CorollaryStatement: AI-adaptedProof: AI-generatedprecheck pass

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every congruence admits a local adapted chart

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $U \subseteq M$ be a nonempty open subset, let $n$ be an observer congruence on $U$ , and let $p \in U$ (Observer congruences, their observer curves and adapted charts). Then there is a chart $(u^{0}, u^{1}, u^{2}, u^{3})$ of $M$ with domain $U^{'} \subseteq U$ containing $p$ such that

$n = \frac{\partial}{\partial u^{0}} on U^{'},$

and this chart is $n$ -adapted: its spatial coordinates $u^{1}, u^{2}, u^{3}$ are constant along every integral curve of $n$ contained in $U^{'}$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , an observer congruence $n$ on an open set $U$ , and a point $p \in U$ .

[L1]

An observer congruence on the nonempty open set $U$ is a smooth vector field $n$ on $U$ with $g_{q} (n_{q}, n_{q}) = - 1$ and $g_{q} (n_{q}, N_{q}) < 0$ for every $q \in U$ ; its maximal integral curves are its observer curves. A chart with nonempty open domain $U^{'} \subseteq U$ is $n$ -adapted when its spatial coordinates are constant along every integral curve of $n ∣_{U^{'}}$ whose parameter domain is an interval; no agreement of spatial labels across exit and re-entry is required (Observer congruences, their observer curves and adapted charts, Definition (a)-(c)).

[L2]

A smooth vector field $X$ on a manifold with $X_{p} \neq 0$ at a point $p$ admits local coordinates $v^{1}, \dots, v^{4}$ near $p$ in which $X = \partial / \partial v^{1}$ (The flow-box theorem).

[L3]

The value $g_{q} (n_{q}, n_{q})$ is a smooth function of $q$ on $U$ , and $g_{q} (n_{q}, n_{q}) = - 1$ implies $n_{q} \neq 0$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L4]

A smooth curve $γ : J \to U^{'}$ on a real interval $J$ is an integral curve of a smooth vector field $X$ on $U^{'}$ when $γ^{'} (s) = X_{γ (s)}$ for every $s \in J$ (Integral curves of a vector field).

Proof

technique · direct

1.1L1L2L3given

The field $n$ is smooth on the open set $U$ and nowhere vanishing: by [L1] it satisfies $g_{q} (n_{q}, n_{q}) = - 1$ at every $q \in U$ , so $n_{q}$ is a nonzero vector by [L3]. Applying [L2] to $n$ at the point $p$ gives a chart $(v^{1}, v^{2}, v^{3}, v^{4})$ with domain $U^{'} \subseteq U$ containing $p$ and $n = \partial / \partial v^{1}$ on $U^{'}$ .

2.1step 1.1L1L4algebra

Define $u^{0} : = v^{1}$ , $u^{1} : = v^{2}$ , $u^{2} : = v^{3}$ , $u^{3} : = v^{4}$ . This is again a chart with domain $U^{'}$ , and $n = \partial / \partial u^{0}$ on $U^{'}$ . For any integral curve $γ : J \to U^{'}$ of $n ∣_{U^{'}}$ as in [L4], its coordinate components satisfy $(u^{0} \circ γ)^{'} = 1$ and $(u^{i} \circ γ)^{'} = 0$ for $i = 1, 2, 3$ , since $\partial / \partial u^{0}$ has components $(1, 0, 0, 0)$ in this chart. Thus each $u^{i} \circ γ$ is constant on the interval $J$ , so by [L1] the chart is $n$ -adapted.

3.1step 2.1∎

The chart produced in step 2.1 has all the stated properties, so a local adapted chart exists near every point of $U$ .

Every congruence admits a local adapted chart

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources