Black Holes and Einstein Field Equations

Question

I recently watched a video where Michio Kaku argues that there must be something fundamentally wrong with the theory of General Relativity because it gives time t=0 as a solution at the centre of a Black Hole. He calls this non-sense and also says that Einstein's Field Equations break down when applied to Black Holes. What exactly does he mean by this?

As far as I've read, Einstein's Field Equations govern relativistic gravity. Why do I also read in multiple articles that they aren't good for Black Holes?

Answer 1

General relativity is a theory that tells us the geometry of spacetime. However it predicts that in some situations the geometry of spacetime is undefined - this happens when we get a singularity. There is a singularity at the centre of a black hole, and the Big Bang was also a singularity.

So we have the odd situation that the theory of general relativity predicts that there are cases where general relativity doesn't work (because a singularity occurs). Because of this it is widely believed that general relativity is an effective theory i.e. it is an approximation to some (even!) more complicated theory. It's widely believed that this deeper theory will be some form of quantum gravity. However we don't know what that new theory is, though there are several contenders.

But this doesn't mean that general relativity is nonsense. It just means there are circumstances in which GR will give a nonsensical description (i.e. singularities), and that happens when we've tried to push general relativity beyond its limits. If we stay away from singularities we expect GR to work eseentially perfectly.

The vast majority of our physical theories are effective theories - indeed I can't think of a single exception. All theories are based on some approximations, and when those approximations break down the theory ceases to be useful. For example Newtonian mechanics is an approximation to special relativity, and special relativity is an approximation to general relativity. However Newtonian mechanics is still extremely useful in everyday life (which is why it's taught at school).