Gravitational redshift recovered from Einstein's equivalence-principle elevator with no redshift formula coded

Does light climbing away from a mass lose frequency — and if a gravitational field is 'just' a force, how could it shift a light wave at all? Einstein's 1907 answer was the equivalence principle: a uniform gravitational field is locally indistinguishable from a frame accelerating at g. Does that identity alone force the redshift Δν/ν = gh/c² that Pound & Rebka weighed with a Mössbauer line on a 22.5 m Harvard tower in 1959 — with no redshift formula assumed?

Units: fractional frequency shift Δν/ν of a 14.4 keV γ climbing the h = 22.5 m Pound–Rebka tower on Earth (g = 9.80665 m/s²): gh/c² = 2.4551e-15; Einstein 1907 predicted Δν/ν = gh/c², Pound & Rebka 1959 / Pound & Snider 1965 measured 0.9990 ± 0.0076 of it

▶ Run this simulationRead how it works

The finding

Gravitational redshift recovered from Einstein's equivalence-principle elevator with no redshift formula coded: tracing wave crests to the receding ceiling of a frame accelerating at g returns the coefficient K = z/(gh/c²) = 1.0000 (Einstein's z = ΔΦ/c²), and applied to Earth's 22.5 m Pound–Rebka tower gives Δν/ν = 2.455×10⁻¹⁵ vs the measured 2.46×10⁻¹⁵; the shift scales z ∝ g·h/c² (log–log slopes +1, +1, −2), a 24×40 Mössbauer Monte-Carlo at 1% noise lands 2.456×10⁻¹⁵ ± 7×10⁻¹⁹ and sits 3400 SE above zero, and the pre-1960 rival 'gravity cannot shift light' (z = 0) — reproduced by zeroing the frame's acceleration — is decisively rejected.

Method

By the equivalence principle the gravitational field becomes a rocket accelerating at g in flat spacetime: floor emitter at z_e(t) = ½gt², ceiling receiver at z_r(t) = h + ½gt², light at constant c. A wave crest emitted from the floor at t_e meets the receding ceiling when ½gt² − ct + (ct_e − ½gt_e² + h) = 0, i.e. arrival(t_e) = [c − √((c−g·t_e)² − 2gh)]/g. Two crests emitted dt = 10⁻⁶·(h/c) apart at t ≈ 0 (frame momentarily at rest, isolating the gravitational part from the frame's velocity Doppler) give the received/emitted period ratio; the crest-arrival difference is computed cancellation-free via P−Q = (P²−Q²)/(P+Q), so z = (2c − g·dt)/(P+Q) − 1 with P = √(c²−2gh), Q = √((c−g·dt)²−2gh). Nondimensionalizing by gh/c² (scoring only) reads the coefficient K; a Richardson pair {2e-4, 1e-4} extrapolates the finite-field elevator's O(gh/c²) artifact away to K = 1. The g/h/c dependence is swept, and a Monte-Carlo 'Mössbauer' block draws 24×40 tower measurements at 1% Gaussian noise (Pound–Snider precision). The closed form gh/c², the number 2.455e-15, the coefficient 1, and the Doppler expression Δv/c appear NOWHERE in the generator. 10 gates, ~0.02 s. ?world=redshift.

The law it recovers

Δν/ν = gh/c² = ΔΦ/c² — recovered, never coded: the coefficient K of z = K·(gh/c²) comes back 1.0000 from the traced crest ratio, and the shift follows z ∝ g·h/c² across independent sweeps (log–log slopes +1 in h, +1 in g, −2 in c). z ∝ h is why the effect needed a 22.5 m tower and a part-per-quadrillion Mössbauer line to see; z ∝ 1/c² is why it is a relativistic effect absent from everyday optics.

Measurements, controls & cross-checks

Recovered uncertainty

7.2300e-19

Recovered note

the MONTE-CARLO 'Mössbauer' value (24 seeds × 40 measurements, 1% fractional noise) = 2.4557e-15 ± 7.2e-19, 0.77 SE from the known 2.4551e-15, and 3396 SE above zero (the z = 0 rival). The NOISELESS coefficient recovery is K = z/(gh/c²) = 1.00015003 at u = 1e-4 (the +3/2·u finite-field elevator term, not numerics) and K = 1.0000000 after the {2e-4, 1e-4} Richardson extrapolation; applied to Earth's tower geometry that gives 2.4551e-15 (rel 1.7e-5 vs the cited gh/c²)

Coefficient scale free

Configs

6

K mean

1.00015

K sd

9.0600e-13

Note

K measured across six independent (g,h,c) at the same u = gh/c² = 1e-4 (g spanning 10⁴×, c = 1 and 100) is identical to the float floor (SD 9e-13) — z is a pure ratio with no absolute-scale dependence, exactly as a dimensionless redshift must be

Higher order 2nd term

Residual at u1e-3

0.001503

Residual ratio u over half u

2.002

Note

the exact elevator trace is z = 1/√(1−2gh/c²) − 1 = gh/c² + (3/2)(gh/c²)² + …; the residual r(u) = K(u) − 1 is positive and r(u)/r(u/2) ≈ 2 (scales as u), the emergent O(gh/c²) term of the finite-field Newtonian-rigid elevator — DISCLOSED as a model artifact, not claimed as GR's exact 2nd order, and vanishing in the weak field where the real experiments live (gh/c² ≤ 2.5e-15)

Perturbation sweep

Slope h

1.00039

Slope g

1.000195

Slope c

-2.000097

Expected

z ∝ g·h/c² → slopes +1, +1, −2

Note

the same elevator crest-tracer, swept independently in h, g, and c, returns the full gravitational-potential-over-c² scaling

Mc mossbauer

Seeds

24

Runs per seed

40

Frac noise

0.01

Recovered

2.4557e-15

Se

7.2300e-19

Se from known

0.77

Sigma above zero

3396

Note

a faithful model of the 1959 experiment — a tiny 2.5e-15 shift measured against noise; the recovered value sits 3396 SE above the z = 0 rival, the detection Pound & Rebka achieved

Rival static

Z

0

Note

setting the frame acceleration to zero (emitter and receiver both STATIC, height h apart — the pre-1960 view that a gravitational force cannot change a light wave's frequency) gives z = 0 to machine precision, sharing the exact same crest-tracer code path. This is the null Pound & Rebka rejected; only the equivalence principle — a gravitational field IS a local acceleration — forces the nonzero shift, and only the accelerating-elevator trace produces it.

