Size of a black hole

Question

If there is no matter inside any of the black holes and all of them have a singularity in the center of them, which is infinitely small and infinitely dense, then how can they differ in size and weight?

Answer 1

The size of a black hole is determined by its energy and angular momentum. For a non-spinning black hole, there is a simple linear relationship between the black hole's mass and its Schwarzschild radius:

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

where $r_s$ is the Schwarzschild radius, $G$ is the universal gravitational constant, $c$ is the speed of light, and $M$ is the black hole's mass. The mass of the black hole comes from all the matter and energy that went into creating the black hole, as well as any additional matter & energy that's fallen into it afterwards.

Strictly speaking, the Schwarzschild solution is for a universe containing nothing but a single eternal black hole. But although it's unphysical it's still a useful solution when we want to model spacetime with spherical spatial symmetry (including planets & stars, not just black holes).

For a rotating black hole, the situation is more complex, and there are several horizons. Please see Wikipedia's article on the Kerr metric for details.

We cannot say exactly what happens at the core of a black hole. As discussed in this answer by Florin, we need a theory that unites General Relativity and Quantum mechanics to answer such questions.

A pure GR black hole has a mathematical singularity at its core, but most astrophysicists believe that's unphysical, and that a proper quantum gravity theory will eliminate that singularity. However, it's likely that the core of a black hole is still very small, since the quantum gravity corrections probably don't kick in until the size gets smaller than an atom. And of course even with a quantum gravity we still won't ever be able to observe black hole cores to validate the theory.

So we cannot currently say exactly what happens to energy and matter once it reaches the core of a black hole. But we do know that once anything crosses the event horizon of a black hole it can no longer affect the universe outside the event horizon, and that anything inside the event horizon must rapidly (in its own proper time) fall towards the core of the black hole, and that no known force can prevent that process.

The gravitational field of a black hole is sometimes described as a "fossil field". All matter & energy falling into the black hole modifies the spacetime curvature as it approaches the event horizon. And once it crosses the event horizon it can no longer change the spacetime curvature outside the horizon, so those curvature changes are preserved (until something else comes along to add its own curvature changes).

So it doesn't really matter what happens at the core of the black hole, since the exact nature of the black hole core doesn't affect what's happening outside the event horizon.