LemmaStatement: AI-generatedProof: AI-generated

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Boundary membership excludes interior focal points from the target

Statement

For the null segment $γ$ in the compact-target hypotheses, reverse its direction and regard it as a past null normal geodesic issued from $c \in C$ . There is no focal point to $C$ strictly between $c$ and $p$ . Focal points here are rank loss of the normal screen Jacobi map, with the direction and affine parameter reversed from the future convention in the null-screen definition.

Statement references

Causal exterior, chronological boundaries and Cauchy data

Null expansions, screen quotients and focal data

A compact chronological target supplies a causal endpoint

A compact-target boundary point has a null geodesic segment

Facts & Assumptions

[A1]

Work in ZFC. The axiom of choice implies dependent choice and countable choice, supplying the choice assumptions of the geometric and causal suppliers used below. The Axiom of Choice AC implies DC implies countable choice

[F1]

The causal relations, boundary, global hyperbolicity and temporal function are relative to $E$ as in the causal-exterior interface. The metric-selected connection is torsion-free and metric-compatible, so differentiation of metric pairings obeys the product rule. Causal exterior, chronological boundaries and Cauchy data Metric compatibility, torsion freedom and the metric-selected connection class

[F2]

The target surface, null normal congruence, positive screen quotient, normal screen Jacobi map and rank-loss definition of focality have the meanings in the null-screen interface. Null expansions, screen quotients and focal data

[F3]

The entire causal segment from $p$ to $c$ lies in $\partial_{E} I_{E}^{-} (C)$ under the compact-target hypotheses specified in [F7]. The entire target segment stays on the chronological boundary

[F4]

The final tangent of a smooth nonconstant future null affine segment from $p$ to $c \in C$ is orthogonal to $T_{c} C$ under those hypotheses; global acausality of $C$ is not required. Endpoint orthogonality without an acausal target assumption

[F5]

For a smooth null normal geodesic from a smooth spacelike surface without boundary, a normal screen Jacobi field with nonzero initial tangent vector and zero screen value at a positive instant puts points sufficiently shortly afterward in $I^{+} (C)$ . Its equation is $D_{u}^{2} J + R J = 0$ , where $R [Z] = [R (Z, L) L]$ ; its initial data are $J (0) = [X]$ and $D_{u} J (0) = B_{0} X$ , with $g (B_{0} X, Y) = χ_{ℓ} (X, Y)$ for $X, Y \in T_{c} C$ . Apply this mathematical theorem in $(E, g, - N)$ when reversing time orientation. Null focal points exclude later horismos generators

[F6]

The geodesic initial-value problem has a unique smooth maximal solution on an open interval, and its evaluation is smooth on an open time-and-initial-data domain. An included smooth finite endpoint extends by this same initial-value theorem. The geodesic initial-value problem and the exponential map of the metric-selected connection

[F7]

The compact-target hypotheses mean that $(E, g, N)$ is a smooth globally hyperbolic time-oriented Lorentzian four-manifold, $C \subset E$ is a nonempty compact smooth embedded spacelike two-surface without boundary, $τ$ is temporal, and $p \in \partial_{E} I_{E}^{-} (C)$ satisfies $τ (p) < \min_{C} τ$ . Under these hypotheses some $c \in C$ satisfies $p \leq_{E} c$ , $p \neq c$ , and $p \notin I_{E}^{-} (C)$ . A compact chronological target supplies a causal endpoint

[F8]

Under [F7], there is a nonconstant future null affine segment $γ : [0, 1] \to E$ from $p$ to some $c \in C$ , smooth up to both endpoints and with nonzero tangent. The argument below applies to every such segment. A compact-target boundary point has a null geodesic segment

[F9]

Around every point of a smooth embedded spacelike two-surface there is a smooth normalized pair of future null normals. The normal plane is Lorentzian with exactly two future null rays; selecting a ray and positive rescaling gives a prescribed nonzero vector on that ray. The positive screen metric, screen derivative and symmetric form $χ_{ℓ} (X, Y) = g (\nabla_{X} ℓ, Y)$ are well-defined. Local null normal pairs and the positive screen quotient

[F10]

For a smooth variation through affinely parametrized geodesics with tangent $L$ and variation field $V$ , $D_{u} V = D_{r} L$ and $D_{u}^{2} V = R (L, V) L$ . These are the geodesic-deviation identities, with the same curvature convention as [F5]. The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields

[F11]

Each chronological past $I_{E}^{-} (c)$ is open. Arbitrary unions of open sets are open by topology axiom (T2). Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison

[F12]

Each $X \in T_{c} C$ is the velocity at zero of a smooth curve in $C$ through $c$ , defined on an open interval about zero. Every tangent vector is the velocity of a smooth curve

[F13]

Closed bounded real intervals are compact: every open cover has a finite subcover. Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line

Proof

Given: The data and hypotheses in the Statement, with the compact-target hypotheses and segment made explicit in [F7] and [F8].

Goal: Exclude rank loss of the reversed normal screen Jacobi map at every $u \in (0, 1)$ .