What it reduces to

Einstein's 1907 gravitational-redshift prediction Δν/ν = ΔΦ/c² (Jahrbuch der Radioaktivität 4, 411), confirmed by Pound & Rebka (Phys. Rev. Lett. 4, 337, 1960) and refined by Pound & Snider (1965) to 0.9990 ± 0.0076 of the prediction, and by Gravity Probe A (Vessot 1980) to 1.4e-4. It VALIDATES, not derives: the accelerating-elevator kinematics are the substrate (floor/ceiling worldlines + light at c), and the redshift law, the coefficient 1, and the 2.455e-15 tower shift are recovered by MEASURING the crest-arrival ratio — the formula appears only in scoring. Non-circularity: the generator maps (g, h, c, dt) → crest-meets-ceiling quadratic → arrival interval → z; no gh/c², no coefficient 1, and no Δv/c Doppler expression exists in that path (the 2gh under the √ is the arrival-quadratic discriminant, not the redshift law), and the coefficient is read only after nondimensionalizing at the scoring step (tamper test: falsifying known_value flips exit to 1 with the recovered 2.4557e-15 unchanged). The decisive control — the static frame (acceleration 0) returning z = 0 through the same code path — establishes that the shift needs the equivalence principle specifically, not merely a gravitational field: it is the g_tt (metric time-dilation) piece that the timelike sibling ?world=schwarzschild advances Mercury's perihelion with and the null sibling ?world=lightbending doubles into light bending. A Newtonian 'photon of mass hν/c² loses kinetic energy climbing' argument happens to reproduce gh/c² to leading order, so it is NOT the falsified rival; the genuine one is that light frequency is a gravitational invariant (z = 0).

Module systematics

The module's on-screen numbers (K = 1.0000, the tower shift 2.455e-15, and the z-vs-h chart) come from the SAME crest-tracer the oracle uses, run at the real Pound–Rebka geometry in RedshiftModule.init() — honest, not formula-fed. The only exaggeration is visual: the DRAWN colour shift of the climbing light pulses (blue floor → red ceiling) and their size 'stretch' are exaggerated ~10¹⁴× so the reddening is visible at all (the true shift is 2.5e-15, invisible on any screen), and the on-screen banner discloses 'colour shift EXAGGERATED on screen; printed numbers are the real trace'. No bias vs the oracle in any reported number — the coefficient K and the 2.455e-15 are the true traced values, and the chart plots the recovered z(h) = K·gh/c², not a hand-drawn line. CERTIFIED (module-honesty gates J–N, zero module edits): the derisk EXECUTES the shipped RedshiftModule.ts (sha256-pinned, 30 exact strip pairs, new Function with Babylon/DOM stubs) — the executed _coeff() === the oracle's gate-A' Richardson K BIT-FOR-BIT (0.999999949976571, one code path, not two agreeing numbers), the executed _z ≡ the oracle's measureZ bit-for-bit at all 28 sweep points including the g=0 rival, 1200 fl(1/120) engine ticks of the real fixedUpdate/render are bit-exact vs a statics-built replica at EVERY tick (accumulated _t, all 7 pulse positions/scales/Lerp colours/alphas, the source's breathing emissive, the %6 HUD/chart write cadence: 201/200 writes), the display is LIVE (pulse 0 wraps at the replica-predicted ticks 545/1090, final screen ≠ first), the HUD and chart SVG === replicas built from the ORACLE's numbers (sha-pinned; shown 2.455e-15 digit-exact vs known yet _prShift !== known — earned, not pasted), and the executed recovery path is answer-free (no 2.455e-15 / tower geometry / gh-over-c² division in _z; scanner proven live by a planted violation). Tamper self-tests: known→2.0e-15 fails C+H+M with recovery byte-unchanged; cert sha flip fails ONLY J; engine_ticks→600 fails ONLY L (execution gates not vacuous).

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- redshift (scripts/redshift-derisk.mjs)

Oracle

scripts/oracles/redshift.reference.json

Sources

A. Einstein, 'Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen', Jahrbuch der Radioaktivität und Elektronik 4, 411 (1907) — the equivalence-principle derivation of Δν/ν = ΔΦ/c². R. V. Pound & G. A. Rebka Jr., 'Apparent Weight of Photons', Phys. Rev. Lett. 4, 337 (1960) — the 22.5 m Harvard tower Mössbauer measurement. R. V. Pound & J. L. Snider, Phys. Rev. Lett. 13, 539 (1964) / Phys. Rev. B 140, 788 (1965) — refined to 0.9990 ± 0.0076 of the Einstein prediction. R. F. C. Vessot et al., Phys. Rev. Lett. 45, 2081 (1980) — Gravity Probe A hydrogen maser, ΔΦ/c² to 1.4e-4. C. M. Will, Living Rev. Relativity 17, 4 (2014). Constants: standard gravity g = 9.80665 m/s², tower h = 22.5 m, SI c.