# Interior material and spacetime of a black hole described from outside

This question is mostly about general relativity. Answers regarding the quantum physics behind it are also appreciated, but first I would prefer solutions from a relativistic viewpoint.

If you fall into a black hole you reach the singularity at $\text{r}=0$ in a finite proper time. In contrast, for a stationary observer on the outside, the infalling object needs an infinite amount of coordinate time to reach the horizon at $\text{r}=\text{r}_s=2\text{G}\text{M}/ \text{c}^2$, in other words, everything that falls into a black hole gets stuck on the horizon and never crosses it.

Relative to a stationary observer outside the black hole the gravitational time dilation on the event horizon $\text{r}_s$ is infinite, so even things moving with $\text{c}$ can not cross the horizon because velocity is distance/time, and if time stands still, there is no movement.

Also, if you try to find the outer coordinate time for a given proper time of an infalling observer who has already passed the horizon, you get physically nonsense results like imaginary numbers. So from the relativistic viewpoint, if someone is inside the black hole, there should not even be any time left on the outside, if I get that right.

That means, for the observer on the outside, the black hole should be hollow with all the stuff plastered outside the horizon (on the "stretched horizon, as Susskind mentions here and here).

Image: sketch of the supposed space-grid at a given coordinate time as viewed from outside the black hole (note the empty place inside the circle in the center of gravity, which would be a hollow shell in 3D)

So my question is:

Right now, speaking from the perspective of a stationary observer on the outside of the black hole, is there even any space inside a black hole, or is the metric of spacetime warped to form a hollow shell with $\text{r}=\text{r}_s$, if you were to describe it from 1 higher dimension? At least that's what it looks like here, here and here.

Image: Susskind saying that even from the perspective of an infalling observer there might not be any spacetime behind the event horizon

Does the singularity in the center even form during the lifespan of an outer observer?

• An observer outside the black hole will never observe the formation of a singularity, the light is red shifted to oblivion so unless we are in-falling observers, the formation of the singularity will never be observed. – Horus Mar 4 '16 at 13:47
• I won't vote to close, but I think this is basically a duplicate of When does a singularity start to exist during a black hole formation?. My answer to that question explains how the horizon forms. As the star collapses external observers see an apparent horizon grow outwards from the centre of the collapsing star. We never see a true horizon, because that would take an infinite time, but for most purposes the apparent and true horizons are indistinguishable. – John Rennie Mar 4 '16 at 18:30
• The other post is related, but it does not explicitly answer the question "Right now, speaking from the perspective of a stationary observer on the outside of the black hole, is there even any space inside a black hole, or is the metric of spacetime warped to form a hollow shell with r=rs, if you were to describe it from 1 higher dimension? " How I understand your comment here, there should not be any space inside the horizon, and all the material, even the one that was initially at the center before the BH formed, should now be outside its horizon, in the system of an outside observer? – Yukterez Mar 5 '16 at 0:46

There are two kinds of black holes. Eternal black holes, and astrophysical black black.

An eternal black hole has always been around. There could still be stuff coming away from it, and at later times you just see stuff coming away from it today that was a tiny bit closer to the horizon than stuff you saw coming away from it yesterday.

An astrophysical black hole formed from the collapse of matter. And for that one, you can still see the center from before an horizon formed. Well, you can see it as well as you could ever see the center of a star.

Right now, speaking from the perspective of a stationary observer on the outside of the black hole, is there even any space inside a black hole,

For an astrophysical black hole, yes. For an eternal blacks hole, doesn't matter, they don't exist.

or is the metric of spacetime warped to form a hollow shell with $\text{r}=\text{r}_s$, if you were to describe it from 1 higher dimension?

For an eternal black hole you could cut out the spacetime beyond the event horizon and do it at the horizon or somewhere inside. Not that the event horizon is as easy to find as that one example of a spherically symmetric static stationary object alone in the universe would have you believe. In fact, being able to cut it off at the event horizon is almost entirely a tautology and completely devoid of content.

You could also use radar distance instead of that higher dimensional embedding and then the event horizon is an infinite radar distance away so it might seem mysterious.

Does the singularity in the center even form during the lifespan of an outer observer?

That's what cosmic censorship is all about. People want to say no, the only problem is the answer is yes for some parameters choices. So you can claim those parameters are "unphysical" and then say no. But it's become another tautology like the event horizon itself.

If so, what happens to the initial material that was at the center before the black hole formed?

You say "if so" but the rest of your question still makes sense. It's all there. Imagine a 3d spacetime. If it were Minkowski spacetime then surfaces of constant $z$ could be surfaces of constant time. Now we could have a circle of infalling matter. Inside the circle you have flat Minkowksi space. Because of time dilation each surface that is spaced inside the circle a certain distance corresponds to a larger spacing farther outside.

As the circle contracts that effect gets stronger. So that last hour before the center hits the horizon has surfaces of constant outer time that have an infinite number of layers stack up.

Does this center also get warped around the Schwarzschild radius and is technically outside the event horizon

No, the inside isn't outside the horizon spatially. It is before the horizon temporally. The interior of the star before the horizon forms gets time dilated to take forever to form the horizon. So you can still see the center from **before* the horizon formed.