Gravitational Redshift

Why gravitational redshift matters

• GPS depends on it. The 24-satellite GPS constellation orbits at 20,200 km altitude, where gravitational potential is less negative than on Earth's surface. By gravitational redshift, satellite atomic clocks tick faster than ground clocks by ΔΦ/c² × (86400 s/day) ≈ 45 µs/day. Combined with special-relativistic slowing from orbital speed (−7 µs/day), the net advance is 38 µs/day. Without compensating, GPS receivers would compute positions wrong by 10 km after one day — turning satellite navigation into noise. Onboard clocks are deliberately detuned at the factory to compensate.
• Atomic clock precision metrology. Modern optical atomic clocks (strontium, ytterbium ion clocks) reach 10⁻¹⁸ fractional frequency precision. At this level, lifting a clock by 1 cm produces a measurable redshift difference (gh/c² ≈ 1.1 × 10⁻¹⁸). This enables 'relativistic geodesy' — using clock comparisons over fiber networks to map the geoid (gravitational equipotential surface) of Earth at cm precision, complementing classical surveying.
• White dwarf mass measurement. Compact objects with strong surface gravity show large gravitational redshift in their spectra. Sirius B, the white dwarf companion of Sirius A, has surface redshift z ≈ 3 × 10⁻⁴, measured spectroscopically since the 1920s. Combined with mass-radius models, this directly constrains white dwarf interior physics — and was Adams' 1925 first astrophysical confirmation of GR. Modern measurements with HST achieve <1% precision.
• Black hole imaging. Light from accretion disks around black holes shows extreme redshift gradient near the horizon — material moving toward us appears Doppler-blueshifted but gravitationally redshifted, the competition gives a characteristic asymmetric profile. EHT image of M87* (2019) reveals this combined effect; spectroscopic observations of relativistic Fe Kα emission lines (e.g., from MCG-6-30-15 and other AGN) confirm Schwarzschild and Kerr geometries by spectral profile fitting.
• Cosmological tests of GR. Light from distant galaxies traveling through the cosmic gravitational potential field exhibits subtle integrated Sachs-Wolfe effects — temperature shifts in the CMB caused by photons climbing out of evolving potential wells. Combined with weak gravitational lensing surveys (DES, KiDS, LSST), these test general relativity on cosmological scales and constrain dark energy properties.
• Pulsar timing. Millisecond pulsars are precision clocks. Pulses arrive at Earth slightly later when the pulsar's photons traverse our solar system's gravitational potential — the Shapiro delay. Combined with binary pulsar systems (Hulse-Taylor PSR B1913+16, double pulsar PSR J0737-3039), this provides multiple GR tests at parts-per-thousand precision over decades, including direct Shapiro and gravitational redshift measurements.
• Gravity Probe A. 1976 NASA mission launched a hydrogen-maser clock on a suborbital trajectory reaching 10,000 km altitude, comparing it to ground masers. Confirmed Einstein redshift prediction to 70 parts per million, the most precise gravitational redshift test until 2018. The mission used a sounding rocket trajectory specifically chosen to maximize the gravitational potential variation.
• Cosmic microwave background. The 2.7 K CMB photons we detect were emitted at z ≈ 1100 redshift — only partly cosmological (Hubble expansion); a small fraction is gravitational redshift from the surface of last scattering's potential field. Distinguishing these contributions in the CMB power spectrum constrains the Universe's geometry, baryon density, and primordial power spectrum.

