Q. How is it possible for time to change inside a black hole?
A. In general relativity, time and space are a set of variables that can be used to parameterize the geometry of space-time and the kinds of geodesics that are possible. But they are not the only kinds of variables that form a set of four coordinates that "span" the dimensionality of space-time. In probing the mathematics of black holes, physicists have discovered other sets of coordinates that are even better. For example, the event horizon appears in the mathematics as a "coordinate singularity" if you use the coordinate set (x, y, z, t) or (t, r, theta, phi), but if you use the "Kruskal-Szekeres" coordinates, it vanishes completely.
There is only one true singularity in a non-rotating Schwartzschild black hole solution, and that is the one at r=0, at the event horizon, the curvature of space is non-infinite. That means that coordinate singularities are not real singularities and can be mathematically transformed away. Now, if you study what happens to the Kruskal-Szekeres coordinate system as you pass inside the black hole event horizon, nothing unusual happens. But in the conventional Newtonian (x, y, z, t) system, if you look at the formula for the so-called "metric," you see that the space and time parts reverse themselves. This means that just inside the horizon, space becomes time-like and time becomes space-like. What we call time does change to something with the mathematical properties we have normally associated with space.
This sounds pretty bizarre, but consider that we are using a non-proper coordinate system in the first place. It is possible that time changes somehow inside a black hole, but that is an experiment we will never be able to test because we can never receive information from inside a black hole.
Monday, June 19, 2017
From the question and answer page at NASA