Notes on Relativity and Cosmology - Physics Department, UCSB

246 CHAPTER 9. BLACK HOLES
finite proper time, this means that any two such geodesics move infinitely far
apart in a finite proper time. It follows that the relative acceleration (a.k.a.
the gravitational tidal force) diverges at the singularity. (This means that the
spacetime curvature also becomes infinite.) Said differently, it would take an
infinite proper acceleration acting on the objects to make them follow (nongeodesic)
paths that remain a finite distance apart. Physically, this means that
it requires an infinite force to keep any object from being ripped to shreds near
the black hole singularity.
9.3.3 Beyond the Singularity?
Another favorite question is “what happens beyond (after!) the singularity?”
The answer is not at all clear. The point is that just as Newtonian physics
is not valid at large velocities and as special relativity is valid only for very
weak spacetime curvatures, we similarly expect General Relativity to be an
incomplete description of physics in the realm where curvatures become truly
enormous. This means that all we can really say is that a region of spacetime
forms where the theory we are using (General Relativity) can no longer be
counted on to correctly predict what happens.
The main reason to expect that General Relativity is incomplete comes from
another part of physics called quantum mechanics. Quantum mechanical effects
should become important when the spacetime becomes very highly curved.
Roughly speaking, you can see this from the fact that when the curvature is
strong local inertial frames are valid only over very tiny regions and from the
fact the quantum mechanics is always important in understanding how very
small things work. Unfortunately, no one yet understands just how quantum
mechanics and gravity work together. We say that we are searching for a theory
of “quantum gravity.” It is a very active area of research that has led to a
number of ideas, but as yet has no definitive answers. This is in fact the area
of my own research.
Just to give an idea of the range of possible answers to what happens at a
black hole singularity, it may be that the idea of spacetime simply ceases to be
meaningful there. As a result, the concept of time itself may also cease to be
meaningful, and there may simply be no way to properly ask a question like
“What happens after the black hole singularity?” Many apparently paradoxical
questions in physics are in fact disposed of in just this way (as in the question
‘which is really longer, the train or the tunnel?’). In any case, one expects that
the region near a black hole singularity will be a very strange place where the
laws of physics act in entirely unfamiliar ways.
9.3.4 The rest of the diagram and dynamical holes
There still remains one region of the diagram (the ‘past interior’) about which
we have said little. Recall that the Schwarzschild metric is time symmetric
(under t → −t). As a result, the diagram should have a top/bottom symmetry,