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Endpoint orthogonality without an acausal target assumption

Statement

Under the same compact-target hypotheses, let $γ : [0, 1] \to E$ be a smooth nonconstant future null affine geodesic with $γ (0) = p$ and $γ (1) = c \in C$ . Then $g (\dot{γ} (1), Y) = 0$ for every $Y \in T_{c} C$ . It suffices that $C$ be smooth and without boundary near $c$ ; no global acausality assumption on $C$ is needed.

Facts & Assumptions

[A1]

Work in ZFC; the axiom of choice implies the countable and dependent choice assumptions of the causal suppliers. The Axiom of Choice AC implies DC implies countable choice

[F1]

The smooth metric, connection, causal relations and time orientation are those of Causal exterior, chronological boundaries and Cauchy data.

[F2]

The target is a smooth embedded spacelike two-manifold without boundary. Null expansions, screen quotients and focal data

[F3]

The initial point satisfies $p \notin I_{E}^{-} (C)$ . A compact chronological target supplies a causal endpoint

[F4]

Such a smooth null affine segment exists under these hypotheses. A compact-target boundary point has a null geodesic segment

[F5]

Every point has arbitrarily small convex normal neighborhoods. Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points

[F6]

On a convex normal neighborhood, $h (q) = g_{r} (\exp_{r}^{- 1} (q), \exp_{r}^{- 1} (q))$ is smooth and $d h_{q} (Y) = 2 g_{q} (P (r, q), Y)$ , where $P (r, q)$ is the final tangent of the radial geodesic on $[0, 1]$ . The differential of the squared-displacement function and the position vector of the reversed geodesic

[F7]

A future timelike radial inverse vector in a convex normal neighborhood gives a future timelike geodesic. Local causality in a convex normal ball: causal reachability is the exponential cone, the Lorentzian length is bounded by the displacement radius, and null endpoints force the null geodesic

[F8]

Causal then timelike reachability implies timelike reachability. Push-up rule

[F9]

At a point with future unit observer $n$ , every nonzero future causal vector $w$ satisfies $g (n, w) < 0$ . A unit timelike vector $w$ is future-directed if and only if $g (n, w) < 0$ . These are clauses (1) and (3) of Pairing of an observer with future causal vectors.

[F10]

Every tangent vector to a smooth manifold is the velocity of a smooth curve through that point. Applied to the manifold $C$ without boundary, the curve is defined on an open interval about zero. Every tangent vector is the velocity of a smooth curve

[F11]

A real function differentiable at an interior point of its domain has derivative zero there if it has a local extremum there. Fermat's interior extremum theorem: if f has a local extremum at a point c interior to its domain and is differentiable at c, then f'(c) = 0

[F12]

For a smooth real function $f$ , $d f_{x} (V) = V ([f])$ ; the velocity derivation of a smooth curve $η$ acts by $\dot{η} (0) ([f]) = (f \circ η)^{'} (0)$ . Thus $d f_{η (0)} (\dot{η} (0)) = (f \circ η)^{'} (0)$ . The differential of a smooth real-valued function The velocity derivation of a smooth curve

[F13]

A future null affine geodesic is a smooth nonconstant curve whose tangent is nonzero, future-directed and null and satisfies $D_{s} \dot{γ} = 0$ at every interior parameter. Only the geodesic definition is used; no completeness condition is imposed. Future null geodesic completeness and incompleteness

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish the asserted conclusion.

1.1A1F1F2F3F5F6F7F8F9F13givenalgebra

Choose by [F5] a strictly convex normal neighborhood $B \subset E$ of $c$ and $s_{0} \in (0, 1)$ sufficiently close to $1$ that the tail of $γ$ from $r = γ (s_{0})$ to $c$ lies in $B$ . Use the inverse domains selected by $B$ and put $v (q) = \exp_{r}^{- 1} (q)$ and $h (q) = g_{r} (v (q), v (q))$ for $q \in B$ . By the affine geodesic definition in [F13] and uniqueness of the joining geodesic in [F5], its radial segment to $c$ is $t \mapsto γ (s_{0} + (1 - s_{0}) t)$ , so $v (c) = (1 - s_{0}) \dot{γ} (s_{0})$ is nonzero future null and $h (c) = 0$ . Normalize $n_{r} = N_{r} / \sqrt{- g_{r} (N_{r}, N_{r})}$ , so $g_{r} (n_{r}, n_{r}) = - 1$ and $g_{r} (n_{r}, N_{r}) < 0$ ; this is a future unit observer. Clause (1) of [F9] gives $g_{r} (v (c), n_{r}) < 0$ . By continuity, restrict to a neighborhood $V \subset B$ of $c$ where $g_{r} (v (q), n_{r}) < 0$ . If $q \in V \cap C$ had $h (q) < 0$ , then $w = v (q) / \sqrt{- h (q)}$ would be unit timelike with $g_{r} (w, n_{r}) < 0$ . Clause (3) of [F9] makes $w$ , and hence its positive multiple $v (q)$ , future-directed. By [F7] the radial geodesic in $B \subset E$ gives $r ≪_{E} q$ . The initial segment of $γ$ gives $p \leq_{E} r$ , so [F8] gives $p ≪_{E} q$ , contradicting [F3]. Consequently $h ∣_{V \cap C} \geq 0$ and $h (c) = 0$ . The causal suppliers are used with the choice assumptions in [A1].

2.1step 1.1F1F2F4F6F10F11F12algebra∎

Fix $Y \in T_{c} C$ . By [F10], since $C$ is smooth without boundary, there is a smooth curve $η : (- ε, ε) \to C$ with $η (0) = c$ and $\dot{η} (0) = Y$ . Shrink $ε > 0$ so that $η$ lies in $V \cap C$ . The smooth real function $f = h \circ η$ has a local minimum at the interior parameter $0$ by step 1.1, so [F11] gives $f^{'} (0) = 0$ . Regard $Y$ as its ambient tangent under the embedded inclusion; [F12] identifies $f^{'} (0) = d h_{c} (Y)$ . By [F6], $0 = d h_{c} (Y) = 2 g_{c} (P (r, c), Y)$ . The radial segment identified in step 1.1 has terminal tangent $P (r, c) = (1 - s_{0}) \dot{γ} (1)$ . Dividing by $2 (1 - s_{0}) > 0$ proves $g_{c} (\dot{γ} (1), Y) = 0$ for every $Y$ . Only endpoints near $c$ were varied, so no global acausality of $C$ is required; [F4] supplies the existence of a segment under the compact-target hypotheses.

Depends on

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