a null generator has nonzero null velocity. Included parameter endpoints have the one-sided derivative convention.

A regular bifurcate null horizon pair near $S$ consists of $O$ and two smooth embedded null hypersurfaces $H_1,H_2\subset O$ without boundary in $O$, intersecting transversely exactly in $S$ and generated by null geodesics orthogonal to $S$. Transverse means $T_p{H}_1+T_p{H}_2=T_{p}M$ for $p\in S$; null means that the restriction of $g$ to each hypersurface tangent space has a one-dimensional radical, the tangent vectors orthogonal to every vector of that tangent space. Each generator meets $S$ exactly once in this neighborhood and is taken on a connected affine-parameter interval with that intersection in its interior. Choose parameter zero there and increasing to the future. Define $H_i^{+}$ and $H_i^{-}$ as the unions of the nonnegative and nonpositive parameter portions of these generators, respectively. Each includes $S$, and $H_i=H_i^{+}\cup H_i^{-}$ with $H_i^{+}\cap H_i^{-}=S$. Only the signs, not any affine scale, enter these subsets.

The local causal-boundary requirement is the following pair of set identities, with all causal curves confined to $O$:

${\mathrm{\partial }}_O{J}_O^{+}(S)=J_O^{+}(S)\setminus I_O^{+}(S)=H_1^{+}\cup H_2^{+}\mathrm{.}$

${\mathrm{\partial }}_O{J}_O^{-}(S)=J_O^{-}(S)\setminus I_O^{-}(S)=H_1^{-}\cup H_2^{-}\mathrm{.}$

Here ${\mathrm{\partial }}_{O}A$ is the boundary relative to $O$: every neighborhood in $O$ of such a point meets both $A$ and $O\setminus A$. These are the future and past half-branches, joined along $S$; each $H_i$ itself is a full smooth sheet through $S$. Require each full $H_i$ to be achronal in $O$, meaning that no two of its points are joined by a timelike curve contained in $O$. These conditions are hypotheses on the supplied branches, not assertions that an arbitrary sphere has such extensions. They concern the causal sets computed in $O$, without asserting equalities for causal sets computed in the ambient $M$.

For a smooth nonzero null generator $L_i$ on a branch and tangent vectors $X,Y$ to a transverse spacelike section $Q$, put ${\chi }_i(X,Y)=g({\mathrm{\nabla }}_X{L}_i,Y)$ and ${\theta }_i={\mathrm{tr}}_{{\gamma }_Q}{\chi }_i$, where ${\gamma }_Q=g{\mathrm{\mid }}_{TQ}$ is the section metric (equal to $\gamma$ on $S$). Call the pair non-expanding when ${\theta }_i=0$ everywhere on each full branch, including both half-branches and $S$. When generator fields are only locally chosen this means the condition in each such chart. No Killing generator, periodic symmetry, surface gravity, or rigidity conclusion is assumed.

Stationary Lorentzian data and tensor conventions

Null expansions, screen quotients and focal data

Geodesic of an affine connection

Causal curves, chronological and causal reachability, and inextendibility

Alexakis–Ionescu–Klainerman, local horizon setup, §§1–2

Given: The supplied sphere, branches and ambient data in the Definition.

Goal: Explain the conditional data and their relative causal-boundary meaning without assuming a Killing generator.

1.1 Embeddedness, transversality and the radical condition are conditions on the indicated tangent spaces. The positive section metric permits its inverse and the trace in the definition of expansion. The affine generator condition uses the supplied connection and geodesic equation. Thus each derivative and contraction has the supplied Lorentzian and null-screen meaning. [given, F1, F3]

2.1 The causal sets use the curve class on the restricted spacetime $O$, including the stated endpoint convention. Future-directed affine parameters based at $S$ define the two halves of each supplied generator; positive affine rescaling preserves their signs. The displayed relative-boundary identities are explicit hypotheses on these subsets, not a theorem deduced from smoothness and transversality. Achronality uses the same curve class in $O$. [given, F2, F3, step 1.1]

3.1 The full-sheet convention matches the geometric setup recorded in F4, while the displayed identities specify exactly which future and past pieces are required. The expansion condition is a pointwise trace equation on both halves and at $S$. Neither these equations nor the set conditions assert existence of a Killing field. This supplies the pre-rigidity data without using a rigidity conclusion as a premise or claiming that arbitrary supplied sheets satisfy the boundary identities. [given, F1, F2, F3, F4, step 1.1, step 2.1] ∎

Only the geometric sheet/parameter convention is taken from AIK, §§1–2, especially (2.11). The displayed boundary equalities specify that convention as conditional data here. The justification is local definitional bookkeeping; no existence or rigidity theorem is imported by this item.