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Future causal position vectors along a curve in a convex normal set

Statement

Assume Countable Choice and Dependent Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain) for the absolutely continuous causal-curve calculus and the supplied causal results used below.

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), let $C \subseteq M$ be a strictly convex normal set (Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points), let $q \in C$ , let $T_{0} > 0$ , and let $σ : [0, T_{0}] \to C$ be a future causal curve in the class of Causal curves, chronological and causal reachability, and inextendibility with $σ (0) = q$ . Then for every $s \in (0, T_{0}]$ the position vector $\exp_{q}^{- 1} (σ (s)) \in T_{q} M$ is future causal or zero. Moreover, there is $δ \in (0, T_{0}]$ such that the position vectors are nonzero for every $s \in (0, δ]$ .

Facts & Assumptions

Given: A strictly convex normal set $C$ , a point $q \in C$ and a future causal curve $σ : [0, T_{0}] \to C$ with $σ (0) = q$ .

[L1]

Strictly convex normal sets are normal neighborhoods of each of their points and every pair of their points is joined by a unique geodesic contained in $C$ ; moreover $C$ is open (Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points clauses (a)-(c), Normal neighborhoods, convex normal sets, the position vector and the squared-displacement function).

[L2]

For a normal neighborhood $U_{q} = \exp_{q} (W_{q})$ of $q$ and a future causal curve $τ$ in $U_{q}$ with position vectors $v (t) = \exp_{q}^{- 1} (τ (t))$ : if $v (t^{*})$ is future timelike, then $v (t)$ is future timelike for all later $t$ with $τ ([t^{*}, t]) \subseteq U_{q}$ (Length monotonicity of the squared displacement: once a future causal curve in a normal neighborhood has a future timelike position vector, the displacement radius grows at least as fast as the accumulated Lorentzian length clause (a)).

[L3]

On a strictly convex normal set $C$ , the inverse velocity $(q^{'}, r^{'}) \mapsto \exp_{q^{'}}^{- 1} (r^{'})$ is smooth on all of $C \times C$ by The differential of the squared-displacement function and the position vector of the reversed geodesic clause (c) and satisfies $\exp_{q^{'}} (\exp_{q^{'}}^{- 1} (r^{'})) = r^{'}$ .

[L4]

The velocity of the geodesic $t \mapsto \exp_{q} (t w)$ is $d (\exp_{q})_{t w}$ -related to $w$ and the geodesic is future timelike for all parameters if $w$ is future timelike, by parallel transport and the continuity of the timelike cone (The geodesic initial-value problem and the exponential map of the metric-selected connection clause (c), Length monotonicity of the squared displacement: once a future causal curve in a normal neighborhood has a future timelike position vector, the displacement radius grows at least as fast as the accumulated Lorentzian length fact [L5]).

[L5]

The closure of the future timelike cone at a point is contained in the union of the future causal cone with ${0}$ : a limit of future timelike vectors is future causal or zero; and the future causal cone is closed (The timelike set has exactly two convex components, described in orthonormal coordinates, Timelike, null, spacelike and causal vectors of a Lorentz vector space).

Proof

technique · direct

1.1L1L4construct

(A past point and an appended timelike geodesic.) Let $W_{q} \subset T_{q} M$ be the star-shaped domain on which $\exp_{q}$ maps diffeomorphically onto $C$ . Choose a future timelike $w \in W_{q}$ and then choose $ε > 0$ so small that $- ε w \in W_{q}$ ; this follows from openness of $W_{q}$ at $0$ . Star-shapedness implies $t w \in W_{q}$ for all $t \in [- ε, 0]$ . Thus $η (t) : = \exp_{q} (t w)$ lies in $C$ on this interval. Put $q^{'} : = η (- ε)$ ; the segment runs from $q^{'}$ to $q$ with future timelike velocity. Concatenate $η$ with the entire future causal curve $σ : [0, T_{0}] \to C$ to obtain a future causal curve $β : [- ε, T_{0}] \to C$ by Chart independence, concatenation, reparametrization and transitivity for causal curves clause (d).

1.2L1L2L4algebra

(Position vectors with respect to $q^{'}$ .) Since $C$ is convex normal and $q^{'} \in C$ , the set $C$ is a normal neighborhood of $q^{'}$ . At $t^{*} : = - ε / 2$ , the curve $β$ lies on the timelike geodesic $η$ , and $\exp_{q^{'}}^{- 1} (β (t^{*}))$ is a positive multiple of the future timelike initial tangent at $q^{'}$ of the segment from $q^{'}$ to $q$ . By Length monotonicity of the squared displacement: once a future causal curve in a normal neighborhood has a future timelike position vector, the displacement radius grows at least as fast as the accumulated Lorentzian length clause (a), applied to $β$ based at $q^{'}$ , its position vector remains future timelike for every $t \in [t^{*}, T_{0}]$ , since the whole curve segment stays in $C$ . In particular $\exp_{q^{'}}^{- 1} (σ (s))$ is future timelike for every $s \in [0, T_{0}]$ .

2.1step 1.2L3L5algebra

(Limit to $q$ .) Fix $s \in (0, T_{0}]$ . For every sufficiently small $ε > 0$ , step 1.2 gives that $v_{ε} : = \exp_{q^{'} (ε)}^{- 1} (σ (s))$ is future timelike, where $q^{'} (ε) = \exp_{q} (- ε w)$ . The pairs $(q^{'} (ε), σ (s))$ converge to $(q, σ (s))$ in $C \times C$ . By [L3] the inverse vectors converge to $\exp_{q}^{- 1} (σ (s))$ . The future causal cone together with zero is closed, so this limit is future causal or zero. This proves the first claim for every $s \in (0, T_{0}]$ .

3.1step 2.1construct

(Nonvanishing near the initial point.) By Causal cone domination in adapted charts and Lipschitz-graph bounds for causal curves there is an adapted chart around $q$ in which the time coordinate is strictly increasing along every future causal curve contained in the chart. Choose $δ \in (0, T_{0}]$ so that $σ ([0, δ])$ lies in this chart. Then $x^{0} (σ (s)) > x^{0} (q)$ for every $s \in (0, δ]$ , so $σ (s) \neq q$ and $\exp_{q}^{- 1} (σ (s)) \neq 0$ . This proves the second claim without inferring that a return at one time makes the entire preceding curve constant.

4.1step 2.1step 3.1∎

Steps 2.1 and 3.1 prove the claim.

Remarks

Why a strictly convex normal set. The proof uses only: $C$ is open, is a normal neighborhood of $q$ and of points $q^{'} ≪ q$ arbitrarily close to $q$ , and contains the appended timelike segment. The strict form supplied by Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points also supplies the smooth pair inverse used when the base point approaches $q$ .
The gain. The lemma converts causality of the curve into causality of geodesic position vectors throughout the contained curve, which is the input of the global maximality argument in 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.

Future causal position vectors along a curve in a convex normal set

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources