# Size of a black hole

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?

• The usual measure of size of a BH is the Schwarzschild radius of its event horizon. – PM 2Ring Jul 16 '19 at 15:44
• Thank you, but it doesn't really answer my question. I did not asked, how do we measure the size of the black holes, I asked how is it possible that they come in different size. – Ba-Lee Jul 16 '19 at 15:46
• Indeed, it's just a comment, not an answer. There's some relevant info here, but also see Ben's answer on the Physics site here, where he says "A singularity in GR is like a piece that has been cut out of the manifold. It's not a point or point-set at all." – PM 2Ring Jul 16 '19 at 17:00
• I imagine it something like that too, that it's more like a physical tear in the fabric of spacetime. – Ba-Lee Jul 16 '19 at 19:04

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.

• Thanks, black holes makes way more sense without a singularity. My favorite theory probably is that our universe was born as a glitch, a phase transition from another universe with different physics rules and it's still lingering within the fabric of our universe, like dark matter and dark energy. Black holes might just somehow open a gateway to that old universe. If the old universe would be 2 dimensional, than we would observe it as infinitely small and tense, just like a singularity. – Ba-Lee Jul 17 '19 at 15:40
• @Ba-Lee In that case, I think you'd enjoy reading Lee Smolin's The Life of the Cosmos. BTW, in pure GR you can't observe a BH singularity even if you're inside the event horizon: the singularity is always in the future of any observer. – PM 2Ring Jul 17 '19 at 15:50
• Thanks, I'll look into that. – Ba-Lee Jul 19 '19 at 4:35
• It is a very detailed and good answer and I'm very grateful that you put in the afford writing it, but as you said it yourself nobody really knows the answer. I'll except it out of gratitude, but would like to see more theories on this one. – Ba-Lee Jul 19 '19 at 15:01
• @Ba-Lee True, we don't know exactly what happens at the core, but as my last 2 paragraphs explain, that doesn't affect the mass and size of the black hole. The spacetime distortion near the black hole is fully determined by what happened outside the event horizon. From an outside observer's perspective, all events inside the event horizon are in the distant future (how far distant depends on what coordinates you want to use). – PM 2Ring Jul 19 '19 at 17:31