ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

Worked time dilation at speed 3c/5

Example

In an inertial chart $x$ with metric $diag (- 1, 1, 1, 1)$ let $v = 3 c / 5$ , $γ = (1 - v^{2} / c^{2})^{- 1 / 2} = 5 / 4$ , and consider the smooth worldline $y : R \to A$ whose $x$ -coordinates are $y (t) = (c t, \frac{3 c}{5} t, 0, 0)$ .

(a) $y^{'} (t) = (c, \frac{3 c}{5}, 0, 0)$ and $η (y^{'}, y^{'}) = - c^{2} + \frac{9 c^{2}}{25} = - \frac{16}{25} c^{2}$ , so $y$ is future timelike and its proper-time rate is $τ^{'} (t) = \frac{1}{c} \sqrt{\frac{16}{25} c^{2}} = \frac{4}{5}$ : between coordinate times $t_{0} < t_{1}$ the model elapsed proper time is $\frac{4}{5} (t_{1} - t_{0}) = (t_{1} - t_{0}) / γ$ , with rate factor $4 / 5 = 1 / γ$ (Proper time along a future timelike worldline, Inertial frames and inertial charts of the affine Minkowski model).

(b) The four-velocity is $U = y^{'} / τ^{'} = (\frac{5 c}{4}, \frac{3 c}{4}, 0, 0)$ with $η (U, U) = - \frac{25 c^{2}}{16} + \frac{9 c^{2}}{16} = - c^{2}$ , and the dimensionless velocity is $u = U / c = \frac{5}{4} (1, \frac{3}{5}, 0, 0) = γ (e_{0} + \frac{3}{5} e_{1})$ for the chart's first coordinate basis vector $e_{0}$ : this is exactly the boost relation of clause (i) of Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation (Four-velocity and four-acceleration of a future timelike worldline, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition).

(c) Described in the boosted chart $x^{'}$ in which the clock is at rest (boost speed $3 c / 5$ along the first axis, The passive Lorentz boost with parameter beta), its $x^{'}$ -time rate equals its proper-time rate, and the inverse statement $Δ t = γ Δ τ$ holds with $γ = 5 / 4$ .

Facts & Assumptions

Given: An inertial chart $x$ with metric $diag (- 1, 1, 1, 1)$ , the numbers $v = 3 c / 5$ , $γ = 5 / 4$ , and the smooth worldline $y : R \to A$ whose $x$ -coordinates are $y (t) = (c t, \frac{3 c}{5} t, 0, 0)$ (Inertial frames and inertial charts of the affine Minkowski model, Proper time along a future timelike worldline).

[L1]

For a $C^{1}$ future timelike worldline the proper-time function satisfies $τ (b) - τ (a) = c^{- 1} \int_{a}^{b} \sqrt{- η (y^{'}, y^{'})} d s$ ; in an inertial chart $η (u, u) = - (u^{0})^{2} + (u^{1})^{2} + (u^{2})^{2} + (u^{3})^{2}$ (Proper time along a future timelike worldline, Inertial frames and inertial charts of the affine Minkowski model).

[L2]

The four-velocity is $U = x^{'} / τ^{'}$ , its norm is $η (U, U) = - c^{2}$ , and the dimensionless velocity $u = U / c$ is future unit (Four-velocity and four-acceleration of a future timelike worldline, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition).

[L3]

Clause (i) of Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation: the worldline $x^{- 1} (λ, \frac{v}{c} λ + a_{1}, a_{2}, a_{3})$ has proper-time rate $1 / γ$ with respect to the coordinate time of $x$ and its tangent is future timelike; in the chart $x^{'}$ of the boost with parameter $v$ the clock is at rest.

Verification

technique · direct

1.1L1L3algebra

The coordinate derivative of $y$ is $y^{'} (t) = (c, \frac{3 c}{5}, 0, 0)$ , and by [L1] $η (y^{'}, y^{'}) = - c^{2} + \frac{9}{25} c^{2} = - \frac{16}{25} c^{2} < 0$ with $y^{' 0} = c > 0$ , so $y$ is future timelike; the proper-time rate is $τ^{'} (t) = \frac{1}{c} \sqrt{\frac{16}{25} c^{2}} = \frac{4}{5}$ . Hence for $t_{0} < t_{1}$ the proper time elapsed is $\frac{4}{5} (t_{1} - t_{0}) = \frac{t_{1} - t_{0}}{5 / 4}$ , i.e. the model proper-time rate is $4 / 5 = 1 / γ$ ; equivalently $Δ t = γ Δ τ = \frac{5}{4} Δ τ$ .

1.2L2L3algebra

The four-velocity is $U = y^{'} / τ^{'} = \frac{5}{4} (c, \frac{3 c}{5}, 0, 0) = (\frac{5 c}{4}, \frac{3 c}{4}, 0, 0)$ , whose norm is $η (U, U) = - \frac{25}{16} c^{2} + \frac{9}{16} c^{2} = - c^{2}$ by [L1], in agreement with [L2]; the dimensionless velocity is $u = U / c = \frac{5}{4} (1, \frac{3}{5}, 0, 0) = γ (e_{0} + \frac{3}{5} e_{1})$ , which is the unit tangent vector of clause (i) of [L3] with $v / c = 3 / 5$ and $γ = 5 / 4$ .

2.1L3step 1.1algebra

In the chart $x^{'}$ of the boost with parameter $v = 3 c / 5$ the spatial coordinates of $y$ are constant, so the clock is at rest in $x^{'}$ ; a clock at rest in its own chart has $τ^{'} = 1$ with respect to that chart's coordinate time, so its $x^{'}$ -time rate equals its proper-time rate, and the relation $Δ t = γ Δ τ$ of step 1.1 is the inverse statement with $γ = 5 / 4$ .

3.1step 1.1step 1.2step 2.1∎

Conclusion: steps 1.1-2.1 verify the three clauses (a), (b), (c) of the Example: the worldline is future timelike with proper-time rate $4 / 5 = (1 / γ)$ , its four-velocity is $(\frac{5 c}{4}, \frac{3 c}{4}, 0, 0)$ of norm $- c^{2}$ , the dimensionless velocity is the inverse passive-boost image $γ (e_{0} + \frac{3}{5} e_{1})$ , and in the rest chart of the clock the time rate is the proper-time rate, so the inverse dilation $Δ t = γ Δ τ$ holds with $γ = 5 / 4$ .

Worked time dilation at speed 3c/5

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources