If the core left by a supernova has more than about 3 solar masses, not even the neutron pressure that holds up a neutron star can support it, and nothing known can stop the collapse. The result is a black hole. The idea grows out of escape speed: the more mass squeezed into a smaller ball, the faster you must go to leave its surface.
Why it works
Escape speed is $v_{\text{esc}} = \sqrt{2GM/r}$. Squeeze a mass $M$ small enough and the escape speed reaches the speed of light $c$. Setting $\sqrt{2GM/r} = c$ gives $$R_s = \frac{2GM}{c^2},$$ the Schwarzschild radius. Einstein's general relativity gives exactly this radius for the event horizon: nothing that crosses it, not even light, ever comes back out. For the Sun's mass, $R_s$ is about $3$ km; it grows in proportion to mass, about $2.95$ km per solar mass. Far outside the horizon, though, a black hole's gravity is just $\frac{GM}{r^2}$, the same as any star of that mass.
Key idea
A black hole's event horizon has radius $R_s = \dfrac{2GM}{c^2}$, about $3$ km per solar mass. Inside it, nothing escapes. Far away, a black hole pulls exactly like any other object of the same mass.
Worked examples
Worked example: A stellar black hole
Find the event-horizon radius of a black hole of $10$ solar masses ($1.99 \times 10^{31}$ kg).
- $R_s = \frac{2GM}{c^2} = \frac{2 \times 6.674 \times 10^{-11} \times 1.99 \times 10^{31}}{(3.00 \times 10^{8})^2}$.
- $R_s = 2.95 \times 10^{4}$ m $\approx 30$ km: ten Suns' worth of mass inside a city-sized sphere.
Answer: About $30$ km.
Worked example: The giant at our galaxy's center
Sagittarius A* has about $4.3 \times 10^{6}$ solar masses. How big is its event horizon, in AU?
- $R_s \approx 2.95$ km $\times 4.3 \times 10^{6} = 1.27 \times 10^{7}$ km.
- In AU: $\frac{1.27 \times 10^{7}}{1.496 \times 10^{8}} = 0.085$ AU, smaller than Mercury's orbit.
- Its mass was found from stars orbiting it, using orbits as scales.
Answer: About $0.085$ AU.
Common mistake
Black holes are not cosmic vacuum cleaners. Outside the horizon, gravity follows the same $GM/r^2$ as always. If the Sun were replaced by a black hole of the same mass, Earth would keep orbiting exactly as it does (in the dark). Only things that come very close, within a few times $R_s$, are in danger. Check: stars orbit Sagittarius A* on stable ellipses, year after year.
Evidence for black holes: X-ray-bright gas swirling into unseen companions of several to about 20 solar masses (such as Cygnus X-1), stars orbiting Sagittarius A*, the gravitational waves from merging black holes first detected by LIGO in 2015, and the Event Horizon Telescope's images of the dark shadows of M87* (2019) and Sagittarius A* (2022).