New Transformation Equations and the Electric Field Four-vector

Transformation of Length

As in special relativity, we find a contraction of lengths in the direction of motion. To show this, we compare lengths in two reference frames in uniform relative motion. The primed frame is in uniform motion with four-velocity relative to the unprimed frame. To compare length measurements in the direction of motion, we use Eqs. (22). Since the motion is in the x-direction, we can take our length in the primed frame to be, simply, . Measurements of length are made instantaneously, so we have . Therefore, the unprimed coordinates are and

But, we are interested only in the length, or x-coordinate in the unprimed frame, so our comparison of lengths in the primed and unprimed frames gives us

Remembering now that

and, since , we have

therefore, Eq. (24) can be written as

The coordinate , in this case, is the proper length, , since all other coordinates in the primed frame are zero. The coordinate is the improper length measured by the observer in the unprimed frame and is less than the primed observer's proper length, . This represents a contraction of length in the direction of motion. The effect is reciprocal, because an observer in the primed frame finds the same contraction when performing the transformation using Eqs. (20).

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