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A complete timelike collar flow is a reference product

Statement

In the complete stationary-collar model, let $V^{\circ}$ be the original reference graph $τ = a (r, y)$ for $0 < r < r_{0}$ . Suppose $K$ is future timelike on this graph for every metric in the family. Then $(s, x) \mapsto ϕ_{s}^{K} (x)$ is a diffeomorphism from $R \times V^{\circ}$ onto $D_{0}$ . It is future timelike along every orbit for every metric in the family. If the original reference section is Cauchy in the original exterior, this product agrees with its unique Cauchy intersection; it does not require the deformed exteriors to be globally hyperbolic.

Statement references

Spacelike sections, maximality and lapse–shift data

Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

Local and global flows generated by a vector field

Killing fields of a Lorentzian metric

Circle-invariant normal coordinates linearize a bifurcate Killing boost

A smooth spacelike reference section is a bounded rapidity graph

The stationary product extends the local boost and circle to a complete collar

Facts & Assumptions

[F1]

The complete boost/circle extension identifies $D_{0}$ with $(0, r_{0}) \times R \times S$ , with $K = \partial_{τ}$ and the reference graph as stated. Circle-invariant normal coordinates linearize a bifurcate Killing boost A smooth spacelike reference section is a bounded rapidity graph The stationary product extends the local boost and circle to a complete collar

[F2]

A spacelike reference section has a future normal. A Cauchy set meets every inextendible causal curve at exactly one parameter value in the original domain. Spacelike sections, maximality and lapse–shift data Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

[F3]

Complete field flows obey the flow law. Local and global flows generated by a vector field

[F4]

The complete-collar extension makes $K$ Killing for each $g_{q}$ . A Killing field is defined by $L_{K} g_{q} = 0$ , where the Lie derivative is the derivative at zero of the metric pulled back by the local flow. The stationary product extends the local boost and circle to a complete collar Killing fields of a Lorentzian metric

[F5]

In the stationary-saturation construction, $t_{0} (r, τ, y)$ is the time at which the formal $T$ orbit meets the reference graph at $x_{0} \in C$ , and $F (r, τ, y) = ϕ_{- t_{0}}^{T} (x_{0})$ . The bounded-root estimate is $∣ τ + t_{0} ∣ \leq A$ . The original stationary-product time $t_{D}$ , normalized by $t_{D} ∣_{C} = 0$ , is its elapsed $T$ -flow time. The stationary product extends the local boost and circle to a complete collar A smooth spacelike reference section is a bounded rapidity graph

Proof

**Given:**The stated data and complete hypotheses.

Goal: Establish the conclusions on the domains and parameter ranges specified.

1.1givenF1F3F4

In the full model the reference-flow map is $(s, r, y) \mapsto (r, a (r, y) + s, y)$ . Its smooth inverse is $(r, τ, y) \mapsto (τ - a (r, y), r, y)$ , so it is a global diffeomorphism onto $D_{0}$ , not just a local transverse flow map. For each $q$ , the flow definition in [F4] and the translation form of the $K$ flow give $0 = (L_{K} g_{q})_{τ τ} = \partial_{τ} (g_{q})_{τ τ}$ . Hence $g_{q} (K, K)$ is constant along each orbit and is negative by its value at the reference point. Continuity in the given time orientation keeps $K$ future-directed along this connected timelike orbit.

2.1F1F2F5step 1.1∎

In the original stationary product, [F5] gives $t_{D} (F (r, τ, y)) = - t_{0} (r, τ, y)$ , since $x_{0} \in C$ has time zero and $F$ is obtained by flowing it for time $- t_{0}$ . Thus the bounded-root estimate gives $∣ t_{D} - τ ∣ = ∣ t_{0} + τ ∣ \leq A$ . Along a $K$ orbit $τ$ tends to both infinities, so $t_{D}$ does too. As $t_{D}$ is a continuous real coordinate on the entire original exterior, the orbit can have no endpoint there in either direction and is therefore inextendible there. By step 1.1 it is timelike for the original metric, so the original Cauchy condition [F2] makes its displayed reference intersection the unique one. This comparison uses only the original metric; no analogous causal-completeness property of a deformed metric has been assumed.

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