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Compactness gives an earliest translated-cylinder causal contact

Statement

Work in ZFC. Let $(E, g, N)$ be a time-oriented smooth globally hyperbolic Lorentzian four-dimensional exterior, and let $τ : E \to R$ be temporal: its metric gradient is past-directed timelike everywhere. Let $A \subset τ^{- 1} (0)$ be a compact spacelike source. For this lemma, 'spacelike source' is used in the set sense of acausality: no two distinct points of $A$ can be joined by a future causal curve in either direction. No submanifold structure or fixed dimension is required of $A$ . This condition follows already from the displayed temporal-level containment. All causal and chronological relations below are computed in $(E, g, N)$ .

Let $ϕ^{T}$ be a local flow of a stationary Killing field $T$ , defined at every $(t, x) \in R \times C_{0}$ for a compact spacelike cut $C_{0} \subset E$ in the sense specified below. Require that the parameter map $Φ : R \times C_{0} \to E$ , $Φ (t, x) = ϕ_{t}^{T} (x)$ , is continuous in the product topology. Put $C_{t} = Φ ({t} \times C_{0})$ and $Γ = ⋃_{t \in R} C_{t}$ . Here the stationary cylinder hypothesis means that $Γ$ is a smooth embedded three-dimensional submanifold swept out by the $T$ -flow, and timelike means that each $T_{p} Γ$ contains a vector $v$ with $g_{p} (v, v) < 0$ . Its compact spacelike cuts mean that every $C_{t}$ is a compact smooth embedded two-dimensional submanifold of $E$ and $g_{p} (v, v) > 0$ for every $p \in C_{t}$ and nonzero $v \in T_{p} C_{t}$ . Assume these cylinder and cut hypotheses, that $T$ is future timelike on $Γ$ , that $A \cap Γ = \emptyset$ , and that $τ (Φ (t, x)) = t + τ (x)$ for all $(t, x) \in R \times C_{0}$ .

Assume finite chronological access to this specified cylinder: there is a real $u$ with $C_{u} \cap I_{E}^{+} (A) \neq \emptyset$ . Then the set ${t \in R : C_{t} \cap J_{E}^{+} (A) \neq \emptyset}$ has a finite minimum $t_{R}$ , and there is $p_{R} \in C_{t_{R}} \cap J_{E}^{+} (A)$ . Every causal contact point $p \in Γ \cap J_{E}^{+} (A)$ has positive temporal-function value $τ (p) > 0$ , in particular $τ (p_{R}) > 0$ . This positivity concerns $τ (p)$ , rather than the flow parameter $t$ indexing $C_{t}$ ; no sign for $t_{R}$ is asserted. Compactness is used for both the source and a bounded-time portion of the cylinder; compactness of the whole unbounded cylinder is not assumed.

Facts & Assumptions

[F1]

In a globally hyperbolic time-oriented smooth Lorentzian four-manifold, the causal future of a compact set is closed, under the supplier's countable and dependent choice assumptions. We use clause (a), with compact source $A$ and all curves in $E$ . In a globally hyperbolic manifold the causal relation is closed and truncated causal cones over compact sets are compact

[F2]

A temporal function is a time function, hence continuous, and strictly increases between the endpoints of every nonconstant future causal curve. We use clauses (b) and (c), with the temporal function $τ$ on the same exterior. Temporal functions are time functions, and time functions forbid closed causal curves

[F3]

Chronological futures are contained in causal futures, by clause (f). The finite time $u$ and the chronological contact on $C_{u}$ are explicit hypotheses of this lemma. Chart independence, concatenation, reparametrization and transitivity for causal curves

[F4]

A continuous image of a compact space, or of a compact subset of its domain, is compact. A continuous real-valued function on a nonempty compact topological space attains a finite maximum and minimum. We use claims 1 and 2, respectively. A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism

[F5]

Products below carry the product topology, generated by inverse images of open sets under coordinate projections. Every closed bounded real interval is compact, and a finite product of compact spaces is compact in its product topology. The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space Heine-Borel by bisection: every closed bounded interval [a,b] is compact A product of finitely many compact spaces is compact in the product topology