1.1A1F1F2F4F7F8F9given

Put $δ (u) = γ (1 - u)$ for $0 \leq u \leq 1$ and reverse the time orientation to $- N$ . Then $δ$ is a smooth nonconstant future null affine geodesic in $(E, g, - N)$ starting at $c$ , with nonzero normal tangent by [F4] and [F8]. Apply [F9] with this reversed orientation. Interchange the two local null-normal labels if necessary so that $δ' (0) = a ℓ_{c}$ for $a > 0$ , and replace the local pair $(ℓ, k)$ by $(a ℓ, a^{- 1} k)$ . It remains normalized and now $ℓ_{c} = δ' (0)$ . This uses only a local field near $c$ , not a global normal pair. The choice assumptions of these suppliers follow from [A1].

2.1step 1.1F1F2F6F10F12F13constructalgebra

By [F6], extend $δ$ slightly past both endpoints. The open smooth geodesic-evaluation domain contains its initial state at every time in the interval $[0, 1]$ , compact by [F13]; a finite subcover therefore gives a neighborhood $U$ of $c$ in $C$ and an open interval $J \supset [0, 1]$ on which the geodesics with initial data $(x, ℓ_{x})$ define a smooth map $F : U \times J \to E$ . Uniqueness gives $F (c, u) = δ (u)$ . Each parameter curve is null by metric compatibility and its null initial tangent, has nonzero tangent by uniqueness, and stays future-directed for $- N$ by continuity. For $X \in T_{c} C$ , choose the curve $x (r)$ in [F12] with $x (0) = c$ and $x' (0) = X$ , and set $V_{X} (u) = \partial_{r} F (x (r), u) ∣_{r = 0}$ . By [F10], $D_{u} V_{X} = D_{r} L$ and $D_{u}^{2} V_{X} = R (L, V_{X}) L$ , where $L = \partial_{u} F$ . Metric compatibility gives $\frac{d}{d u} g (V_{X}, L) = g (D_{r} L, L) = \frac{1}{2} \partial_{r} g (L, L) = 0$ . Its initial value is $g (X, ℓ_{c}) = 0$ , so $V_{X}$ is orthogonal to $L$ throughout. The same calculation at every $x \in U$ verifies the orthogonality requirement in [F2], hence $F$ is an admissible local null-normal congruence even at rank loss.

3.1step 2.1F1F2F5F9F10algebra

Put $J_{X} = [V_{X}] = A_{u} X$ in the positive screen along $δ$ . Projecting the deviation equation from step 2.1 gives $D_{u}^{2} J_{X} + R J_{X} = 0$ with the convention of [F5], since $R (L, V_{X}) L = - R (V_{X}, L) L$ . At $u = 0$ , $J_{X} (0) = [X]$ and $D_{u} J_{X} (0) = [\nabla_{X} ℓ]$ . At $c$ , normality gives $T_{c} C \subset ℓ_{c}^{⊥}$ . If $a ℓ_{c} \in T_{c} C$ , its squared norm is zero, so positivity of $g ∣_{T_{c} C}$ forces $a ℓ_{c} = 0$ . Thus the quotient map $ι : T_{c} C \to ℓ_{c}^{⊥} / R ℓ_{c}$ , $Y \mapsto [Y]$ , is injective. Nondegeneracy of the ambient metric and $ℓ_{c} \neq 0$ give $\dim ℓ_{c}^{⊥} = 3$ , so its quotient by $R ℓ_{c}$ has dimension two, equal to $\dim T_{c} C$ ; hence $ι$ is onto. By the quotient metric in [F2], $\overset{ˉ}{γ} (ι Y, ι Z) = g (Y, Z)$ , so $ι$ is an isometry. Metric compatibility in [F1] gives $g (\nabla_{X} ℓ, ℓ_{c}) = \frac{1}{2} X (g (ℓ, ℓ)) = 0$ , making $[\nabla_{X} ℓ]$ a screen vector. For every $Y \in T_{c} C$ , $\overset{ˉ}{γ} ([\nabla_{X} ℓ], ι Y) = g (\nabla_{X} ℓ, Y) = χ_{ℓ} (X, Y) = g (B_{0} X, Y) = \overset{ˉ}{γ} (ι (B_{0} X), ι Y)$ . Since $ι$ is onto and the screen metric is positive definite, $[\nabla_{X} ℓ] = ι (B_{0} X)$ , which is the identification used in [F5]. Thus $J_{X}$ has exactly the equation and initial data required in [F5]. In particular rank loss of $A_{u_{0}}$ supplies $X \neq 0$ with $J_{X} (u_{0}) = 0$ , which is a focal instant in that theorem's sense.

4.1A1step 1.1step 3.1F1F3F5F11∎

Suppose $u_{0} \in (0, 1)$ were such a rank-loss instant. Step 3.1 makes [F5] applicable in $(E, g, - N)$ , so for some $u$ with $u_{0} < u < 1$ , sufficiently near $u_{0}$ , one has $δ (u) \in I_{E, - N}^{+} (C) = I_{E, N}^{-} (C)$ . On the other hand [F3] puts $δ (u)$ in $\partial_{E} I_{E, N}^{-} (C)$ . The set $I_{E, N}^{-} (C) = ⋃_{c \in C} I_{E, N}^{-} (c)$ is open by [F11], so by the boundary definition in [F1] it is disjoint from its boundary. This is a contradiction. Hence no focal point occurs strictly between $c$ and $p$ . No exclusion of a focal point at $p$ is asserted.

Depends on

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