What the Schwarzschild radius actually is
The Schwarzschild radius is the radius at which the escape velocity from a mass reaches the speed of light. Compress any object inside that radius and no signal from the interior can reach the outside: the surface at rs becomes an event horizon and the object is a black hole.
Karl Schwarzschild derived it in 1916, within weeks of Einstein publishing the field equations, as the exact vacuum solution outside a spherically symmetric non-rotating mass. The remarkable feature of the result is how little it needs: the horizon radius depends on the mass and on nothing else. Not on composition, not on temperature, not on how the mass got there. Two black holes of the same mass are identical objects.
It is worth being clear about what the number does not mean. The Schwarzschild radius of the Sun is 2.95 km, but the Sun is not a black hole and has no horizon — it is a 696,000 km ball of plasma, and the Schwarzschild solution applies only outside the matter. The 2.95 km figure tells you how small you would have to squeeze the Sun before a horizon formed. For an object that genuinely is a black hole, the same number is the physical size of the horizon.
The horizon is also not a surface in the ordinary sense. Nothing special happens locally as you cross it; an infalling observer in free fall notices no wall. What changes is causal structure: inside, every future-directed path leads to smaller r. That is why the horizon is defined by a radius rather than by a material boundary.
Why the formula is 2GM/c²
The full derivation comes from the Schwarzschild metric, where the coefficient of the time component vanishes at r = 2GM/c2. But a Newtonian argument reaches the same expression and shows where each factor comes from. Set the escape speed equal to c: ½mv² = GMm/r gives v = √(2GM/r), and putting v = c and solving for r returns r = 2GM/c². The agreement is a coincidence of the algebra rather than a justification — the Newtonian argument uses concepts that general relativity discards — but it makes the structure legible.
Mass enters linearly. Double the mass, double the horizon. That single fact drives most of the counter-intuitive behaviour further down this page: area goes as M2, volume as M3, and mean density therefore as M−2.
c2 in the denominator is why horizons are small. With c2 = 8.988×1016 m2/s2, the whole constant 2G/c2 works out to 1.4852×10−27 metres per kilogram. Every kilogram contributes about 1.5 quintillionths of a nanometre.
In practice you rarely want to multiply by 10−27. The useful working constant is the solar-mass scaling: rₛ = 2.95325 km × (M / M☉). A 10-solar-mass stellar remnant has a 29.5 km horizon; a 4.3-million-solar-mass hole like Sagittarius A* has a 1.27×107 km horizon, roughly 17 times the radius of the Sun.
Two companion formulas complete the picture. The horizon area A = 4πrₛ² is the quantity that Hawking proved can never decrease in classical processes, and that Bekenstein tied to entropy. The Hawking temperature T = ℏc³/(8πGMkₖ) is inversely proportional to mass, which is the origin of the strange thermodynamics: a black hole gets hotter as it loses energy.
Worked example: a 10-solar-mass black hole
Take a stellar-mass black hole of 10 M☉, a typical remnant of a massive star and close to the masses LIGO detects in mergers.
- Convert to kilograms. M = 10 × 1.98841×1030 = 1.98841×1031 kg.
- Apply the constant. 2G/c2 = 2 × 6.67430×10−11 ÷ 8.98755×1016 = 1.48523×10−27 m/kg.
- Multiply. rs = 1.48523×10−27 × 1.98841×1031 = 2.95325×104 m, or 29.53 km. The horizon is 59.07 km across — smaller than most metropolitan areas.
- Horizon area. A = 4π(29532.5)2 = 1.0960×1010 m2, about 10,960 km2.
- Mean density. Volume = (4/3)π(29532.5)3 = 1.0789×1014 m3, so ρ = 1.98841×1031 ÷ 1.0789×1014 = 1.843×1017 kg/m3, comparable to the density of an atomic nucleus (about 2.3×1017 kg/m3) — if anything, slightly less dense.
- Hawking temperature. Scale from the solar value: T = 6.1703×10−8 K ÷ 10 = 6.17×10−9 K. That is nine orders of magnitude colder than the 2.725 K microwave background, so in the real universe this hole absorbs far more than it emits.
- Evaporation lifetime. t = 5120πG2M3/(ℏc4) = 2.10×1070 years in an otherwise empty universe.
Check the chain: 1.48523×10−27 × 1.98841×1031 = 2.9533×104, and 4π × (2.95325×104)2 = 4π × 8.7217×108 = 1.0960×1010. Each step reproduces the line below it.
How to read the results
The radius is the only figure that scales gently. Across the 40 orders of magnitude from a kilogram to M87*, the horizon radius tracks mass exactly proportionally, so a mass ratio you can state in words gives a radius ratio you can state in the same words.
Mean density is the figure that surprises people. Because ρ = 3c6/(32πG3M2), doubling the mass quarters the mean density. A solar-mass hole averages 1.8×1019 kg/m3; M87*, at 6.5 billion solar masses, averages 0.44 kg/m3 — about a third the density of air at sea level. This is not a statement about anything you could touch, because the mass is not spread through the horizon volume; it is the ratio of two numbers that happen to scale differently.
