Gravitational time dilation

This is the quick and dirty answer to Diggy's question last night regarding why time slows down in the presence of a gravitational field.

Basically, it comes down to the constancy of the speed of light, the bending of spacetime, and the equivalence principle.  The equivalence principle states that you cannot distinguish between an observation made in the presence of a strong gravitational field and one made while accelerating.

Einstein said (and this has been pretty well verified by various observations and experiments) that the speed of light in a vacuum is a constant for all observers, no matter your frame of reference.

Bending of spacetime:  General relativity states that gravity is not a force, exactly.  Instead, what we perceive as the gravitational force (objects being attracted to more massive objects) comes about because any object with mass will tend to bend space around it.  A visual representation of this is to imagine a trampoline.  Put a baseball in the center of the trampoline and it will cause the center of the trampoline to bend toward the ground just a tiny bit.  Now, remove the baseball and put a bowling ball--the bowling ball causes the surface of the trampoline to bend toward the ground more than the baseball.  A more massive object would cause an even deeper dip, and so on.  The gravitation attraction comes because as another object enters this region of a dip in the trampoline, it tends to fall toward the bottom of the dip.

So, imagine that you are looking at a clock that is moving toward a black hole or some other region with a large gravitational field.  The dip in the trampoline of space caused by a black hole is basically infinitely deep (the size of the black hole only refers to how close you have to be before you fall in).  The clock emits a light pulse with a frequency of 1 pulse/second when it leaves you. You are very far away from the clock and not affected by the gravitational field.  As the clock enters the region of high gravity (starts to fall toward the bottom of the dip in the trampoline), the light coming off the clock seems has to travel up the dip with each pulse, and because the speed of light is always c (it can't start traveling faster), it looks to you that the clock has slowed down with each pulse--the time interval between pules increases with each pulse.  Eventually, the clock gets so far down the dip that it takes an infinitely long time for the pulse to reach you and it looks to you like the clock has stopped.  However, as far as the clock is concerned, it is emitting pulses every second and time hasn't slowed down at all.

What this has to do with the equivalence principle:  Instead of the clock moving with a constant velocity toward a black hole, imagine that you and the clock are very far away from any source of gravity.  To keep with the trampoline analogy, you and the clock are on a flat, stiff trampoline and there are no dips anywhere near you.  Now, the clock starts moving away from you, with a constantly increasing acceleration.  As the clock speeds up, the distance between you and the clock increases with each pulse, so the light has farther to travel between pulses.  Again, it looks to you as if the clock has slowed down.  Eventually, as the clock's acceleration reaches infinite, the distance between you and the clock also approaches infinity and the clock's pulse will never reach you, so it looks like the clock has stopped.  But again, as far as the clock is concerned, nothing has happened and it keeps emitting pulses every second.