The formula and the numbers

• Weak-field formula. For modest gravitational fields (|Φ|/c² ≪ 1), gravitational redshift between emission point at potential Φ_e and observation at Φ_o is z = Δλ/λ ≈ (Φ_o − Φ_e)/c². For a photon climbing out of a well (Φ_e < Φ_o), Δλ > 0 — wavelength stretches. Pure Newtonian expression in the limit of small relativistic corrections.
• Schwarzschild exact formula. In the Schwarzschild metric, 1 + z = √(g_00(emitter)/g_00(observer)) = √((1 − r_s/r_o)/(1 − r_s/r_e)). For observer at infinity (r_o → ∞), this becomes 1 + z = (1 − r_s/r_e)^(−1/2), diverging as r_e → r_s. At r_e = 1.5 r_s (photon sphere), z = √3 − 1 ≈ 0.732. At r_e = 2 r_s, z = √2 − 1 ≈ 0.414.
• Sun. z = GM/(Rc²) for solar surface. M = 1.99 × 10³⁰ kg, R = 6.96 × 10⁵ km. z = (6.67 × 10⁻¹¹ × 1.99 × 10³⁰)/(6.96 × 10⁸ × 9 × 10¹⁶) ≈ 2.12 × 10⁻⁶. Solar Fraunhofer lines are redshifted by ~600 m/s in equivalent Doppler velocity. Tested to 1% by 2020 HST observations.
• Earth. z = gh/c² for terrestrial heights. At h = 1 km altitude, z ≈ 1.1 × 10⁻¹³. Pound-Rebka used h = 22.5 m, giving z = 2.46 × 10⁻¹⁵ — measurable only because the Mössbauer effect provides ultra-narrow γ-ray lines.
• Sirius B (white dwarf). Mass M ≈ 1.0 M_sun, radius R ≈ 5,800 km. z = GM/(Rc²) ≈ 2.5 × 10⁻⁴, equivalent to a Doppler velocity of 75 km/s. Adams (1925) measured ~21 km/s; modern measurements give 80.4 ± 4.8 km/s, in excellent agreement with theory and confirming Sirius B's mass.
• Neutron star. M ≈ 1.4 M_sun, R ≈ 12 km. z ≈ 0.18 (linear weak-field) or, exact Schwarzschild, z ≈ 0.21. Compact-object surface emission shows this directly in absorption-line redshifts, when atmospheric models can identify the rest-frame transitions.
• Sgr A* (supermassive black hole). M = 4.15 × 10⁶ M_sun. At star S2's perihelion (~120 AU = 1.8 × 10¹³ m), the redshift contribution is ~200 km/s in equivalent Doppler velocity — measured by GRAVITY interferometer in 2018, confirming GR. Light from material at the photon sphere (~1.5 r_s ≈ 1.8 × 10¹⁰ m) is redshifted by z ≈ 0.732.
• GPS detailed numerics. Earth potential difference between surface and orbit: ΔΦ/c² = (GM_E/r_surface − GM_E/r_orbit)/c². With r_surface = 6378 km, r_orbit = 26,578 km: ΔΦ ≈ 4.77 × 10⁷ m²/s². Divide by c²: 5.31 × 10⁻¹⁰. Per day (86,400 s): 4.58 × 10⁻⁵ s = 45.8 µs/day faster at altitude. Special-rel kinematic time dilation: −γv²/(2c²) × 86,400 ≈ −7.2 µs/day. Net: +38.6 µs/day, applied as factory pre-tune to satellite atomic clocks.

Two derivations

• Photon-as-particle (heuristic). Photon energy E = ℏω, 'gravitational mass' E/c². Climbing height h against acceleration g does work ΔE = (E/c²) gh. Conservation: ΔE/E = gh/c² = ΔΦ/c² in weak-field, so Δω/ω = −ΔΦ/c² (frequency loss). Quick and gives the right answer in weak fields, but is conceptually flawed because photons don't have rest mass and the Newtonian picture mixes frames.
• Equivalence-principle (Einstein 1911). Apply to a uniformly accelerating elevator (equivalent to rest in uniform gravity). Photon emitted at floor with frequency ω₀ travels to ceiling, taking time h/c. By that time, the ceiling has accelerated to velocity v ≈ gh/c. Doppler-shifted reception: ω_received = ω₀ × (1 − v/c) ≈ ω₀(1 − gh/c²). Rigorous in the weak-field limit, derived from equivalence alone.
• GR full calculation. Schwarzschild metric: ds² = −(1 − r_s/r)c²dt² + … For a stationary observer at radius r, proper time Δτ = Δt √(1 − r_s/r). A photon emitted with frequency ω_e (per unit emitter-proper-time) takes time Δt to reach an observer at infinity, where it is detected with frequency ω_o per unit observer-proper-time. Frequency ratio: ω_o/ω_e = √(1 − r_s/r_e), which becomes 1 + z = (1 − r_s/r_e)^(−1/2). Exact, no weak-field assumption.
• From metric to redshift. The general formula 1 + z = √(−g_00(observer)/g_00(emitter)) for static observers. Generalizes to non-static cases via the inner product of the photon 4-momentum with the observer's 4-velocity at each end. Cosmological redshift (from Universe expansion) and gravitational redshift are both captured by this formula in the appropriate metric.
• Connection to time dilation. Clocks at different gravitational potentials tick at different rates: ω_clock ∝ √(−g_00). A photon's frequency, set by the source clock, is then 'transcribed' to the observer's clock rate at the receiver. So gravitational redshift is exactly equivalent to comparing clock rates between two locations — they describe the same metric structure.

