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Compact timelike-length bounds and local temporal functions

Statement

Assume countable choice for the usual Lebesgue-measure convention. Let $B$ be an open subset of a smooth four-manifold without boundary, $g$ a $C^{2}$ time-oriented Lorentz metric, and $N$ a continuous timelike orientation field. Suppose every compact $C \subset B$ admits a finite $L_{C}$ bounding $L_{g} (c) = \int_{a}^{b} \sqrt{- g (\dot{c}, \dot{c})} d t$ for every future Lipschitz curve $c : [a, b] \to C$ with $g (\dot{c}, \dot{c}) \leq - ϵ_{c} < 0$ almost everywhere for some curve-dependent $ϵ_{c} > 0$ . The curves, length and quantifiers have exactly the compact-length convention below. Then every compact $Q \subset B$ has an open neighborhood $O \subset B$ and a smooth real function $τ$ on $O$ whose $g$ -gradient is past timelike. For every compact $K \subset O$ , the same $d τ$ remains past-timelike after raising with any sufficiently small $C^{0}$ perturbation of $g$ on $K$ , oriented by $N$ . Here smallness means uniform coefficient closeness in a finite compact chart cover; no property of a deformed metric away from $K$ is inferred. The length hypothesis must hold for every compact subset of the open domain, not merely one selected family of curves.

Statement references

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Causal exterior, chronological boundaries and Cauchy data

The inverse metric, musical isomorphisms and index raising

Compact sets admit relatively compact open neighborhoods and finite chart pieces

Compact timelike-length bounds restrict to open subdomains

A temporal covector has uniform compact negativity and orientation margins

Inversion is uniformly stable on compact chart families

A compact temporal covector persists under metric perturbation

The Axiom of Countable Choice (AC_ω)

Facts & Assumptions

[A1]

Countable choice is assumed for the standard Lebesgue measure in the compact-length convention used in [F2] and step 1.1. The Axiom of Countable Choice (AC_ω)

[F1]

The Lorentzian-object interface supplies its pointwise conventions: $g_{p}$ is symmetric and nondegenerate of signature $(-, +, +, +)$ , timelike means $g_{p} (v, v) < 0$ , and $N_{p}$ selects the future timelike cone by $g_{p} (v, N_{p}) < 0$ ; the opposite cone is past. The causal and inverse-metric interfaces supply the temporal-gradient and index-raising conventions. Only these pointwise definitions are used for the stated $C^{2}$ metric and continuous orientation field; the suppliers' smooth-field hypotheses and any stationarity requirement are not imposed. Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model Causal exterior, chronological boundaries and Cauchy data The inverse metric, musical isomorphisms and index raising

[F2]

Compact sets admit nested relatively compact neighborhoods, and the exact compact-length hypothesis restricts to open subdomains. Compact sets admit relatively compact open neighborhoods and finite chart pieces Compact timelike-length bounds restrict to open subdomains

[F3]

Compact margins, uniform inverse-metric continuity and temporal-covector persistence hold as locally proved in the three prerequisite helpers. A temporal covector has uniform compact negativity and orientation margins Inversion is uniformly stable on compact chart families A compact temporal covector persists under metric perturbation

[F4]

Radu Lemma B.2 is an imported existence input under exactly the stated $C^{2}$ , time-orientation and all-compact length hypotheses; the External input section gives its scope.

External input

Radu, Lemma B.2 and (B.2), supplies the following external existence input: a $C^{2}$ time-oriented Lorentz metric on an open domain, with a uniform length bound on all uniformly timelike future Lipschitz curves in every compact subset (allowing a curve-dependent positive timelikeness constant), admits a smooth function with past timelike gradient on a neighborhood of each compact subset. This is imported as a source statement. No proof of this existence theorem is claimed locally.

Proof

Given: The hypotheses in the Statement.

Goal: Establish the conclusions in the Statement.

1.1A1F1F2F4

Choose $Q \subset V ⋐ W ⋐ B$ using [F2]. Under [A1], the length hypothesis restricts to $W$ with exactly the same curve and compact-set quantifiers. Apply the precise external existence input [F4] on $W$ to the compact set $Q$ . It supplies an open neighborhood $O$ of $Q$ , restricted to $W$ if necessary, and a smooth $τ$ whose $g$ -gradient is past timelike.

2.1F1F3step 1.1∎

On any compact $K \subset O$ , set $α = d τ$ . Then $g^{- 1} (α, α) < 0$ and $α (N) > 0$ . The three compact-control helpers in [F3] yield a coefficient threshold preserving these inequalities for the perturbed inverse metric and preserving that $N$ is timelike. Raising $d τ$ with that metric gives the asserted past-timelike gradient on $K$ . This proves both conclusions, with existence supplied externally and perturbation stability proved locally.

Depends on

Used by · 2

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

A. F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Appendix B, Lemma B.2, equation (B.2)

Let g be a C² time-oriented Lorentz metric on an open domain B. If every compact C in B has a constant L_C bounding the g-length of every Lipschitz future curve c:[a,b]→C with g(ċ,ċ)≤−ε_c<0 almost everywhere, then every compact Q compactly contained in B has a neighborhood admitting a smooth function with past timelike gradient.

Hypotheses: The bound is uniform over all eligible curves in each compact C, for every compact C⊂B. The positive timelikeness constant ε_c may depend on the curve. The C² metric is time oriented; all curves and compact sets are in that open domain.

Reading limits: Fetched the versioned HTML and read the complete Lemma B.2 statement, (B.2), and Appendix B introductory context. Located and read excerpts of the later Proof of Lemma B.2; its displacement argument was not independently reconstructed or verified. This item imports the exact stated existence result. Minguzzi was not independently consulted for this resolution and is not used as an alternative proof.

Sources

A. F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Appendix B, Lemma B.2, equation (B.2)