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

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.

Right-wedge Rindler clocks, radial rays and frequency ratio

Physical scope

Exact coordinate and proper-time calculations in the right wedge of flat Minkowski spacetime for the stated uniformly accelerated congruence and idealized radial null rays. The clock ratio is conditional on the adopted ideal-clock postulate. The instantaneous received-frequency ratio is conditional on the additional ideal local phase-reception postulate; its finite-apparatus use carries the short-observation and local ray-regime limits, without a numerical accuracy claim. No measurement, material clock model, global accelerated frame, or claim about real clocks or light is asserted.

Statement

Use the flat affine Minkowski model with signature $(-, +, +, +)$ and SI coordinates $(x^{0}, x^{1}, y, z) = (c t, x^{1}, y, z)$ . Fix $a > 0$ in metres per second squared and define

$Φ_{a} (T, ρ, y, z) = (ρ \sinh (a T / c), ρ \cosh (a T / c), y, z), T \in R, ρ > 0.$

Then $Φ_{a}$ is a smooth chart of the right wedge $W = {x^{1} > ∣ x^{0} ∣}$ , with $ρ$ in metres and $T$ in seconds. Its pullback metric, the proper-time rate of a stationary Rindler observer, and that observer's proper-acceleration magnitude are

$d s^{2} = - \frac{a^{2} ρ^{2}}{c^{2}} d T^{2} + d ρ^{2} + d y^{2} + d z^{2}, \frac{d τ}{d T} = \frac{a ρ}{c^{2}}, α = \frac{c^{2}}{ρ} .$

For ideal radial vacuum rays with $y, z$ fixed, $d ρ / d T = \pm a ρ / c$ . A ray joining positive radii $ρ_{E}, ρ_{R}$ therefore has one-way Rindler travel time $Δ T = (c / a) ∣ \log (ρ_{R} / ρ_{E}) ∣$ , and stationary ideal clocks at those radii satisfy $Δ τ_{R} / Δ τ_{E} = ρ_{R} / ρ_{E}$ over any common nonzero $T$ interval. The clock statement is conditional on the ideal-clock adoption in Adopted ideal-clock and ideal-vacuum-light interpretation.

For a future null plane-wave covector $k$ directed along the same radial ray, define $ω_{n} = - k (U_{n})$ using Null wave covectors, frame frequency data and the observer frequency of an ideal plane wave. Under the additional adopted local reception rule Ideal local phase reception by stationary Rindler observers, the instantaneous received angular frequencies at the two events obey $ω_{R} / ω_{E} = ρ_{E} / ρ_{R}$ . The ratio is exact within that ideal model and inherits its local ray-regime and finite-apparatus qualifications, with no numerical error bound or empirical validation. Clause (iv) of Adopted ideal plane-wave and frequency-reading model retains its accelerated-reception exclusion; this theorem uses the separately adopted rule. The coordinate chart excludes $ρ = 0$ ; no global rigid accelerated frame is asserted.

Facts & Assumptions

Given: The flat affine model, $c > 0$ , $a > 0$ , the map $Φ_{a}$ , stationary observers with fixed $ρ > 0, y, z$ , ideal vacuum radial rays, and the conditional ideal-clock and wave-covector readings specified below.

[F1]

Inertial frames and inertial charts of the affine Minkowski model: an inertial frame is an $η$ -orthonormal basis with matrix $diag (- 1, 1, 1, 1)$ , and its chart gives the four inertial coordinates.

[F2]

SI units and coordinate conventions of the special-relativity framework: $x^{0} = c t$ , all inertial coordinates carry metres, $c$ has metres per second, and $η$ has the stated diagonal matrix in inertial coordinates.

[F3]

The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance: the right-wedge map has range $x^{1} > ∣ x^{0} ∣$ , inverse $ρ = \sqrt{(x^{1})^{2} - (x^{0})^{2}}$ , $θ = \frac{1}{2} \log ((x^{1} + x^{0}) / (x^{1} - x^{0}))$ , pullback metric $- ρ^{2} d θ^{2} + d ρ^{2} + d y^{2} + d z^{2}$ , and stationary observer proper acceleration $c^{2} / ρ$ when $θ = c τ / ρ$ .

[F4]

Adopted ideal-clock and ideal-vacuum-light interpretation: an adopted ideal clock accumulates Minkowski proper time, and ideal vacuum light rays are future null straight lines with speed $c$ in inertial coordinates.

[F5]

Norms of four-velocity and four-acceleration, and the observer rest-space decomposition: a future timelike inertial-chart worldline has $∣ v ∣ < c$ , $U = γ (c, v)$ , and four-acceleration given by proper-time differentiation of this four-velocity.

[F6]

Proper acceleration of a massive worldline and its agreement with the acceleration of an observer congruence: proper acceleration is the nonnegative square root $\sqrt{η (A, A)}$ of the four-acceleration norm.

[F7]

Null wave covectors, frame frequency data and the observer frequency of an ideal plane wave: for future null wave covector $k$ and observer four-velocity $U_{n}$ , the angular-frequency functional is $ω_{n} = - k (U_{n})$ .

[F8]

Adopted ideal plane-wave and frequency-reading model: the adopted affine plane-wave phase has constant future null covector and clause (iv) excludes accelerated-observer reception.

