I wanted to check my understanding of momentarily co-moving inertial frames, so I came up with this example:(adsbygoogle = window.adsbygoogle || []).push({});

Consider an inertial frame (with unprimed coordinates), about whose origin a clock moves in a circular path with constant speed, ## v ##. What is the time elapsed on the moving clock after one period of the motion when the time elapsed on the stationary clock is ##T ##?

Construct a series of inertial frames, each one of which instantaneously spatially coincides with the moving clock at a different point along its trajectory. Moreover, each momentarily co-moving inertial clock reads the same time as the orbiting clock at the moment the two coincide. Then the elapsed time interval on any one of the co-moving inertial clocks and the stationary clock are related by

## \Delta t' = \Delta t \sqrt{1-\frac{v^2}{c^2}} ##.

If ## T ## is the period of the orbiting clock as viewed from the stationary frame, then the time elapsed on the orbiting clock after one period is therefore

## T' = \int_0^T \sqrt{1-\frac{v^2}{c^2}} dt = T\sqrt{1-\frac{v^2}{c^2}} ##.

Is all this correct?

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# Momentarily co-moving inertial frames

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