The size of a black hole
In 1916, only weeks after Einstein published general relativity, Karl Schwarzschild found an exact solution to its equations for a spherical mass. It contains a special distance, now called the Schwarzschild radius:
rs = 2 × G × M ÷ c²
where G is the gravitational constant, M the mass and c the speed of light. If a mass is compressed inside this radius, nothing can escape it, not even light. The sphere of radius rs is the black hole’s event horizon.
Worked example
For one solar mass:
| Result | Value |
|---|---|
| Schwarzschild radius | 2.954 km |
| Event horizon diameter | 5.908 km |
| Average density inside, compared with an atomic nucleus | about 80 times (1.84 × 10¹⁹ kg/m³) |
| Hawking temperature | 6.17 × 10⁻⁸ K |
The whole mass of the Sun would fit in a sphere smaller than a small town.
Reference values
| Object | Mass | Schwarzschild radius |
|---|---|---|
| Earth | 5.97 × 10²⁴ kg | 8.9 mm |
| Sun | 1 solar mass | 2.95 km |
| Typical stellar black hole | 10 solar masses | 29.5 km |
| Sagittarius A* (Milky Way center) | 4.3 million solar masses | 12.7 million km |
| M87* (first black hole imaged) | 6.5 billion solar masses | about 128 times the Earth–Sun distance |
The radius grows in direct proportion to the mass, so the average density inside the horizon falls with the square of the mass. A supermassive black hole like M87* has an average density lower than air.
Hawking radiation
In 1974 Stephen Hawking showed that quantum effects make black holes glow faintly, as if they had a temperature inversely proportional to their mass. A solar-mass black hole is far colder than the 2.7 K cosmic microwave background, so in practice it absorbs more than it emits. A solar-mass black hole would need about 2 × 10⁶⁷ years to evaporate, and the time grows with the cube of the mass; only tiny black holes, if they exist, would evaporate on observable timescales.
Things to keep in mind
- The formula is for a non-rotating, uncharged black hole. Rotating (Kerr) black holes have a smaller horizon, down to half this radius for the fastest spin.
- Ordinary objects are far larger than their Schwarzschild radius, so the formula only says how much they would have to be compressed.
- Near the horizon, time slows dramatically for an outside observer, and tidal forces near small black holes would stretch any object apart.