New Transformation Equations and the Electric Field Four-vector

8. Transformation of Time

We can use similar methods to compare 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 that . Using (6.4), we get and

But since we are comparing only time measurements, we have

or, from (7.4),

The coordinate in this case, is the proper time,. The coordinate is the unprimed observer's improper measurement of the elapsed time. 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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