[F9]

Addition formulas, identities, parity, and derivatives of the hyperbolic functions: $\cosh^{2} θ - \sinh^{2} θ = 1$ , $\sinh^{'} = \cosh$ , and $\cosh^{'} = \sinh$ .

[F10]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log$ is differentiable on $(0, \infty)$ with derivative $1 / s$ .

[F11]

The six hyperbolic functions and their natural domains: $\sinh θ = (e^{θ} - e^{- θ}) / 2$ and $\cosh θ = (e^{θ} + e^{- θ}) / 2$ .

[A2]

Ideal local phase reception by stationary Rindler observers: an ideal comoving point receiver on the stationary Rindler congruence assigns the instantaneous angular frequency $- k (U_{n})$ at reception. This is an additional adopted local phase-reading model; finite-apparatus use is limited to its local ray regime and short observation intervals, with no claimed numerical accuracy or empirical validation.

Proof

1.1givenF1F2F3F11algebra

Put $θ = a T / c$ . The exponential definitions [F11] give $x^{1} + x^{0} = ρ e^{θ}$ and $x^{1} - x^{0} = ρ e^{- θ}$ , both positive, so the image lies in $W$ . Conversely, [F3] supplies the unique smooth Rindler inverse $ρ = \sqrt{(x^{1})^{2} - (x^{0})^{2}}$ , $θ = \frac{1}{2} \log ((x^{1} + x^{0}) / (x^{1} - x^{0}))$ ; setting $T = c θ / a$ gives the stated inverse, which is smooth because $a, c > 0$ . The range agrees with [F3], and units and coordinates are those of [F1, F2].

2.1step 1.1F1F2F4F9algebra

Differentiating the map with [F9] gives $d x^{0} = \sinh θ d ρ + (a ρ / c) \cosh θ d T$ and $d x^{1} = \cosh θ d ρ + (a ρ / c) \sinh θ d T$ . Substitution into the inertial metric of [F1, F2], using $\cosh^{2} θ - \sinh^{2} θ = 1$ , cancels the mixed terms and yields the displayed pullback metric. For $d ρ = d y = d z = 0$ , the proper-time line element and adopted clock rule [F4] give $d τ / d T = a ρ / c^{2}$ .

3.1step 2.1F1F3F5F6F9algebra

Along a stationary worldline set $χ : = c τ / ρ = a T / c$ . Its inertial coordinates are $(ρ \sinh χ, ρ \cosh χ)$ . Differentiating twice with respect to $τ$ by [F9] gives $U = (c \cosh χ, c \sinh χ)$ and $A = (c^{2} \sinh χ / ρ, c^{2} \cosh χ / ρ)$ . Thus $η (U, U) = - c^{2}$ and $η (A, A) = c^{4} / ρ^{2}$ by [F1]; $U$ is future timelike and has the proper-time normalization of [F5]. The definition [F6] makes the proper-acceleration magnitude $c^{2} / ρ$ , consistently with [F3].

3.2step 2.1F4F10algebra

For a radial null curve, set $d y = d z = 0$ and $d s^{2} = 0$ in the metric of step 2.1. Since $a, ρ, c$ are positive, this gives $d ρ / d T = \pm a ρ / c$ . By [F10], $d (\log ρ) / d T = \pm a / c$ ; integration between its endpoint radii gives $Δ T = (c / a) ∣ \log (ρ_{R} / ρ_{E}) ∣$ , finite for positive finite radii. [F4] identifies these null curves with the ideal vacuum rays.

3.3step 2.1F4algebra

Over a common nonzero Rindler-time interval $Δ T$ , step 2.1 gives $Δ τ_{E} = (a ρ_{E} / c^{2}) Δ T$ and $Δ τ_{R} = (a ρ_{R} / c^{2}) Δ T$ . Their ratio is $ρ_{R} / ρ_{E}$ . If the radii coincide the two increments agree; for distinct positive radii the ratio is positive and finite.

4.1step 2.1step 3.2F7F8F11A2algebra∎

For an outward radial wave take dimensionless phase $ϑ = κ (x^{1} - x^{0}) = κ ρ e^{- a T / c}$ with $κ > 0$ in inverse metres; for an inward wave take $ϑ = - κ (x^{1} + x^{0}) = - κ ρ e^{a T / c}$ , using [F11]. Each is an inertial affine phase with future null covector as in [F8], and its $T$ derivative is constant along its corresponding null ray because $ρ e^{\mp a T / c}$ is constant there by step 3.2. A stationary observer has $U_{n} = (c^{2} / (a ρ)) \partial_{T}$ by step 2.1, so [F7] yields $ω_{n} = - k (U_{n}) = - (c^{2} / (a ρ)) \partial_{T} ϑ$ , proportional to $1 / ρ$ with the same positive proportionality constant at the two events. Hence $ω_{R} / ω_{E} = ρ_{E} / ρ_{R}$ as a contraction identity. Under [A2] these contractions are the instantaneous received angular frequencies; their finite-apparatus interpretation retains the local ray-regime and short-observation qualifications of that postulate. This is separate from [F8] and does not extend its scope.

Right-wedge Rindler clocks, radial rays and frequency ratio

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources