Gravitational Redshift

The intuition: a photon paying a toll to escape gravity

Throw a ball straight up and it slows down — it trades kinetic energy for gravitational potential energy. A photon cannot slow down: it always moves at c. So when light climbs out of a gravitational well, it has to pay the toll some other way. It pays with frequency. A photon emitted at the surface of a star arrives at a distant telescope with a slightly lower frequency, a slightly longer wavelength, and slightly less energy. Blue light leaves a little redder than it arrived. That is gravitational redshift.

The cleanest way to see it is energy bookkeeping. A photon has energy E = hf and, by mass-energy equivalence, an effective gravitational mass E/c². Lifting that "mass" through a potential difference ΔΦ costs E ΔΦ/c² of energy, which must come out of the photon itself. So Δf/f = −ΔΦ/c². This heuristic gives exactly the right weak-field answer, even though a photon has no rest mass — a coincidence that pointed Einstein toward the deeper truth: gravity is not a force acting on the photon, but a warping of time itself.

The mechanism: it's really gravitational time dilation

The energy-conservation story is a useful crutch, but the correct picture is that clocks run at different rates at different depths in a gravitational field. The source emitting at frequency f is really a clock ticking f times per second by its own time. If the source's clock runs slow relative to the receiver's clock, the receiver counts fewer crests per of its own seconds — a lower frequency, a redshift. Gravitational redshift and gravitational time dilation are two descriptions of one fact.

In the Schwarzschild geometry around a non-rotating mass M, the proper time elapsed on a static clock at radius r relates to coordinate time by

dτ = dt · √(1 − r_s/r),   where  r_s = 2GM/c²  (Schwarzschild radius)

A clock deeper in the well (smaller r) accumulates less proper time per coordinate second — it ticks slow. The ratio of frequencies received versus emitted between two static radii r_emit and r_obs is

f_obs / f_emit = √( (1 − r_s/r_emit) / (1 − r_s/r_obs) )

1 + z = √( (1 − r_s/r_obs) / (1 − r_s/r_emit) )

For an observer at infinity (r_obs → ∞) receiving light from radius r, this collapses to the headline formula

1 + z = 1 / √(1 − r_s/r)

which diverges as r → r_s: light emitted right at the event horizon is redshifted to infinite wavelength and zero energy — it never escapes. That divergence is precisely why the horizon is a horizon.

The weak-field limit and the numbers that matter

For everyday gravity — stars, planets, towers — r_s/r is tiny and the square root linearises. Writing the Newtonian potential Φ = −GM/r, the shift between two points becomes

z ≈ Δλ/λ ≈ ΔΦ/c² = (Φ_obs − Φ_emit)/c²
For a uniform field over a height H:   z ≈ gH/c²

The denominator c² = 9 × 10¹⁶ m²/s² is enormous, so the effect is minuscule until the potential is deep. Some real magnitudes:

Source → observer	Potential / setup	Fractional shift z	How it shows up
Earth's surface → infinity	GM⊕/R⊕c²	≈ 7.0 × 10⁻¹⁰	0.7 ns/s clock offset
22.5 m tower (Pound-Rebka)	gH/c², H = 22.5 m	≈ 2.46 × 10⁻¹⁵	Mössbauer null shift
1 m height difference (optical clock)	g·(1 m)/c²	≈ 1.1 × 10⁻¹⁶	Optical-clock comparison, 2010s
Sun's surface → Earth	GM☉/R☉c²	≈ 2.12 × 10⁻⁶	633 m/s equivalent line shift
Sirius B (white dwarf)	GM/Rc², R ≈ 5,800 km	≈ 2.7 × 10⁻⁴	+80 km/s line shift, HST 2005
Neutron star surface	r_s/r ≈ 0.3–0.4	z ≈ 0.2–0.35	Spectral line / X-ray burst
Photon at event horizon	r → r_s	z → ∞	Light cannot escape

Note how the effect spans nine orders of magnitude in the same equation, from a part in 10¹⁶ between two tabletop optical clocks separated by one metre to an infinite shift at a black-hole horizon. White dwarfs occupy a sweet spot: compact enough (z ~ 10⁻⁴, tens of km/s) to be measurable spectroscopically, yet weak-field enough that the linear formula still works well.

Why it must be true: the equivalence principle

Einstein derived gravitational redshift in 1907 — eight years before the full field equations — from nothing but the equivalence principle. Imagine a rocket in deep space accelerating upward at g. A laser at the floor fires a pulse of frequency f toward a detector on the ceiling, a height H away. The light takes time H/c to arrive, and during that time the ceiling has accelerated to a speed v = gH/c. The detector is therefore receding from the emission event, so it measures a Doppler-shifted, lower frequency: Δf/f = −v/c = −gH/c².

The equivalence principle says a uniformly accelerating frame is locally indistinguishable from a uniform gravitational field. So in a real gravitational field, light climbing a height H must be redshifted by exactly the same gH/c². No metric, no field equations — just acceleration plus the Doppler effect. This is why gravitational redshift is often called the most fundamental test of general relativity: a theory of gravity that did not predict it would already be inconsistent with the equivalence principle and special relativity together.

Worked example: the Sun, and a sanity check on GPS

Solar redshift. Light leaving the Sun's photosphere and reaching Earth (whose potential is negligible by comparison) is shifted by

z = GM☉ / (R☉ c²)
M☉ = 1.989 × 10³⁰ kg,  R☉ = 6.96 × 10⁸ m,  G = 6.674 × 10⁻¹¹,  c² = 8.988 × 10¹⁶
z = (6.674e−11 × 1.989e30) / (6.96e8 × 8.988e16)
  = 1.327e20 / 6.256e25
  = 2.12 × 10⁻⁶

Expressed as an equivalent Doppler velocity, v = zc = 2.12 × 10⁻⁶ × 3 × 10⁵ km/s ≈ 0.64 km/s ≈ 633 m/s. This solar redshift was finally confirmed cleanly in 2020 (González Hernández et al.) using HARPS observations of the solar spectrum reflected off the Moon, which measured 638 ± 6 m/s against a GR prediction near 633 m/s — agreement at the percent level once convective blueshifts in the photosphere are modelled out.

GPS. A GPS satellite sits at orbital radius r ≈ 26,560 km (altitude ~20,200 km). The gravitational potential there is shallower than at Earth's surface, so the satellite clock runs fast:

Δf/f|grav = (Φ_sat − Φ_ground)/c² = GM⊕ (1/R⊕ − 1/r)/c²
        ≈ +5.3 × 10⁻¹⁰   →  +45.7 µs/day  (clock runs fast)
Δf/f|SR  = −v²/2c²,   v ≈ 3.87 km/s
        ≈ −0.83 × 10⁻¹⁰  →  −7.2 µs/day   (clock runs slow)
Net      ≈ +38.5 µs/day

An uncorrected 38 µs/day error maps to c × 38 µs = 11.4 km of ranging error accumulating per day — utterly fatal for a navigation system that needs metre accuracy. Engineers pre-tune the satellite oscillators to 10.22999999543 MHz instead of the nominal 10.23 MHz so the clocks tick at the correct ground rate once in orbit. GPS is the most widely deployed everyday verification of gravitational redshift on the planet.

How it is observed and measured

The challenge is always that the shift is tiny and competes with ordinary Doppler shifts from motion. The experimental tradition splits into terrestrial precision tests and astrophysical line-shift measurements.

• Mössbauer spectroscopy (Pound-Rebka, 1959–64). Iron-57 emits a 14.4 keV gamma with a fractional linewidth near 10⁻¹², narrow enough to resolve a 10⁻¹⁵ shift. Pound and Rebka nulled the redshift by moving the source at a few mm/s to add a compensating Doppler shift, then read off the velocity. The 22.5 m drop in Harvard's Jefferson tower gave 2.46 × 10⁻¹⁵; they matched it to 10% (1960), and Pound–Snider tightened it to 1% (1964).
• Atomic-fountain and hydrogen-maser clocks. The Gravity Probe A experiment (Vessot & Levine, 1976) flew a hydrogen-maser clock to 10,000 km on a Scout rocket and compared it with a ground maser, confirming the redshift to 7 × 10⁻⁵ — for decades the most precise test.
• Optical lattice and ion clocks. By 2010 an aluminium-ion clock resolved the redshift over a 33 cm height change in the lab. By 2022, an optical-lattice clock measured the frequency gradient across a 1 mm-tall sample of strontium atoms — gravitational redshift inside a single atomic cloud.
• Stellar spectroscopy. White dwarfs like Sirius B and 40 Eridani B show z ~ 10⁻⁴; the Sun shows z ~ 2 × 10⁻⁶ buried under convective motions. Gaia DR3 (2023) statistically detected the solar-type gravitational redshift across tens of thousands of stars.
• Strong-field tests. The ESA Galileo satellites GSAT-0201/0202, accidentally launched into eccentric orbits in 2014, modulated their gravitational redshift by ±0.4 µs over each orbit; two 2018 analyses confirmed GR to ~2.5 × 10⁻⁵. That same year GRAVITY caught the gravitational + transverse Doppler redshift of star S2 at the Galactic-centre black hole.

Three kinds of redshift, side by side

"Redshift" names three physically distinct phenomena that happen to share one formula, 1 + z = λ_obs/λ_emit. Conflating them is the single most common conceptual error.

Property	Doppler redshift	Gravitational redshift	Cosmological redshift
Cause	Relative velocity	Difference in gravitational potential	Expansion of space in transit
Depends on	v / c (line of sight)	ΔΦ / c² (well depth)	Scale factor ratio a(now)/a(then)
Formula (weak)	z ≈ v/c	z ≈ ΔΦ/c²	1 + z = a₀/a_emit
Needs motion?	Yes	No (static source & observer)	No (comoving frames)
Sign can flip?	Yes (approach = blueshift)	Yes (light falling IN = blueshift)	Only redshift (expanding universe)
Distance dependence	None intrinsically	On endpoint potentials only	Grows with look-back distance
Canonical example	Binary star wobble	White dwarf, GPS, Pound-Rebka	Hubble flow, CMB at z ≈ 1100

A real astronomical spectrum often contains all three at once. Untangling them is exactly the work the GRAVITY team did at Sagittarius A*: subtract the Newtonian orbital Doppler, and the leftover ~200 km/s residual at closest approach is the gravitational-plus-transverse-Doppler relativistic signature.

Discovery and the people who tested it

Einstein first wrote down gravitational redshift in his 1907 review article on relativity and gravity, refined it in 1911 (predicting the solar shift), and embedded it in general relativity in 1915–16. For two decades it resisted clean measurement: the solar line shift is swamped by pressure shifts and convective blueshifts, and early attempts gave ambiguous results that nearly undermined confidence in GR.

• 1907 / 1911 / 1916 — Einstein. Derived the effect from the equivalence principle, then from the full metric. The 1911 paper predicted z ≈ 2 × 10⁻⁶ at the Sun.
• 1925 — Walter Adams. Claimed to measure the redshift of the white dwarf Sirius B and called it a triumph for GR. Modern analysis shows his number was contaminated by scattered light from Sirius A; the "confirmation" was partly luck, vindicated only by the cleaner 2005 Hubble measurement (z giving +80.4 ± 4.8 km/s).
• 1959–1964 — Pound, Rebka, Snider. The decisive terrestrial test, using the Mössbauer effect over a 22.5 m tower; 10% then 1% agreement with GR.
• 1976 — Vessot & Levine (Gravity Probe A). A hydrogen maser on a sounding rocket to 10,000 km; 7 × 10⁻⁵ agreement, the gold standard for decades.
• 1978 onward — GPS / NAVSTAR. The first NAVSTAR satellites flew clocks with switchable relativistic correction; turning the correction off demonstrably broke the system, an engineering confirmation at planetary scale.
• 2018 — GRAVITY & ESA Galileo. Gravitational redshift detected near a 4-million-solar-mass black hole (S2 star) and, independently, refined to 2.5 × 10⁻⁵ from two eccentric-orbit Galileo satellites.

Variants and close relatives

• Gravitational blueshift. The same effect with the sign reversed — light falling into a potential well gains energy and shifts blue. CMB photons falling into galaxy-cluster potential wells and climbing out again give the integrated Sachs-Wolfe effect; if the well decays during transit, a net blueshift remains.
• Transverse (second-order) Doppler shift. A purely special-relativistic redshift from time dilation of a moving source, z ≈ v²/2c². Near compact objects it adds to the gravitational redshift; the S2 measurement detected the two together because both scale as the same small parameter at closest approach.
• Shapiro time delay. The temporal cousin: light passing through a deep potential is delayed, not frequency-shifted, because coordinate time runs differently along the path. Same metric coefficient, different observable.
• Cosmological gravitational redshift / Wojtak effect. Photons climbing out of the potential wells of galaxy clusters carry a measurable ~10 km/s gravitational redshift, statistically detected across thousands of clusters (Wojtak et al. 2011) as a test of GR on megaparsec scales.
• Black-hole horizon redshift. The r → r_s divergence is the same physics taken to its limit; an infalling object appears to freeze and fade at the horizon as its emitted light is redshifted toward infinite wavelength.

Common misconceptions and subtleties

• "The photon slows down as it climbs out." No. Locally measured speed is always c. Only frequency, wavelength, and energy change; the speed is invariant.
• "It needs the source or observer to be moving." No — gravitational redshift is fully present between two perfectly static clocks at different potentials. That is exactly what distinguishes it from Doppler shift.
• "It depends on the distance the light travels." Only the endpoint potentials matter, not the path length or how long the photon is in flight. A photon emitted at the Sun and absorbed at Earth carries the same z whether the path is straight or bent by intervening masses (to leading order).
• "Energy isn't conserved — the photon lost energy." Energy is conserved once you include the gravitational field (or, more carefully, recognise that there is no global energy conservation law in a general curved spacetime, only a local one). The "missing" energy is bookkeeping against the field, exactly as for a rising ball.
• "The 1925 Sirius B result proved GR." Adams's measurement was largely an artifact of contamination; the real white-dwarf confirmation came decades later. A reminder that a "confirmation" matching the right number can still be wrong for the wrong reasons.
• "Gravitational and cosmological redshift are the same because both involve gravity." The mechanisms differ: gravitational redshift is a static potential difference; cosmological redshift is the metric expanding while the photon is in transit. They only coincide in sharing the definition of z.