Infinite Curvature & Hawking Radiation Explained

[Castty]

Hi,

i have a question which i can't solve myself, as i am not a student of physics:

I have heard of the infinite space curvature which occurs when matter collapses into a black hole.

On the other hand i have heard, that a black hole radiates energy away.

Now i see a contradiction: When the space curvature has a value that is infinite, how can u substract an finite value from it so that the curvature flattens out in the end? Is my understanding of infinite in that sense wrong? Thats must be, otherwise, due to hawking radiation, we would have "nacked" space curvature without mass.

Id be glad, if someone could help me out here.

Thanks and regards.

 

[Nugatory]

Hawking radiation comes from the spacetime at the event horizon, where the curvature is finite.

The infinite curvature that you’re thinking about is at the singularity at the “center” of the black hole.

 

[Castty]

Thanks for the answer but i read, that a black hole will radiate completely away in a far far far future. If that's true, then someday even the singularity must be touched by this. Thats why my questions raised.

 

[PeterDonis]

the infinite space curvature which occurs when matter collapses into a black hole

At the singularity at $r = 0$, yes, as @Nugatory has said. More precisely, this is a prediction of classical General Relativity.

i have heard, that a black hole radiates energy away

Yes, but that prediction is not a prediction of classical General Relativity. It is a prediction of quantum field theory in curved spacetime, which is the best we have done so far at combining quantum field theory with General Relativity. But that theory is still incomplete, and when we have a full quantum theory of gravity, it might also change the prediction of what happens inside a black hole so there is no longer infinite curvature at $r = 0$. We don't know for sure because we don't have a full quantum theory of gravity yet.

When the space curvature has a value that is infinite, how can u substract an finite value from it so that the curvature flattens out in the end?

In the quantum field theory in curved spacetime model that predicts that black holes radiate (more precisely, the original model that Hawking came up with in the 1970s), the curvature does not "flatten out" inside the hole. When we say the hole radiates away, what we really mean is that the region of spacetime inside the event horizon, which has a singularity at $r = 0$, ends up becoming disconnected from the region of spacetime outside the event horizon. But the region containing the singularity is still part of the overall spacetime, even though it ends up disconnected from the outside region. So the curvature at the singularity never has to "flatten out".

I should emphasize, though, that we don't know whether this original model of Hawking's is what actually happens. It's a mathematical model that may or may not reflect reality. There are other, different proposed mathematical models that make different predictions, but as I said, until we have a complete quantum theory of gravity, we won't know which, if any, of the models we have now describe what actually happens.

 

[Castty]

Thanks for the answer! That helps a lot. Exciting topic. Studied the wrong subject :)