The temperature tells you which way energy flows. The Hawking temperature crosses the 2.725 K cosmic microwave background at about 4.5×1022 kg, roughly two thirds of the mass of the Moon. Above that mass a hole is colder than the background, absorbs more than it radiates, and grows. Below it, the hole is hotter than the background and shrinks. The calculator flags which side of that line your input falls on.
Evaporation lifetimes are theoretical. The M3 scaling means a 1011 kg hole — about the mass of a mountain, with a horizon smaller than a proton — would evaporate in roughly the current age of the universe, which is why that mass is the interesting one in primordial black hole searches. Anything of stellar mass survives past 1067 years, long after the background has cooled below its temperature.
Spin is the biggest omission. Real holes rotate. For a maximally spinning Kerr black hole the outer horizon sits at GM/c2, exactly half the Schwarzschild value, so treat the number here as the non-rotating upper bound. Measured spins for stellar-mass holes span nearly the full allowed range, so the correction is not small.
Schwarzschild radius across the mass scale
| Object | Mass (kg) | Schwarzschild radius | Mean density (kg/m3) | Hawking temperature (K) |
|---|---|---|---|---|
| 1 kg | 1.000×100 | 1.485×10−27 m | 7.29×1079 | 1.227×1023 |
| The Moon | 7.342×1022 | 0.1090 mm | 1.35×1034 | 1.671 |
| Earth | 5.972×1024 | 8.870 mm | 2.04×1030 | 2.054×10−2 |
| Jupiter | 1.898×1027 | 2.819 m | 2.02×1025 | 6.464×10−5 |
| The Sun | 1.988×1030 | 2.953 km | 1.84×1019 | 6.170×10−8 |
| Stellar remnant, 10 M☉ | 1.988×1031 | 29.53 km | 1.84×1017 | 6.170×10−9 |
| Sagittarius A*, 4.3×106 M☉ | 8.550×1036 | 1.270×107 km | 9.97×105 | 1.435×10−14 |
| M87*, 6.5×109 M☉ | 1.292×1040 | 1.920×1010 km | 0.436 | 9.493×10−18 |
Masses for Sagittarius A* and M87* are the Event Horizon Telescope figures. Mean density is M divided by the Euclidean volume of a sphere of radius rₛ, a conventional bookkeeping quantity rather than a physical density.
Which constants this page uses
G = 6.67430×10−11 m3 kg−1 s−2 and ℏ = 1.054571817×10−34 J·s are the CODATA 2018 recommended values; c = 299,792,458 m/s and kB = 1.380649×10−23 J/K are exact by SI definition. The solar mass is taken as 1.98841×1030 kg, which is the IAU 2015 nominal GM☉ = 1.32712440018×1020 m3/s2 divided by G. Astronomers quote GM rather than M because GM is measured to nine significant figures while G alone is known to only about five — which is why rs for the Sun is far better determined than the solar mass itself.
Mistakes and limits worth knowing
- Confusing radius with diameter. Published "size of the black hole" figures are sometimes diameters. This calculator returns the radius; double it before comparing.
- Assuming the horizon is what a telescope sees. The bright ring in Event Horizon Telescope images is the photon ring at about 2.6 rs for a non-spinning hole, not the horizon itself. The dark central region is roughly 2.6 times larger in radius than rs.
- Ignoring spin. The Kerr outer horizon shrinks from 2GM/c2 toward GM/c2 as spin rises to the maximum. Use the Schwarzschild figure as the non-rotating case.
- Treating mean density as a physical density. It is mass divided by the flat-space volume of a sphere of radius rs. Spacetime inside the horizon is not flat and that volume is not a proper volume.
- Quoting evaporation lifetimes as forecasts. The lifetime formula assumes an isolated hole radiating into empty space. For anything above about 4.5×1022 kg the microwave background is warmer than the hole today, so the hole grows instead.
- Applying it to ordinary stars. The Schwarzschild solution holds in vacuum, outside the mass. For the Sun the exterior solution applies only above 696,000 km, and there is no horizon anywhere.
Where this sits among related calculations
The Schwarzschild radius is the entry point to a family of horizon-scale quantities. The photon sphere at 1.5 rs is the radius where light orbits in a circle; the innermost stable circular orbit at 3 rs marks where an accretion disc must end, which sets the efficiency of accretion power. The shadow that a distant observer sees has an apparent radius of √27 GM/c2 ≈ 2.6 rs. All three are fixed multiples of the number this page computes.
For ordinary orbital work outside the horizon you want Kepler's third law, which relates orbital period to semi-major axis and total mass, or the geostationary orbit altitude calculator for the same relation applied to a synchronous orbit. Surface gravity for a body you can stand on is a different formula entirely — see the planetary surface gravity calculator. If you are converting the enormous distances involved into travel times, the light travel time calculator handles light years, parsecs and astronomical units in one place, and the angular size calculator converts a physical horizon size at a known distance into the microarcseconds an interferometer has to resolve.
Historically, the idea predates relativity. John Michell in 1783 and Pierre-Simon Laplace in 1796 both noted that a sufficiently massive, sufficiently compact body would have an escape velocity above the speed of light and would appear dark. Their Newtonian argument gives the same 2GM/c2, but for the wrong reasons and with a completely different physical picture: light was imagined to slow, rise and fall back, rather than to be trapped by causal structure.