Gravitational redshift · Pound–Rebka 1959

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 said gravity is indistinguishable from acceleration; does that alone force the redshift Δν/ν = gh/c² that Pound & Rebka measured on a Harvard tower?

▶ Run the simulationSee the measured result

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

How the lab tests it

The equivalence principle turns the gravitational field into a rocket accelerating at g in flat space: a light crest emitted from the floor toward the ceiling (height h) flies for ≈ h/c while the ceiling accelerates away, so the received light is Doppler-shifted. The lab TRACES successive wave crests to the receding ceiling (the crest-meets-ceiling quadratic, evaluated cancellation-free) and reads the received/emitted period ratio — with no redshift formula coded. The dimensionless coefficient K = z/(gh/c²) is Richardson-extrapolated to the weak field, the g/h/c dependence is swept, and a Monte-Carlo 'Mössbauer' measurement at 1% noise recovers the tower shift.

What it checks

light REDDENS as it climbs, by exactly Δν/ν = gh/c² — the coefficient comes back K = 1.0000 (Einstein's z = ΔΦ/c²), recovered from the traced crest ratio with no formula fed, and applied to Earth's 22.5 m tower it gives 2.455×10⁻¹⁵, matching Pound & Rebka's 1959 measurement (Pound–Snider 1965 refined it to 0.9990 ± 0.0076 of the prediction). The shift scales z ∝ g·h/c² (log–log slopes +1, +1, −2), which is why it needed a tall tower and a part-per-quadrillion nuclear line to see. The decisive rival is the pre-1960 view that a static gravitational field, being merely a force, cannot shift a light wave's frequency (z = 0): setting the frame's acceleration to zero gives exactly z = 0, and the measured 2.455×10⁻¹⁵ sits thousands of standard errors above it — rejected. This is the g_tt (time-dilation) piece of the metric, the sibling of ?world=lightbending's g_rr light bending and ?world=schwarzschild's perihelion advance. The on-screen colour shift is exaggerated (~10¹⁴×) so the reddening is visible; the printed K = 1.0000 and 2.455×10⁻¹⁵ are the true traced values.

Gravitational redshift, time dilation, and the spelling of the exact formula that destroys the answer

A photon climbing out of a gravitational field arrives with less frequency than it left with, and the whole of the weak-field answer is the potential it climbed divided by c squared. Einstein got that in 1907 out of nothing but the equivalence principle — a uniform field IS a frame accelerating at g, so the receiver is running away from the light and the shift is an ordinary Doppler shift in flat spacetime — and this page runs that argument as arithmetic rather than quoting its result. The simulation above traces wave crests to the receding ceiling of an accelerating elevator with no redshift formula anywhere in it; this page takes the same published geometry, three square roots and one Richardson pair, and lands on 0.99999994997657105, which is the module's own certified executed coefficient to the last bit. Note that it is not 1: a page that had typed the answer would print 1, and the residual is the extrapolation's own second-order leftover, which the generated binomial series then predicts in closed form. Four directions cover what people actually come here to compute — the 22.5 m Harvard tower shift of 2.455e-15 that Pound and Rebka weighed with a Mössbauer line in 1959, the exact static ratio between two radii, the 77 km/s surface redshift of Sirius B against its measured 80.42 ± 4.83, and the 45.7 microseconds a day a GPS clock gains by being 20 000 km higher in the potential. But the direction worth the visit is the one about the formula itself. Spelled the way every textbook prints it, √(A/B) − 1, the EXACT general-relativistic answer returns 2.4425e-15 at the tower against the true 2.4585e-15 — 0.65% wrong, about two correct digits out of sixteen, three orders of magnitude worse than the weak-field approximation it exists to correct. Nothing is wrong with the physics: A/B is 1 + 4.9e-15 there, and subtracting 1 from a double that close to 1 throws away every digit the field was hiding in. Over 200000 random arguments the two spellings of 1/√(1−x) − 1 disagree at 98.062% of them and by up to 1.7e+12 ulps. The cure is algebra rather than precision, and the page prints both spellings everywhere it computes one, each checked against a bound DERIVED from the expression — 2·ULP/z — instead of a tolerance somebody chose. Everything here is +, −, ×, ÷ and a square root, all five correctly rounded by IEEE-754, so every figure prints in full and means the same thing in every browser. Four things this page will not do: velocity time dilation, which has the opposite sign and belongs to ?world=muon; anything cosmological, since a static field has no expansion history and no redshift here becomes a distance; orbits weighed against r_s, which belong to ?world=schwarzschild; and correcting one number the simulation above recovered.