Question: In response to my previous question; "Will you ever stop falling into a black hole?" - Wouldn't the 103 seconds relative to the person falling actually be infinity to the person watching. Therefore doesn't this mean that the person falling in would only reach the singularity if the universe and the black hole would last for infinity?

Cesar Lopez-Monsalvo answered on 17 Jun 2011:

This is an excellent point. In fact, that is a problem most university students taking the General Relativity course need to show.

The answer is as follows. Say that it takes 100 seconds for the person falling to reach the horizon, so that exactly the 100Th flash will take an infinite amount of time to reach the observer (the other person). This does not imply that the universe and the black hole last forever. This is simply exhibiting one of the key features of relativity, that is, the difference between “proper time” and “coordinate time”. If you are the observer, then you stay away from the black hole and you use a “set of coordinates” to describe the situation. In those coordinates, the flashes seem to arrive more spaced each time. However, the truly meaningful time here is not the one you measure (the separation between two consecutive flashes), but the one the actual person falling reads from its own watch. For him, every flash happens at a rate of a second. Now, the person falling will not notice anything while emitting the 100Th flash, when he crosses the horizon. From his point of view there is nothing special there, but for the outside observer that is the last boundary it can actually see (hence the horizon name). The rest of the flashes, i.e. 101, 102 and 103, will never reach the observer because those happen INSIDE the black hole, so they cannot escape.

It is a matter of calculating the “proper time” of the person falling into the black hole to see that it will reach the centre of the black hole in a finite amount of time, say at the 104Th second. Once he reaches that point, he “hits” the singularity and basically that is the end. We call that an “end point”, for obvious reasons.

The fact that for a single observer it seems that falling into a black hole takes an infinite amount of time, does not mean that it actually does. That is just telling us that the means of translating your clock readings with his simply fails around the horizon. However, proper time is a quantity that is “invariant” for every observer and you can calculate it, so even if you never see it happening, you would know that 104 seconds after you two separated, your fellow’s “worldline” has come to an end.