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This observer-independent quantity, the change in reading on the stop-watch, can be computed by some formulae from observations made in a different frame in which the stop-watch may be (and usually is) moving. So we'd assign coordinates (usually t,x,y,z) to every point along the clock's path through space-time, and we can relate the changes in the clock reading to the changes in the coordinates via some mathematical formula (I'll avoid the details for now) involving the changes in coordinates.

Wiki has a writeup of proper time and the necessary formulas, <<link>> but I doubt it's the best place to start learning about the concept.

If you grasp the principle of computing the proper time in SR, you can read about the relatively minor modifications (involving the metric coefficients) that are needed to compute proper time in GR. If you take these formula as a given, using the Schwarzschild metric coefficients, you can compute the proper time interval that experienced by a stop-watch falling into a black hole - unfortunately, you need also to either know or be able to figure out which path a free-falling clock takes. You can find some paths that are not free-fall that do take an infinite time, so it's not a trivial question.

Most GR textbooks will do some version of this problem, though, and you can fact-check the result that the time on the clock is finite.