LemmaStatement: AI-generatedProof: AI-generated

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The normal future and past quadrants are the local causal sets of the sphere

Statement

In the normal tube and boundary setting of the preceding lemmas, write $α = a + b$ , $β = a - b$ . For the locally absolutely continuous timelike and causal curve conventions, let $I_{O}^{\pm} (S)$ and $J_{O}^{\pm} (S)$ denote the corresponding futures and pasts of $S$ witnessed by curves lying entirely in $O$ . One has $I_{O}^{+} (S) = {α > 0, β > 0}$ and $J_{O}^{+} (S) = {α \geq 0, β \geq 0}$ , where $J$ includes $S$ by reflexivity. The past versions have both inequalities reversed. Consequently the two null axes $H_{+} \cup H_{-}$ form exactly the local null boundary sheets of the causal future and past of $S$ .

Statement references

Stationary Lorentzian data and tensor conventions

The geodesic initial-value problem and the exponential map of the metric-selected connection

Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed

A compact spacelike surface has a normal exponential tube

Smooth transverse horizon branches are the null axes of the normal tube

Causal curves from a common chronological boundary obey two barriers

Causal curves, chronological and causal reachability, and inextendibility

Facts & Assumptions

[F1]

The normal tube is diffeomorphic and star shaped in its normal variables. Its event boundaries and chronological-set sides are the stated axes and quadrants. A compact spacelike surface has a normal exponential tube Smooth transverse horizon branches are the null axes of the normal tube

[F2]

Curves from $S$ obey the two chronological boundary barriers. Causal curves from a common chronological boundary obey two barriers

[F3]

Geodesics preserve their metric norm and future/past character, and the normal radial geodesic has its prescribed initial tangent. The geodesic initial-value problem and the exponential map of the metric-selected connection Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed Stationary Lorentzian data and tensor conventions

[F4]

Timelike and causal curves are locally absolutely continuous, with nonzero appropriately directed timelike or causal velocity almost everywhere. The sets $I^{\pm} (S)$ and $J^{\pm} (S)$ are unions of pointwise chronological and causal reachability sets over $S$ , and causal reachability also allows equality. The subscript $O$ restricts the witnessing curves to $O$ . Causal curves, chronological and causal reachability, and inextendibility

Proof

**Given:**The data and hypotheses in the Statement.

Goal: Establish the stated conclusions under exactly those hypotheses.

1.1F1F2F4

A future timelike curve from any $p \in S$ ends outside $\overline{G}$ and inside $F_{+}$ by [F2]. The side equations [F1] give $β > 0$ and $α > 0$ . For a future causal curve, the weak barriers give $β \geq 0$ and $α \geq 0$ . These are necessary conditions for the local chronological and causal future.

2.1F1F3F4step 1.1∎

Conversely, if both coordinates are positive then the initial normal vector $a n + b ν$ has $a > ∣ b ∣$ , so is future timelike. Its radial geodesic lies in $O$ by [F1], remains future timelike by [F3], and joins its base point in $S$ to the specified endpoint. For weak inequalities the same vector is future causal or zero, so its future causal radial geodesic (timelike in the interior and null on a nonzero axis), or reflexive reachability at $S$ when the vector is zero, gives membership in $J_{O}^{+} (S)$ by [F4]. The past assertions follow by changing the signs of $a, b$ . Since $Ξ$ is a homeomorphism, the boundaries of these quadrants are exactly their null axes in the respective future/past halves.

Depends on

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