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I've been trying to understand gravitational time dilation by considering a light-clock of length $l$ undergoing an equivalent acceleration $a$ from rest along the direction of the bouncing light pulse.

I find that the time $t$ that the light pulse takes to travel to the forward receding mirror and then back to the approaching back mirror is given by

$$t \approx \frac{2l}{c}-\frac{al^2}{c^3}$$

Is this result obviously wrong as it implies that the light clock is taking less time to tick in a gravitational field than the time $2l/c$ that it takes without any gravitational field?

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