Is time dilation a form of dimensional compactification?
As a probe approaches a black hole, toward a point on the equator of the event horizon, does general relativity predict that the time dimension is compactified like the extra dimensions of string theory? Will this compactification continue until the uncertainty principle becomes significant?
Is time dilation a form of dimensional compactification?
As far as I know, no, it isn't. Compactified dimensions have a finite extent, and as used in string theory they're typically periodic. For example, the Earth's surface has two compactified dimensions: you can't travel more than a certain distance in any one direction in that two-dimensional surface without returning to where you started. But that's not true of the time dimension, which is generally considered to be infinite. There is a sense in which time ends at a black hole singularity, much like the set of negative numbers ends at zero, but still, it extends infinitely far back in the other direction, and infinitely far forward outside the black hole.