Key experiments

• Pound-Rebka 1959 / Pound-Snider 1964. Mössbauer-effect Fe-57 γ-rays at the 22.5 m Jefferson Tower, Harvard. Linewidth narrow enough that 2.46 × 10⁻¹⁵ shift is detectable. 1959: 10% precision. 1964 refinement: 1% agreement with GR.
• Gravity Probe A 1976. NASA suborbital sounding rocket carrying a hydrogen maser to 10,000 km altitude. Direct comparison with ground masers gave 70-ppm agreement with GR — for decades the most precise direct redshift test.
• Adams 1925 (Sirius B). First astrophysical detection of gravitational redshift, ~30 km/s velocity-equivalent shift (later refined). Confirmed Eddington's interpretation of the white dwarf as having a few-thousand-km radius for solar mass.
• Solar limb measurements. Hubble Space Telescope (Henley et al. 2020) used precise stellar comparisons to measure solar gravitational redshift to 1% — demonstrating modern instrumental precision matches the predicted 600 m/s effect.
• GRAVITY at S2 perihelion 2018. The S2 star orbits Sgr A* with 16-year period, perihelion at ~120 AU. The 2018 closest approach showed ~200 km/s gravitational redshift in spectroscopic observations of S2, the first detection of GR effects in a black hole's strong-field environment.
• Optical clock comparisons. Strontium and ytterbium clocks at NIST, NPL, and PTB compared via fiber links demonstrate gravitational redshift over height differences of a few meters (e.g., Chou et al. 2010 measured 30 cm differences). These tests confirm GR at 10⁻⁵ level and enable relativistic geodesy.
• Galileo satellites 2014–2018. Two Galileo-IOV satellites accidentally launched into elliptical orbits became serendipitous redshift testbeds. Their varying altitude over an orbit makes their clock rates oscillate, allowing a direct 0.86 ppm test of gravitational redshift over several years (Delva et al., Herrmann et al., 2018), the most precise space-based test currently.
• Quasar spectroscopy. Distant quasar emission lines arrive both cosmologically redshifted and gravitationally redshifted by the host galaxy's deep potential. Disentangling these requires careful modeling of line profiles, but provides supplementary GR tests at high redshift.

Common misconceptions

• "The energy lost by photons goes somewhere." No — there is no universal energy conservation in general relativity for a photon climbing a static potential. The frequency change is what one observer measures vs. another; energy is observer-dependent in curved spacetime. The 'work-against-gravity' picture is a heuristic that gives correct numbers in weak fields, not a literal energy bookkeeping. Photons traveling through a static gravitational field do not lose energy in any local sense — different observers along the path measure different frequencies.
• "Only visible light is gravitationally redshifted." Wrong — the effect is universal across electromagnetic frequencies, from radio to gamma rays, and applies equally to gravitational waves and matter waves. Pound-Rebka used 14 keV gamma rays; GPS uses microwaves; CMB observations use mm-wavelengths; pulsar timing uses radio. The fractional shift Δλ/λ is the same regardless of wavelength.
• "Tiny effect, irrelevant in practice." Wrong. Despite the small fractional shifts on Earth, gravitational redshift accumulates: GPS satellites' 38 µs/day net advance corresponds to ~10 km/day positioning error if uncorrected. The system is unusable without compensating. Likewise, atomic clock comparisons at the 10⁻¹⁸ level, financial high-frequency-trading network synchronization, and SpaceX inter-satellite ranging all depend on careful application of gravitational time dilation.
• "Gravitational redshift = cosmological redshift." Different effects, often confused. Cosmological redshift comes from the expansion of the universe (metric stretching during photon travel); gravitational redshift comes from photons being emitted in a gravitational potential. In our universe both contribute to high-redshift observations. They have similar expressions in special cases but distinct physical origins; only general relativity unifies them via the metric.
• "Photons fall in gravity like baseballs." Roughly correct for trajectory bending in weak fields, but the quantitative comparison fails for redshift. A baseball gains kinetic energy falling; a photon gains frequency falling (blueshift). The two pictures predict the same fractional shift in weak fields but for different reasons. The correct picture is metric-based: clocks at different potentials tick at different rates.
• "Doppler and gravitational redshift are different mechanisms." They have different physical origins (relative motion vs. metric tilt) but both follow from the same general formula in GR: ω_observed/ω_emitted = (k·u)_observer/(k·u)_emitter, where k is the photon 4-momentum and u is the observer 4-velocity. Doppler arises from differing u; gravitational arises from differing metric coupling. Mathematically unified.
• "You can't escape an event horizon because of redshift." Backward causality. The horizon is a one-way membrane because of the metric's signature change at r = r_s — light cones tilt to point inward. Infinite redshift from the horizon is the consequence, not the cause. Physically: the metric component (1 − r_s/r) flips sign, so the radial direction becomes timelike inside, forcing all worldlines toward smaller r.
• "Pound-Rebka measured 'photon mass'." No — there is no such thing as photon mass in standard physics (photons are massless). Pound-Rebka measured the gravitational shift of frequencies, which is consistent with massless photons in a curved metric. The 'photon weight' E/c² is a calculational convenience, not a real mass.