New Transformation Equations and the Electric Field Four-vector

7. Transformation of Length

As in special relativity, we find a contraction of lengths in the direction of motion. To show this, we compare lengths measured by observers 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 (6.4). Since the motion is in the x-direction, we can take our lengths in the primed frame to be, simply, and . Measurements of length are made instantaneously, so we have . Therefore, from (6.4), 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 from (5.5) that

and since , in this case, we have

therefore, (7.2) 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 we find, by using (6.2), that an observer in the primed frame measures the same contraction..

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