What determines the diameter of a black hole?

Question

I am wondering what determines the diameter of a black hole. Is it just the amount of mass it has consumed? If so why does that determine the diameter?

Answer 1

A black hole in a stationary state is believed to be specified by only three parameters: its mass, charge, and angular momentum. (This is the famous “no-hair conjecture”.) These three parameters determine everything about the black hole, including where its event horizon is.

In the case of an uncharged, non-rotating hole, the event horizon is a sphere determined only by the mass $M$. In Schwarzschild coordinates, the horizon is at radial coordinate

$$r_s=\frac{2GM}{c^2},$$

giving a simple linear relationship between the “radius” and the mass. (Here $G$ is Newton’s — and Einstein’s — gravitational constant, and $c$ is the speed of light.)

For a one-solar-mass black hole, this Schwarzschild radius is about 3 kilometers.

I put “radius” in quotes because this radial coordinate is dependent on a particular choice of coordinate system. This means that different observers disagree on the radius of the horizon. However, the area of the horizon,

$$A=4\pi r_s^2=\frac{16\pi G^2M^2}{c^4}$$

turns out to be independent of which coordinate system you choose, so all observers agree on this area.