LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

How statement and proof provenance work

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.

Length-integral invariance under monotone C^1 maps and proper-time invariance under regular increasing maps

Statement

Let $c > 0$ , let $x : I \to A$ be a $C^{1}$ future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space), let $a < b$ with $[a, b] \subseteq I$ , and let $c_{0} < d_{0}$ . Let $φ : [c_{0}, d_{0}] \to [a, b]$ be a strictly increasing $C^{1}$ surjection, with one-sided endpoint derivatives as in Curves, regular causal curves and future timelike worldlines in affine Minkowski space.

Then $φ (c_{0}) = a$ , $φ (d_{0}) = b$ and $φ^{'} \geq 0$ . Put $y = x \circ φ$ and $f (s) = \sqrt{- η (x^{'} (s), x^{'} (s))}$ . The $C^{1}$ curve $y$ has tangent

$y^{'} (u) = φ^{'} (u) x^{'} (φ (u)),$

which is future timelike exactly where $φ^{'} (u) > 0$ and is the zero vector where $φ^{'} (u) = 0$ . Its nonnegative length integrand

$\sqrt{- η (y^{'} (u), y^{'} (u))} = φ^{'} (u) f (φ (u))$

is defined even when $y$ is not a worldline, and the broad substitution identity

$\int_{c_{0}}^{d_{0}} \sqrt{- η (y^{'} (u), y^{'} (u))} d u = \int_{a}^{b} f (s) d s = c (τ_{x} (b) - τ_{x} (a))$

holds. If $φ^{'} (u) > 0$ at every point, including the one-sided endpoint derivatives, then $y$ is a $C^{1}$ future timelike worldline, its proper-time function $τ_{y}$ of Proper time along a future timelike worldline is defined, and $τ_{y} (d_{0}) - τ_{y} (c_{0}) = τ_{x} (b) - τ_{x} (a)$ ; more generally $τ_{y} (u) - τ_{y} (v) = τ_{x} (φ (u)) - τ_{x} (φ (v))$ for all $v, u \in [c_{0}, d_{0}]$ , so $τ_{y}$ and $τ_{x} \circ φ$ differ by a base-point constant. No $τ_{y}$ is asserted for a zero-tangent curve outside that definition.

For a strictly decreasing $C^{1}$ surjection $φ$ onto $[a, b]$ one has $φ (c_{0}) = b$ , $φ (d_{0}) = a$ and $φ^{'} \leq 0$ ; the tangent of $y$ is past timelike where $φ^{'} < 0$ and zero where $φ^{'} = 0$ , so $y$ is not a future timelike worldline, while the nonnegative length integral still equals $\int_{a}^{b} f$ : oriented substitution gives $\int_{c_{0}}^{d_{0}} φ^{'} f (φ) = \int_{b}^{a} f$ , and $∣ φ^{'} ∣ = - φ^{'}$ reverses that sign.

These are mathematical curve-integral identities; physical clock reading is a separate premise. No inverse of $φ$ and no nonvanishing derivative are needed for the broad integral identities.

Facts & Assumptions

Given: A real $c > 0$ ; a $C^{1}$ future timelike worldline $x : I \to A$ ; reals $a < b$ with $[a, b] \subseteq I$ ; reals $c_{0} < d_{0}$ ; a strictly increasing (respectively strictly decreasing) $C^{1}$ surjection $φ : [c_{0}, d_{0}] \to [a, b]$ with one-sided endpoint derivatives; the composite $y = x \circ φ$ ; and $f (s) = \sqrt{- η (x^{'} (s), x^{'} (s))}$ .

[L1]

Curves and reparametrizations (Curves, regular causal curves and future timelike worldlines in affine Minkowski space, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): a strictly increasing surjection $[c_{0}, d_{0}] \to [a, b]$ maps endpoints to endpoints, so $φ (c_{0}) = a$ and $φ (d_{0}) = b$ (respectively $φ (c_{0}) = b$ , $φ (d_{0}) = a$ in the decreasing case); a $C^{1}$ increasing function has $φ^{'} \geq 0$ , a $C^{1}$ decreasing one has $φ^{'} \leq 0$ ; and the tangent of a curve is the limit of difference quotients over the relevant side.

