Proving the Absence of Length Contraction Perpendicular to Velocity

When teaching conceptual physics, students almost always ask why length contraction only occurs parallel to velocity and not perpendicular to it. That’s a meaty conceptual question and one that always leaves me looking for a convincing, non mathematical explanation. Earlier this semester, I finally found what I think is the best one I’ve ever heard by user “cepheid” in this thread on PhysicsForums.com and decided to present it in class. It is a rather simple thought experiment.

Consider a train with a window, moving alongside a smooth vertical wall. In the frame of the wall, there is a blue marker mounted precisely at the window’s center such that as the train moves, the marker leaves a horizontal blue line on the smooth wall. An observer in the moving train is at a different, but identical, window with a red marker mounted precisely at this window’s center.

Now, let’s assume the existence of vertical length contraction in a frame in which a “vertical stick” is observed moving, with the “stick” marking the height of the windows’ centers from the tracks. The observer in the train will see the “stick” moving (it’s attached to the wall) and should thus see its height, marked by the red line made on the wall, contracted. The observer on the tracks will see the “stick” moving (it’s attached to the train) and should thus see its height, marked by the blue line made on the wall, contracted. One observer will report that the red line on the wall must be above the blue line. The other observer will report that the blue line on the wall must be above the red line. While both observers agree the red and blue lines are parallel, they disagree on which one is above the other. This is logically impossible, and we have a apparent paradox. The resolution is that both the red and blue lines are the same height in both frames, leading to the conclusion that vertical length contraction doesn’t occur.

We actually acted this out in class and I was really happy with the results. Unfortunately, I don’t think any of the students could recreate the thought experiment from scratch, but there’s always next time. I still think this is a good explanation for the absence of vertical length contraction and will certainly use it again in the future.

As always, feedback and comments are welcome.

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