Where does matter go after reaching singularity in uncharged black holes?

Question

I am a layman in physics and just read about black holes on the internet. I read that matter encounters geodesic incompleteness in the singularity in an uncharged black hole.

I heard an analogy of geodesic incompleteness as a straight line on a paper reach a hole on the paper, so it cannot continue. But in this analogy, isn't the straight line possible to continue into 3D (continue down the paper)? So, if matter reaches the singularity, is it possible too (to reach another dimension)?

I also heard that the matter is annihilated when reaching the singularity, does it mean it disappear from this world, and violates conservation of energy?

Answer 1

The answer is we don't know. I think it is uncontroversial to say that the prediction of singularities by GR is a sign that the theory is failing: we don't expect there actually to be singularities. But we don't have a theory which works (which does not predict singularities in other words) where GR predicts singularities -- such a theory would need to unify QM and GR -- so currently the best we can say is that we don't know.

Answer 2

Strictly speaking geodesic incompleteness doesn't mean the worldline of the particle ends at the singularity, but rather that we can't predict what happens to it. The trajectory of a freely falling particle is given by an equation called the geodesic equation:

$$ \frac{d^2x^\alpha}{d\tau^2} = -\Gamma^\alpha_{\,\,\mu\nu}\frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} $$

It's a scary looking equation but you don't need to understand all the details to see what the problem is. What happens at the singularity in a black hole is that some of the parameters $\Gamma^\alpha_{\,\,\mu\nu}$ become infinitely large and we're left with an equation that has infinity on the right hand side. Since we can't do arithmetic with infinity (because it's not a number) we have no way to calculate the trajectory of the particle at the singularity.

Incidentally much the same happens when we try to work backwards in time towards the Big Bang, and that's why it's commonly said that time started at the Big Bang. See my answer to How can something happen when time does not exist? for more on this.

Anyhow, the upshot is that GR cannot tell us what happens to matter falling into a black hole when it hits the singularity. However most of us believe that general relativity ceases to be a good description of the physics when we get close to the singularity and some form of quantum gravity theory will take over. The trouble is that we currently have no theory of quantum gravity.

Answer 3

General relativity does not tell us much. In the Schwarzschild metric the singularity is a spatial surface. I use a diagram for the truncated Schwarzschild metric with hawking radiation. The $r~=~0$ singularity is continuous with the exterior region when the black hole quantum evaporates or explodes.

Fava and others have proposed the singularity of a black hole as a condensate of tachyons. Tachyons are these oddities which are thought in some ways as particles which move faster than light. However, it is more that they are a case of a completely unstable vacuum. The invariant interval is given by $$ (mc^2)^2~=~E^2~-~(pc)^2. $$ I will set $c~=~1$. For standard relativistic motion we have $m^2~\ge~0$, where it is zero for massless particles. For the tachyon $m^2~<~0$.

We now consider the Klein-Gordon equation from the invariant interval with $E~\rightarrow~i\hbar\partial/\partial t$ and $\vec p~=~-i\hbar\nabla$ so the invariant interval is the field equation $$ \frac{\partial^2\phi}{\partial t^2}~-~\nabla^2\phi~=~m^2\phi. $$ We then have that $m^2$ is a sort of potential and for the tachyon it is negative. This means the vacuum is completely unstable. This may then be a model for the singularity of a Schwarzschild black hole. It is completely unstable and explodes in the exterior region as Hawking radiation.

Answer 4

Nobody knows what's real, and you can do worse than thinking about an analogy involving sheets of paper, so let's just fix that analogy.

as a straight line on a paper reach a hole on the paper [...] But in this analogy, isn't the straight line possible to continue into 3D (continue down the paper)?

Don't think of it as "punch a hole into a sheet of paper". Your image should be more of a very deep, in fact infinitely deep well made by the sheet of paper. It goes ever on and on (in the 3D space into which the paper is embedded).

In this analogy, your line does not have to end, it also does not have to re-appear on the other side of the well (which is infinitely deep), and it will not leave the paper either, and nothing in particular will happen to it.

On a side note, this analogy extends to give an idea how wormholes could be imagined - instead of an infinitely deep well, curve it around and attach it to another bit of your paper...

Again, I have to stress that these are all just imagery or illustrations to guide your mind into having anything to think about. It would be very weird if the complex real world would function as simply as a sheet of paper.

By the way, I like another analogy more than the paper: a reasonably big lake of water with fish with it and a hole in the middle through which the water can flow out freely. Very near the hole, the water flows faster than the fish can swim (="speed of light") with all the consequences you can imagine. Obviously, nobody knows what happens to the fish when it is finally sucked in. :)