If I have two synchronized (at the station while waiting for me) clocks on a train, one at front and the other one at back, and I start to accelerate the train perpendicular to Earth's gravitational field then do those clocks go out-of-sync? Say, I accelerate at rate 0.1 m/s^2 till the speed equals 20 m/s and the distance between the clocks is 100 m.
Are those clocks out-of-sync when I travel comfortably after the acceleration?
Edit: Using gravitational time dilation for solving the problem sounds odd when I'm not accelerating along a gravitational field.
Edit 2: If I accelerated perpendicularly to a gravitational field I should have, according to the equivalence principle, experienced out-of-sync for my clocks. Does there exist any experimental evidence for the phenomenon? Theory isn't a proof.