Gravitational Redshift

What gravitational redshift is

Send a photon straight up, away from a massive body. Nothing about it slows down — light always travels at c. Instead, the photon pays its toll in wavelength: it arrives at the top redder than it left the bottom. Its frequency drops, its energy E = hf drops with it, and the spectral lines printed in the starlight shift toward longer wavelengths. That is gravitational redshift.

Crucially, this is not the Doppler effect. The source and the observer can be sitting perfectly still relative to each other. The shift comes entirely from the difference in gravitational potential between where the light is emitted and where it is detected. Deeper well, more energy lost on the climb, redder the arriving light.

The exact general-relativistic factor for a static, spherical mass (the Schwarzschild metric) is

1 + z = 1 / √(1 − r_s/R),   with   r_s = 2GM/c².

When the radius R is much larger than the Schwarzschild radius r_s — true for any ordinary star — this collapses to the famous weak-field approximation z ≈ GM/(Rc²). As R approaches r_s, the denominator goes to zero and the redshift diverges: light emitted at a black hole's event horizon is redshifted to infinite wavelength, which is one way of saying it never escapes at all.

Two pictures: time dilation and the equivalence principle

There are two equivalent ways to see why light loses energy climbing out of gravity, and both are worth holding in your head at once.

Picture one — gravitational time dilation. A clock deep in a potential well runs slow compared with a clock far away. The Schwarzschild factor √(1 − 2GM/Rc²) that slows the deep clock is exactly the factor that lowers the received frequency. A wave is just a clock: count the crests. If the emitter's "clock" ticks slow, it produces crests at a slower rate as measured by the distant observer, and slower crests means longer wavelength. Redshift and time dilation are the same statement read two different ways.

Picture two — the equivalence principle. Einstein's 1907 insight was that a uniformly accelerating elevator in deep space is indistinguishable from sitting still in a gravitational field. Fire a light beam from the floor of an upward-accelerating elevator toward the ceiling. By the time the light arrives, the ceiling has sped up, so it sees the light Doppler-redshifted. Swap the acceleration for gravity and you have predicted gravitational redshift before ever writing down the field equations. This is why the effect was derived in 1907, almost a decade before the full theory of general relativity arrived in 1915–1916.

How big is it — the numbers that matter

The whole story is set by the single dimensionless number GM/(Rc²), the "compactness" of the emitting surface. The more mass you cram into a smaller radius, the deeper the well and the larger the shift. The table below walks from the everyday to the extreme. The velocity column is the equivalent Doppler velocity, cz, which is how astronomers usually quote the shift.

Object	Mass	Radius	z = GM/Rc²	Equivalent velocity
Earth surface	1 M⊕	6,371 km	7×10⁻¹⁰	0.21 m/s
Pound–Rebka tower	(22.5 m of height)	—	2.46×10⁻¹⁵	0.74 µm/s
The Sun	1 M☉	696,000 km	2.12×10⁻⁶	636 m/s
White dwarf (Sirius B)	1.02 M☉	~5,800 km	~2.6×10⁻⁴	~80 km/s
Neutron star	1.4 M☉	~12 km	~0.2–0.4	(strong field)
Black hole horizon	any	rs	∞	light never escapes

Two things jump out. First, the effect is absurdly small in everyday gravity — the entire 22.5-metre Harvard tower changes a gamma ray's frequency by about two parts in a thousand trillion. Second, it explodes once gravity becomes strong: a white dwarf, just an Earth-sized ball of degenerate matter, produces a shift a hundred times the Sun's, and a neutron star's surface light is reddened by tens of percent, which is precisely why measuring a neutron star's redshift pins down its mass-to-radius ratio and so its equation of state.

The white-dwarf laboratory

White dwarfs are the historical workhorse of this measurement. A white dwarf supports itself against gravity with electron degeneracy pressure rather than fusion, so it can pack a star's worth of mass into a radius the size of Earth. That gives it a surface compactness about 100× the Sun's — large enough that the redshift stands out from the star's ordinary line-of-sight motion.

Walter Adams attempted the measurement on Sirius B in 1925 and reported a shift consistent with general relativity, though his value was badly contaminated by scattered light from the dazzling primary star Sirius A, eight thousand times brighter and only a few arcseconds away. The clean modern measurement waited until the Hubble Space Telescope resolved Sirius B's spectrum in 2005, yielding 80.6 ± 4.8 km/s — beautifully consistent with a 1.02 M☉, 0.0084 R☉ white dwarf. The nearby, more isolated 40 Eridani B gave Daniel Popper a cleaner early value of about 21 km/s back in 1954.

Tests, from Earth towers to the Galactic centre

• Pound–Rebka (1959). The decisive lab test. Iron-57's 14.4 keV gamma ray, made razor-sharp by the recoil-free Mössbauer effect, was sent up a 22.5 m tower at Harvard. The predicted 2.46×10⁻¹⁵ shift was confirmed to ~10%; the 1965 Pound–Snider follow-up reached 1%.
• Gravity Probe A (1976). A hydrogen-maser clock flown to 10,000 km on a rocket confirmed the prediction to 1.4×10⁻⁴ — still one of the most precise tests.
• Galileo satellites 5 & 6 (2018). Two GPS-style atomic clocks accidentally launched into elliptical orbits became a gift: their periodic altitude change modulated the redshift, confirming GR to 2.5×10⁻⁵.
• S2 at the Galactic centre (2018–2020). The GRAVITY collaboration tracked the star S2 as it whipped past the 4-million-solar-mass black hole Sgr A* at 7,650 km/s; the gravitational redshift in its spectrum matched GR and ruled out pure Newtonian gravity at high confidence.
• GPS, every day. Satellite clocks run fast by ~45 µs/day from the weaker field at 20,200 km altitude (partly offset by ~7 µs/day of special-relativistic slowing). Without the correction, positions would drift ~10 km per day.

Common misconceptions

• "Gravity slows the light down." No — the photon always moves at c. Only its wavelength and frequency change.
• "It's a Doppler shift." No — source and observer can be at rest. The shift comes from the potential difference, not relative velocity.
• "It's the same as cosmological redshift." No — cosmological redshift is space itself expanding in transit; gravitational redshift is climbing a static well.
• "The photon decelerates and re-accelerates." No — energy is lost monotonically as it climbs, recovered as blueshift only if it falls back down.
• "You need general relativity to derive it." No — the equivalence principle gives the leading term in 1907, before GR existed.
• "It only matters near black holes." No — GPS, atomic clocks and solar spectroscopy all confront it routinely.