A while back ( https://www.physicsforums.com/threa...-transforms-like-direction-of-a-stick.730373/ ), I posted a discussion here of the following seemingly mysterious relationship between two transformation rules:(adsbygoogle = window.adsbygoogle || []).push({});

(1) If a field is purely electric in a particular frame, then its direction in other frames is given by scaling of ##\tan \theta## by ##\gamma##.

(2) If a stick has a certain orientation in its own rest frame, then its direction in other frames obeys the same rule.

The methods I knew for proving these facts were completely different, so I couldn't figure out whether there was some simple or deeper reason for the correspondence. I think I finally figured out a simple explanation.

Consider a stick with charges +q and -q fixed at the ends. The stick is nonrotating and moving inertially. In the stick's rest frame K, there is a field line originating from +q and terminating on -q which coincides with the stick.

Now consider frame K' moving in some direction relative to the stick. The field due to each charge points toward or away from its present instantaneous position in K' as well as K (a counterintuitive fact that can be proved in various ways, e.g., see Purcell section 5.6). Therefore each field, at the stick, is parallel to the stick, and we again have a field line in K' that coincides with the stick. Since the transformation of the field is independent of how the field was created, this holds for any field that is purely electric in the original frame.

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# Direction of E field transforms like direction of a stick

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