Gravitational Redshift: Light Climbing Out of a Well

Climbing out of a gravity well costs a photon energy

General relativity predicts that a photon climbing out of a gravitational well loses energy, and since a photon's energy is E = hf, losing energy means dropping in frequency — its light shifts toward the red end of the spectrum. This is gravitational redshift, and unlike the Doppler redshift of a receding source, it happens even for a perfectly static emitter and observer; it is purely a statement about how gravity warps the flow of time itself.

The cleanest way to see it is through gravitational time dilation. A clock deep in a gravity well ticks slower than a distant clock, as measured by comparing signals between them. Light emitted at frequency f by the deep clock, when counted by the distant, faster-ticking clock, arrives at a lower frequency — not because anything happened to the photon in transit, but because the two observers' own definitions of a second disagree.

The Schwarzschild factor

For a non-rotating spherical mass M, the exact redshift between a static emitter at radius r and a distant observer follows directly from the Schwarzschild metric:

1 + z = f_emit / f_observed = 1 / sqrt(1 - rs/r)

rs = 2GM/c²      (the Schwarzschild radius)
G  = gravitational constant, M = mass, c = speed of light, r = emission radius

As r approaches rs, the factor sqrt(1 − rs/r) approaches zero and the redshift diverges — light emitted exactly at the event horizon is stretched to infinite wavelength before it can escape, which is precisely why nothing, light included, escapes from inside. Far from the mass, where r ≫ rs, this exact formula reduces to the familiar weak-field approximation z ≈ GM/(rc²), the version used for anything short of a black hole or neutron star.

Measured on Earth, not just in theory

The effect is tiny at everyday gravitational strengths but not too tiny to measure. The 1959 Pound–Rebka experiment used the Mössbauer effect to detect the fractional frequency shift of gamma rays climbing just 22.5 metres up a tower at Harvard — a shift of about 2.5×10⁻¹⁵, confirmed to within 10% of general relativity's prediction using 1960s equipment. GPS satellites experience the effect at a scale that actually matters operationally: sitting roughly 20,000 km up, their clocks run about 45 microseconds per day faster than clocks on the ground due to weaker gravity there, partially offset by about 7 microseconds per day slower from their orbital speed (special-relativistic time dilation). Left uncorrected, the net 38 microsecond/day drift would make GPS position errors grow by roughly 10 km per day.

Redshift, blueshift and the direction of the effect

The rule is symmetric: light climbing out of a well redshifts, light falling into one blueshifts, and the size of the shift depends only on the difference in gravitational potential between emission and observation, not on the path taken. A photon emitted near a white dwarf and observed far away is measurably redshifted — this was one of the earliest astrophysical confirmations of the effect, observed in the spectral lines of Sirius B in the 1920s, well before Pound and Rebka's laboratory version. Astronomers today use gravitational redshift routinely to weigh compact stars: measure a spectral line's shift, and if the star's radius is known independently, the Schwarzschild formula gives its mass.

Frequently asked questions

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