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Physics, Astronomy & Cosmology / Re: Does time move slower at the equator than at the poles?
« on: 05/11/2016 17:06:38 »The shape of the Earth is determined by the balancing of these two potentials so that when combined, all points of the Earth's surface at sea level are at an equal potential. The clocks can't tell the difference between the two potentials (Equivalence principle), so they will run the at the same rate.
Quite right. Centripetal force cancels oblateness. But an equatorial observer travels through space at a significant speed. A polar observer just spins in place. It is therefore tempting to invoke special relativity, but that's a mistake because the observers are actually in the same (rotating) reference frame.
Clocks in the same inertial frame will always agree, but this is not the case with all clocks in an non-inertial frame such as a rotating one. Let's take an example. The equation of time dilation for an orbiting clock is
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If we put a clock at geosynchronous orbit over the equator, r will be equal to 42164 km And the time dilation factor will be 0.9999999998424.
The time dilation for a clock resting on the Equator will be 0.9999999993042. The clock in orbit will run 1.0000000005382 times faster than the surface clock.
Now since these clocks share the same rotating frame, then if we assume that clocks in the same rotating frame run at the same rate, then the only difference between these two clock rates should be due to gravitational time dilation. But if you compare the clock rates by just this factor, it turns out that the orbital clock should run 1.0000000005895 times faster than the surface clock. A small difference from that gotten above, but a definite difference. This difference is due to the two clocks being at different radii from the axis of a rotating frame.
Or we can look at it like this: Set up a line of clocks in a string such that one clock is at the center of the rotating frame with others at various distances from the axis of rotation. Add another clock on the axis that does not share in the rotation of the other frames. According to this clock, all the clocks in the rotating frame are moving at different speeds relative to it and ticking at different rates, with the the one sharing the axis with it showing no time dilation. But this is in contradiction to the idea that all the clocks in the rotating frame run at the same rate.