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

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Static-clock redshift, proper acceleration, and a controlled weak-field expansion in Schwarzschild

Physical scope

Conditional model prediction for ideal static clocks, observer energy/frequency ratios in geometric optics, and the proper acceleration required to remain static in the Schwarzschild exterior. The weak-field statement is a uniform mathematical remainder bound. No clock, spectral, acceleration, or gravitational-field measurement is asserted.

Statement

Adopt the Schwarzschild exterior vacuum model datum $(M, g, N)$ of The Schwarzschild exterior vacuum model: field law, static clock readings and the mass parameter with $m_{*} > 0$ , $s = 2 G_{N} m_{*} / c^{2} > 0$ and $f (r) = 1 - s / r$ . Let $r_{E}, r_{R} > s$ be the emitter and receiver radii of static observers in this model. Let $Δ t \neq 0$ be a common Schwarzschild coordinate-time duration over which ideal clocks on these static worldlines accrue proper-time increments $Δ τ_{E}$ and $Δ τ_{R}$ . For the frequency ratio, let $η : J \to M$ be a specified smooth, affinely parametrized future-directed null geodesic in the exterior and choose $λ_{E} < λ_{R}$ in $J$ so that $η (λ_{E})$ and $η (λ_{R})$ lie on the static worldlines at radii $r_{E}$ and $r_{R}$ , respectively. Let $ω_{E}$ and $ω_{R}$ be the ideal geometric-optics frequencies assigned by the corresponding static observers at these endpoint events, using the model's geometric-optics identification with observer energy. With $n = f^{- 1 / 2} \partial_{0}$ the unit static congruence and $U = c n$ the corresponding four-velocity, the ideal-clock and geometric-optics predictions are

$\frac{Δ τ_{R}}{Δ τ_{E}} = \frac{ω_{E}}{ω_{R}} = \sqrt{\frac{f (r_{R})}{f (r_{E})}}, \frac{ω_{R}}{ω_{E}} = \sqrt{\frac{f (r_{E})}{f (r_{R})}} .$

The proper four-acceleration magnitude required to remain static at radius $r > s$ is

$a (r) = \sqrt{g (\nabla_{U} U, \nabla_{U} U)} = \frac{c^{2} s}{2 r^{2} \sqrt{f (r)}} = \frac{G_{N} m_{*}}{r^{2} \sqrt{1 - s / r}} .$

For any $0 \leq ρ < 1$ and radii satisfying $0 < s / r_{E}, s / r_{R} \leq ρ$ , define $x_{i} = s / r_{i}$ and write $\log : = L$ for the natural integral logarithm of The integral logarithm $L (x) : = \int_{1}^{x} \frac{d t}{t}$ for $x > 0$ . The exact logarithmic ratio has the uniform expansion

$\log \frac{ω_{R}}{ω_{E}} = \frac{s}{2} (\frac{1}{r_{R}} - \frac{1}{r_{E}}) + R, ∣ R ∣ \leq \frac{x_{E}^{2} + x_{R}^{2}}{4 (1 - ρ)} .$

These are conditional deductions of the idealized model; no measured redshift or acceleration is asserted.

Facts & Assumptions

Given: The Schwarzschild vacuum exterior model with $s > 0$ , static observers at $r_{E}, r_{R} > s$ , an ideal-clock reading of proper time, and the geometric-optics interpretation of the observer-energy ratio.

[A1]

Ideal clocks on future timelike worldlines record proper time, including clocks on accelerated static worldlines; these observers are constrained, not freely falling. Ideal rays follow future null geodesics in the geometric-optics approximation, and the observer-energy ratio is read as their frequency ratio. Adopt these rules in the smooth test-field regime, with no back-reaction, self-force, finite-size or tidal corrections, dispersion, dissipation or quantum effects (The Schwarzschild exterior vacuum model: field law, static clock readings and the mass parameter clauses (b), (d), Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clauses (i), (iii), (iv)); the massive free-fall rule in clause (ii) is not used for static observers.

[F1]

The static congruence is $n = f^{- 1 / 2} \partial_{0}$ , with $g (n, n) = - 1$ , $Δ τ = \sqrt{f (r)} Δ t$ , and $\nabla_{n} n = (s / (2 r^{2})) \partial_{r}$ (The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios clauses (a), (c), (e)).

[F2]

For a specified future null geodesic between static observers, $ω_{R} / ω_{E} = \sqrt{f (r_{E}) / f (r_{R})}$ under the geometric-optics interpretation (Restricted redshift prediction of the Schwarzschild vacuum model: the observer-energy ratio between static observers clauses (b), (d)).

[F3]

In the Schwarzschild exterior $g_{r r} = f^{- 1}$ , $f = 1 - s / r$ , and $s = 2 G_{N} m_{*} / c^{2}$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation).

