Gravitational Redshift Calculator

Calculate how much light is gravitationally redshifted escaping from radius R around a mass M, z = (1-r_s/R)^(-1/2) - 1, with the observed wavelength, a naive equivalent velocity for comparison, a climbing-photon wavelength-stretch diagram, and a redshift-versus-radius chart that diverges at the horizon.

Gravitational redshift calculator

Light climbing out of a mass's gravity well loses energy on the way out — pure general relativity, no motion required: z = (1 − r_s/R)^(−1/2) − 1, r_s = 2GM/c². This assumes a spherical, non-rotating mass and a static emitter/observer. It's only defined for R > r_s — at or inside the Schwarzschild radius, there's no escaping light left for this formula to describe.

Mass (M)

Emission radius (R)

Emitted wavelength (λ) — optional

z ≈ 2.1217 × 10⁻⁶

R = 2.357 × 10⁵× the Schwarzschild radius (2.953 km) · λ_obs ≈ 550 nm

If this redshift were (wrongly) attributed to velocity alone, it would suggest ≈ 0.63606 km/s — but no motion is involved here at all.

A photon's wavelength stretches continuously as it climbs outward — shown here to the real, computed redshift factor (1+z ≈ 1) at each point along the way, on a log scale of distance from the surface.

z diverges to infinity as R → r_s (left edge) — there is no finite redshift for light escaping from arbitrarily close to the horizon, let alone from at or inside it.

Light doesn’t need to be moving away from anything to redshift. Climbing straight up and out of a gravity well costs energy — the same way throwing a ball upward costs kinetic energy — and for a photon, losing energy means stretching to a longer wavelength. No velocity, no expanding space, just gravity itself doing the redshifting. It’s one of general relativity’s cleanest, earliest-tested predictions.

The formula

z = \left(1 - \frac{r_s}{R}\right)^{-1/2} - 1, \qquad r_s = \frac{2GM}{c^2}

R is the radius where the light is emitted, and r_s is the Schwarzschild radius of the mass M. This describes a spherical, non-rotating, uncharged mass with a static emitter and a static observer arbitrarily far away — a spinning (Kerr) mass, an orbiting emitter, or a nearby (rather than infinitely distant) observer would each need additional terms this simple relation doesn’t include.

From negligible to enormous

The Sun’s surface: z ≈ 2.1 × 10⁻⁶ — the effect predicted for sunlight, first tentatively confirmed in the 1960s once instruments became precise enough, and now routinely measured.
A white dwarf: a solar mass squeezed into roughly Earth’s size gives z of a few×10⁻⁴ — corresponding to a naive “equivalent velocity” of order tens of km/s. This was actually observed decades before general relativity’s other predictions were tested with high precision, since white dwarfs made it measurable with 1920s-era spectroscopy.
A neutron star: with a solar-mass-plus object packed into ~10 km, z reaches tens of percent — light climbing away loses a genuinely substantial fraction of its energy.
Near a black hole’s horizon: z grows without bound as the emission radius approaches r_s. At the photon sphere (R = 1.5 r_s, the radius where light itself can orbit), z = √3 − 1 ≈ 0.732 exactly.

Why R must be strictly greater than r_s

This calculator refuses to compute anything for R ≤ r_s, and that’s deliberate, not a limitation to work around. At the Schwarzschild radius itself, every outward light-cone has folded over to point inward — there’s no “escaping light” left for a redshift formula to describe. Asking “what’s the redshift of light emitted at the event horizon” is a bit like asking how fast something needs to travel to escape from where escape velocity already equals the speed of light: the premise of the question has already broken down.

The “naive equivalent velocity”

Because both Doppler shifts and gravitational redshift shift wavelengths the same directional way, it’s tempting to translate a gravitational z into “the velocity that would produce this redshift.” This calculator does compute that number for context — but it’s worth being explicit that no velocity is actually involved in a purely gravitational redshift. The comparison is useful for intuition (a white dwarf’s redshift really is comparable in size to modest stellar radial velocities), not a claim about motion.

Reading the visuals

The climbing-photon diagram shows a wave whose local wavelength is the actual, computed redshift factor accumulated between the surface and each point along its outward path — not just an illustrative before/after jump. For the Sun, nearly all the stretching happens within the first stellar radius or two; for a near-horizon case, it keeps visibly growing much farther out.
The z-versus-radius chart plots redshift against distance in units of r_s on a log-log scale, where the divergence toward the horizon is directly visible — landmarks from the Sun to a neutron star to just outside a black hole show the same relation spanning many orders of magnitude.