[L2]

Chart calculus (Inertial frames and inertial charts of the affine Minkowski model, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral): in an inertial chart with coordinate path $y_{A} = x_{A} \circ x$ the frame isomorphism $E$ is linear, an $A$ -valued curve is differentiable exactly when each coordinate component is, with componentwise derivatives, and its tangent is $E$ applied to the coordinate derivative; a map is $C^{1}$ exactly when all components are $C^{1}$ .

[L3]

Chain rule (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ): for real functions, $(g \circ φ)^{'} (u) = g^{'} (φ (u)) φ^{'} (u)$ whenever $g$ is differentiable at $φ (u)$ and $φ$ at $u$ , with the one-sided reading at endpoints.

[L4]

Bilinearity (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space): $η (λ v, λ v) = λ^{2} η (v, v)$ for real $λ$ .

[L5]

Substitution (Substitution: if $φ$ is differentiable on $[c, d]$ with $φ^{'}$ integrable and $f$ is continuous on an interval containing $φ ([c, d])$ , then $\int_{φ (c)}^{φ (d)} f = \int_{c}^{d} (f \circ φ) φ^{'}$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Proper time along a future timelike worldline): if $φ : [c_{0}, d_{0}] \to [a, b]$ is differentiable with $φ^{'}$ integrable and $g$ is continuous on an order-convex set containing $φ [[c_{0}, d_{0}]]$ , then $\int_{φ (c_{0})}^{φ (d_{0})} g = \int_{c_{0}}^{d_{0}} (g \circ φ) φ^{'}$ ; the derivative of a $C^{1}$ function on a compact interval is continuous and hence integrable; and $f$ is continuous on $I$ with $τ_{x}$ as in Proper time along a future timelike worldline.

[L6]

Proper time (Proper time along a future timelike worldline): for $u < v$ in a worldline's domain, $τ (v) - τ (u) = c^{- 1} \int_{u}^{v} \sqrt{- η (x^{'}, x^{'})}$ through the oriented integral; $τ_{y}$ is defined only when $y$ is a $C^{1}$ future timelike worldline.

[L7]

The future and past cones (The timelike set has exactly two convex components, described in orthonormal coordinates, Time orientation, future and past timelike/null/causal vectors, and the closed cones, Timelike, null, spacelike and causal vectors of a Lorentz vector space): in a future $η$ -orthonormal basis the future timelike set is $C^{+} = {v : v^{0} > ∣ v_{s p} ∣}$ and the past timelike set is $C^{-} = {v : v^{0} < - ∣ v_{s p} ∣}$ ; both are closed under multiplication by positive, respectively negative, scalars: $λ v \in C^{+}$ for $λ > 0$ and $v \in C^{+}$ , and $λ v \in C^{-}$ for $λ < 0$ and $v \in C^{+}$ .

[L8]

Continuity of composites (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs, Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function): components of continuous vector-valued maps are continuous, composites of continuous real functions are continuous, and products and sums of continuous real functions are continuous.

Proof

technique · direct

1.1givenL1

A strictly increasing surjection carries the least point to the least point and the greatest to the greatest point of the target interval, so $φ (c_{0}) = a$ , $φ (d_{0}) = b$ and (in the decreasing case) $φ (c_{0}) = b$ , $φ (d_{0}) = a$ ; the derivative of a $C^{1}$ monotone function has the sign of the monotonicity, $φ^{'} \geq 0$ or $φ^{'} \leq 0$ , with the one-sided reading at the endpoints.

2.1step 1.1L2L3L8

In an inertial chart, the coordinate path of $y$ is $y_{A} \circ φ$ ; each component $(y_{A})^{j} \circ φ$ is differentiable with derivative $((y_{A})^{j^{'}} \circ φ) φ^{'}$ by the chain rule [L3], and the derivatives are continuous by [L8] because $(y_{A})^{j^{'}}$ and $φ$ are continuous; hence $y$ is $C^{1}$ with $y^{'} (u) = φ^{'} (u) x^{'} (φ (u))$ by the componentwise characterisation and linearity of $E$ [L2].

3.1step 2.1L4

Bilinearity [L4] gives $- η (y^{'} (u), y^{'} (u)) = φ^{'} (u)^{2} (- η (x^{'} (φ (u)), x^{'} (φ (u)))) = φ^{'} (u)^{2} f (φ (u))^{2}$ , so the nonnegative length integrand is $\sqrt{- η (y^{'}, y^{'})} = ∣ φ^{'} (u) ∣ f (φ (u))$ , which equals $φ^{'} (u) f (φ (u))$ in the increasing case and $- φ^{'} (u) f (φ (u))$ in the decreasing case.

