New Transformation Equations and the Electric Field Four-vector

Transformation of Time

We can use similar methods to find the comparison of elapsed times in two reference frames. The frames are in uniform motion with four-velocity relative to each other. An observer at rest in the primed frame makes his time measurement, , at the same place in his frame, so . Using Eqs. (22), we have and

But since we are comparing only time measurements, we have

The coordinate in this case, is the proper time,. The coordinate is the unprimed observer's improper measurement of the elapsed time in the primed frame. This constitutes a dilation of time. In other words, an observer in the unprimed frame says that the rate at which clocks run in the primed frame is slower than in his own frame. Again, the effect is reciprocal.

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