[F4]

The natural logarithm is the integral logarithm $L (y) = \int_{1}^{y} d t / t$ for $y > 0$ , and a continuous integrand has its integral function as a primitive (The integral logarithm $L (x) : = \int_{1}^{x} \frac{d t}{t}$ for $x > 0$ , 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).

[F5]

If two continuous real functions satisfy $g \leq h$ on an interval, their integrals preserve that order (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ).

[F6]

The mean value theorem applies to continuous functions on closed intervals that are differentiable on their interiors; in particular, a function differentiable on an open interval with derivative zero is constant there (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

technique · combine the exact static observer identities and redshift ratio, scale the acceleration from unit observer to four-velocity, and bound the logarithmic remainder by an integral inequality

1.1F1F3algebra

By [F1], $U = c n$ is the static observer four-velocity because $g (U, U) = - c^{2}$ . Since $c$ is constant, $\nabla_{U} U = c^{2} \nabla_{n} n = c^{2} s \partial_{r} / (2 r^{2})$ . With $g (\partial_{r}, \partial_{r}) = f^{- 1}$ from [F3], its squared norm is $c^{4} s^{2} / (4 r^{4} f)$ , so its positive magnitude is $c^{2} s / (2 r^{2} \sqrt{f})$ . Substituting $s = 2 G_{N} m_{*} / c^{2}$ gives the displayed Newton-constant form.

1.2A1F1F2algebra

Over the same coordinate-time interval, [F1] gives $Δ τ_{R} / Δ τ_{E} = \sqrt{f (r_{R}) / f (r_{E})}$ . [F2] gives the inverse ratio for received to emitted ideal-ray frequency, so the two displayed exact ratios follow. Each ratio is a conditional model result for the specified static observers and ideal ray.

1.3F2F4F6algebra

By [F4], $L^{'} (y) = 1 / y$ for $y > 0$ and $L (1) = 0$ . For fixed $q > 0$ , the function $a \mapsto L (a / q) - L (a)$ has derivative $1 / a - 1 / a = 0$ on $(0, \infty)$ by the chain rule. It is constant by [F6], and its value at $a = q$ is $- L (q)$ , so $L (a / q) = L (a) - L (q)$ . Likewise $y \mapsto L (y^{2}) - 2 L (y)$ has derivative $2 / y - 2 / y = 0$ for $y > 0$ and value zero at $1$ , hence vanishes by [F6]. Setting $y = \sqrt{a / q} > 0$ yields $L (\sqrt{a / q}) = \frac{1}{2} [L (a) - L (q)]$ . Taking $a = 1 - x_{E}$ and $q = 1 - x_{R}$ in the positive frequency ratio of [F2] establishes the logarithmic ratio identity used below.

2.1step 1.3F2F4F5F6givenalgebra

For $0 < x \leq ρ < 1$ , the chain rule and [F4] show that $d \log (1 - x) / d x = - 1 / (1 - x) = - 1 - x / (1 - x)$ . By [F4] the function $\log (1 - x) + x + \int_{0}^{x} t / (1 - t) d t$ has derivative zero for $0 < x < 1$ and is continuous at $0$ , where it vanishes. Applying [F6] on $[0, x]$ gives $\log (1 - x) = - x - \int_{0}^{x} \frac{t}{1 - t} d t, 0 \leq \int_{0}^{x} \frac{t}{1 - t} d t \leq \frac{1}{1 - ρ} \int_{0}^{x} t d t = \frac{x^{2}}{2 (1 - ρ)} .$ For $t \in [0, x]$ , $0 \leq t / (1 - t) \leq t / (1 - ρ)$ . Thus [F5] and $\int_{0}^{x} t d t = x^{2} / 2$ give the displayed bound. By [F2] and step 1.3, $\log (ω_{R} / ω_{E}) = \frac{1}{2} [\log (1 - x_{E}) - \log (1 - x_{R})]$ . Inserting the integral identity separates the linear term $(x_{R} - x_{E}) / 2 = s (1 / r_{R} - 1 / r_{E}) / 2$ . The remaining terms have absolute value at most half the sum of their nonnegative integral bounds, namely $(x_{E}^{2} + x_{R}^{2}) / (4 (1 - ρ))$ .

3.1step 1.1step 1.2step 1.3step 2.1A1∎

The exact ratios, acceleration magnitude and uniform remainder bound are all consequences of the stated model identities; no empirical input is used.

Remarks

The static congruence is accelerated. The divergence of its required proper acceleration as $r ↓ s$ is not a divergence of the Schwarzschild curvature invariant and does not make the exterior-coordinate boundary a curvature singularity.
The frequency ratio assumes an ideal geometric-optics ray. Propagation through matter, finite wavelength, moving endpoints and measurement apparatus require additional models.

Static-clock redshift, proper acceleration, and a controlled weak-field expansion in Schwarzschild

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources