Gravitational Redshift

Core Idea

Gravitational redshift is the decrease in frequency (increase in wavelength) of light as it climbs out of a gravitational potential well, and the corresponding blueshift as light falls in. For the Schwarzschild metric, a photon emitted at radius r_e with frequency ν_e is observed at radius r_o with frequency ν_o = ν_e √[(1 - r_s/r_e)/(1 - r_s/r_o)], where r_s = 2GM/c². In the weak-field limit, the fractional shift is Δν/ν ≈ ΔΦ/c², where ΔΦ is the difference in Newtonian gravitational potential. Equivalently, clocks at lower gravitational potentials tick more slowly than clocks at higher potentials. Gravitational redshift was predicted by Einstein from the equivalence principle alone (before the full theory was complete), confirmed by the Pound-Rebka experiment (1959), and is an essential correction in the GPS satellite system.

Explainer

Gravitational redshift can be understood through the equivalence principle without any detailed knowledge of the Schwarzschild metric. Consider a photon emitted upward in a uniformly accelerating elevator. By the time the photon reaches the ceiling, the ceiling is moving faster than the floor was when the photon was emitted (the elevator accelerated during the photon's transit). The ceiling detector therefore sees the photon Doppler-shifted to a lower frequency. By the equivalence principle, the same effect occurs in a uniform gravitational field: a photon climbing up through height Δh acquires a redshift Δν/ν = gΔh/c². This argument was Einstein's original derivation, published in 1907 — eight years before the full field equations.

In the full Schwarzschild geometry, the redshift formula is exact: ν_o/ν_e = √[(1 - r_s/r_e)/(1 - r_s/r_o)]. For an observer at infinity (r_o → ∞) receiving light from radius r_e, this becomes ν_∞/ν_e = √(1 - r_s/r_e). At the event horizon (r_e = r_s), the redshift becomes infinite — photons emitted at the horizon are infinitely redshifted to zero frequency by the time they reach a distant observer. This infinite redshift is why a distant observer sees an infalling object fade and freeze at the horizon rather than crossing it. The formula reduces to the weak-field approximation Δν/ν ≈ ΔΦ/c² when r_s/r << 1, which is the regime relevant for Earth (r_s/R_Earth ≈ 1.4 × 10⁻⁹) and the Sun (r_s/R_Sun ≈ 4.2 × 10⁻⁶).

The equivalence between gravitational redshift and gravitational time dilation is exact. A clock at radius r in the Schwarzschild geometry ticks at a rate dτ = √(1 - r_s/r) dt relative to coordinate time t. Two clocks at different radii r₁ and r₂ therefore have a proper-time ratio dτ₁/dτ₂ = √[(1 - r_s/r₁)/(1 - r_s/r₂)], which is exactly the frequency ratio of light exchanged between them. The lower clock ticks slower, and photons emitted by it arrive at the higher clock with a proportionally lower frequency. These are not two separate effects but one phenomenon viewed from two perspectives: the photon's frequency is determined by the emitter's clock rate as seen by the receiver.

Experimental confirmation of gravitational redshift spans an impressive range of precision. The Pound-Rebka experiment (1959) measured the redshift of ⁵⁷Fe gamma rays over 22.6 meters in the Harvard physics building, confirming the predicted shift of about 2.5 × 10⁻¹⁵ to 10% accuracy. Hydrogen maser clocks flown on rockets (Gravity Probe A, 1976) confirmed the redshift to 7 × 10⁻⁵ precision. Most dramatically, the GPS satellite system provides a continuous, operational test: satellite atomic clocks at 20,200 km altitude gain about 45 μs/day from gravitational time dilation (offset by -7 μs/day from special-relativistic velocity effects), requiring a net correction of about 38 μs/day. Without this relativistic correction, GPS position errors would accumulate at roughly 10 km per day. Modern optical atomic clocks can detect the gravitational redshift between two laboratories separated by just 30 cm of vertical height, pushing precision to the 10⁻¹⁸ level.