[F6]

A continuous map has closed preimages of closed sets, by clause (c), and a closed subspace of a compact space is compact, by claim 1. For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and f(Ā) ⊆ f(A) A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact

[F7]

Coordinate projections are continuous, by claim 1. In particular the time coordinate on a product with a real interval is continuous as a real-valued map: an open set $V \subset R$ has preimage $([ℓ, u] \cap V) \times C_{0}$ , which is open in the product. A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice

[F8]

A local flow has an open domain, a smooth parameter map and the integral-curve and flow laws of the flow definition. Its domain containing $R \times C_{0}$ and the continuity of $Φ$ on that product are explicitly supplied here, so no completeness of $T$ on the whole exterior is inferred. The exterior, reachability and temporal-function conventions are those of the causal-data definition. The cylinder and cut conditions are the explicit tangent-space and embedded-submanifold predicates in the Statement; embedded submanifolds are understood through slice charts. Local and global flows generated by a vector field Causal exterior, chronological boundaries and Cauchy data Embedded submanifolds and slice charts

[A1]

Work in ZFC, including the countable and dependent choice assumptions carried by the causal and temporal-function suppliers in [F1]–[F3]. The finite extrema and finite-product arguments introduce no additional choice requirement. The Axiom of Choice

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish the finite earliest contact and positivity of the temporal-function value at every contact point.

1.1givenA1F2F3F4choosealgebra

If two distinct source points were causally related, [F2] would strictly order their equal temporal values, which is impossible. Thus the stated set-level spacelike condition holds. Fix the access time $u$ . By [F3], $C_{u} \cap J_{E}^{+} (A)$ is nonempty, so both $A$ and $C_{0}$ are nonempty. By [F2] and [F4], the continuous function $τ ∣_{C_{0}}$ has a finite maximum $m$ . If $z = Φ (t, x) \in J_{E}^{+} (A)$ , choose $a \in A$ causally preceding $z$ . Disjointness gives $a \neq z$ , so their causal relation is represented by a nonconstant future causal curve entirely in $E$ . By [F2], $τ (z) > τ (a) = 0$ . Hence $t + τ (x) > 0$ , and $t > - τ (x) \geq - m$ . Put $ℓ = - m$ . Thus every contact time is greater than $ℓ$ , and in particular $ℓ < u$ .

2.1givenA1F1F4F5F6F8step 1.1construct

By [F5], $P = [ℓ, u] \times C_{0}$ is compact. The restriction $Φ_{P} : P \to E$ is continuous by the supplied continuity of $Φ$ in [F8], so its image is a compact bounded-time portion of the cylinder by [F4]. By [F1], $J_{E}^{+} (A)$ is closed in $E$ ; therefore $D = Φ_{P}^{- 1} (J_{E}^{+} (A))$ is closed in $P$ by [F6], and is compact by the same fact. The contact at time $u$ from step 1.1 gives a point of $D$ , so $D$ is nonempty.

3.1F4F7step 1.1step 2.1chooseconstructalgebra∎

Let $π : P \to R$ be $π (t, x) = t$ , continuous by [F7]. By [F4], $π (D)$ is compact and nonempty, and its identity real-valued function attains a minimum $t_{R}$ . By the definition of this image there is $x_{R} \in C_{0}$ with $(t_{R}, x_{R}) \in D$ ; set $p_{R} = Φ (t_{R}, x_{R})$ . Then $p_{R} \in C_{t_{R}} \cap J_{E}^{+} (A)$ , and step 1.1 gives $τ (p_{R}) > 0$ . Any contact time at most $u$ is greater than $ℓ$ by step 1.1, hence lies in $π (D)$ and is at least $t_{R}$ ; a contact time greater than $u$ is also at least $t_{R}$ , since $t_{R} \leq u$ . Thus $t_{R}$ is the minimum over all real contact times.

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