TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Proper-time parameter and normalized tangent

Statement

Let $x : I \to M$ be a $C^{2}$ future timelike curve on a nonempty open interval $I$ , with parameter $s$ and tangent $T = d x / d s$ . Fix $c > 0$ , $s_{0} \in I$ , and $τ_{0} \in R$ , and define $ℓ (s) = \sqrt{- g (T, T)}$ and $τ (s) = τ_{0} + c^{- 1} \int_{s_{0}}^{s} ℓ (r) d r$ . Then $τ^{'} (s) = ℓ (s) / c > 0$ , so $τ$ is an increasing $C^{2}$ coordinate on its open image $J = τ (I)$ . The tangent with respect to this parameter is $U = d x / d τ = c T / ℓ$ and satisfies $g (U, U) = - c^{2}$ . This is a mathematical reparametrization result; by itself it does not identify $τ$ with a physical clock reading.

Facts & Assumptions

Given: A $C^{2}$ future timelike curve and the definitions in the Statement.

[F1]

For a future timelike worldline, the proper-time density is $h = \sqrt{- g (T, T)} > 0$ and its integral defines proper time (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold).

[F2]

The fundamental theorem gives the derivative of an integral of a continuous density (The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive); a $C^{1}$ map with nonzero derivative has a local $C^{1}$ inverse (The Euclidean inverse function theorem).

[F3]

The chain rule gives $(x \circ τ^{- 1})^{'} = T / (τ^{'})$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F4]

The mean value theorem applies on every compact subinterval of $I$ (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

Proof

1.1givenF1F2F4algebra

Since $T$ is future timelike and continuous, $- g (T, T)$ is a continuous positive function, so $ℓ$ is continuous and strictly positive. By [F1] and [F2], $τ$ is differentiable with $τ^{'} = ℓ / c > 0$ . By [F4], the mean value theorem on each compact subinterval makes $τ$ strictly increasing.

2.1step 1.1F2F3algebra∎

At each point the nonzero derivative gives a local inverse; injectivity makes those local inverses agree, so they define a $C^{1}$ inverse on the open image $J$ . Because $T$ is $C^{1}$ and $g$ is smooth, $ℓ$ is $C^{1}$ and $τ$ is $C^{2}$ . Its inverse is first $C^{1}$ by [F2], and $(τ^{- 1})^{'} = 1 / (τ^{'} \circ τ^{- 1})$ is $C^{1}$ , so $τ^{- 1}$ is $C^{2}$ . By [F3], $U = T / τ^{'} = c T / ℓ$ , and hence $g (U, U) = c^{2} g (T, T) / ℓ^{2} = - c^{2}$ .

Proper-time parameter and normalized tangent

Statement

Facts & Assumptions

Proof

Depends on

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