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Let us continue assuming that the Principle of Equivalence is true. What else
can we deduce from it? We can carry out our reasoning in a frame where there
is uniform acceleration, and then deduce that the same thing
must happen in a uniform gravitational field.
Consider a rocket which is accelerating with a uniform acceleration of g. At the top of the rocket, there is a clock connected to a device which emits a light beam every second.
At the bottom of the rocket, there is an observer who receives these light pulses, and compares them with the ticks on her local clock. (Before the rocket blasted its engines and created the acceleration, they agreed that their clocks were identical, ticking at the same rate.)
The observer at the bottom will get the ticks at intervals a bit shorter than 1 second, because the acceleration upwards will allow them to be caught a little earlier. So the clock at the top appears to the observer at the bottom to be running a little fast.
We therefore deduce that if the Principle of Equivalence is true, clocks at lower heights in a downward pointing uniform gravitational field run slower than clock at higher heights.
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