3.2step 2.1L7

At every $u$ , $x^{'} (φ (u))$ is future timelike because $φ (u) \in [a, b] \subseteq I$ ; by the cone description [L7], $y^{'} (u) = φ^{'} (u) x^{'} (φ (u))$ is future timelike where $φ^{'} (u) > 0$ , is the zero vector where $φ^{'} (u) = 0$ , and is past timelike where $φ^{'} (u) < 0$ .

4.1step 1.1step 3.1L5L6

Since $φ^{'}$ is continuous on $[c_{0}, d_{0}]$ , hence integrable, and $f$ is continuous on the order-convex $I$ containing $φ [[c_{0}, d_{0}]]$ , substitution [L5] applied to the increasing case gives $\int_{c_{0}}^{d_{0}} φ^{'} f (φ) = \int_{φ (c_{0})}^{φ (d_{0})} f = \int_{a}^{b} f = c (τ_{x} (b) - τ_{x} (a))$ by [L6]; combined with step 3.1 this proves the broad identity.

4.2step 3.1step 3.2L6

Where $φ^{'} = 0$ the tangent $y^{'}$ is zero, so $y$ is not a worldline of Curves, regular causal curves and future timelike worldlines in affine Minkowski space and no proper-time function is defined for it by [L6]; the length integrand of step 3.1 remains defined, which is exactly the content of the broad identity.

4.3step 3.1step 3.2L5L6

In the strictly decreasing case, step 3.2 makes every tangent of $y$ past timelike or zero, so $y$ is not a future timelike worldline; and substitution [L5] with oriented endpoints gives $\int_{c_{0}}^{d_{0}} φ^{'} f (φ) = \int_{φ (c_{0})}^{φ (d_{0})} f = \int_{b}^{a} f = - \int_{a}^{b} f$ , which combined with $\sqrt{- η (y^{'}, y^{'})} = - φ^{'} f (φ)$ from step 3.1 yields $\int_{c_{0}}^{d_{0}} \sqrt{- η (y^{'}, y^{'})} = \int_{a}^{b} f = c (τ_{x} (b) - τ_{x} (a))$ .

5.1step 4.1step 3.2L6

If $φ^{'} > 0$ at every point including the one-sided endpoint derivatives, then by step 3.2 every tangent of $y$ is future timelike, so $y$ is a $C^{1}$ future timelike worldline and $τ_{y}$ is defined; its endpoint increment is $τ_{y} (d_{0}) - τ_{y} (c_{0}) = c^{- 1} \int_{c_{0}}^{d_{0}} \sqrt{- η (y^{'}, y^{'})} = τ_{x} (b) - τ_{x} (a)$ by step 4.1.

6.1step 4.1step 5.1L5L6

For $v < u$ in $[c_{0}, d_{0}]$ , substitution [L5] on $[v, u]$ gives $τ_{y} (u) - τ_{y} (v) = c^{- 1} \int_{φ (v)}^{φ (u)} f = τ_{x} (φ (u)) - τ_{x} (φ (v))$ by [L6], so the based functions $τ_{y}$ and $τ_{x} \circ φ$ differ by the base-point constant.

7.1step 4.1step 4.3∎

No step forms an inverse of $φ$ , divides by $φ^{'}$ , or assumes a derivative-limit hypothesis at the endpoints; and no object is selected from a family, so no choice principle is used.

Remarks

The broad identity is the substitution theorem, not a proper-time theorem. The left-hand integrand is the nonnegative square root $∣ φ^{'} ∣ f \circ φ$ , which exists for every $C^{1}$ $φ$ ; only when $φ^{'}$ is everywhere positive is this integrand the length integrand of an actual worldline $y$ , and only then is $τ_{y}$ defined. The zero-tangent case is deliberately separated from the worldline case.
Everything is a one-dimensional substitution. The curve enters only through the continuity of $f$ and the chain rule for its coordinate components; no derivative-limit hypothesis, no improper integral and no choice principle is used.

Length-integral invariance under monotone C^1 maps and proper-time invariance under regular increasing maps

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources