# Time dilation derived from first principles vs. “width contraction”

Wikipedia describes the usual derivation of time dilation with a moving light clock: In this and similar derivations nothing is ever said about a possible "width contraction" and why it does not happen. In principle, the clock would not run slower, if it would undergo sufficient width contraction from $$L$$ to $$L'=\alpha L$$ with $$\alpha<1$$ such that the path $$D=\sqrt{(1/2 v\Delta t')^2 + L'^2}$$ is again equal to $$L$$.

Can width contraction be ruled out theoretically or does it need measurement and observation?

• The length contraction will be only along the direction of relative motion. – UKH Apr 14 at 7:00

## 1 Answer

Can width contraction be ruled out theoretically or does it need measurement and observation?

You're right that possible width contraction must be excluded. There is a simple theoretical proof for that: the "truck and tunnel" argument. I'll only sketch the idea, leaving for you to have fun completing the proof.

There is a truck which just fits in a narrow tunnel. If a width contraction existed, you could reason that it only fits thanks to that effect. But then observing the same scenario from a frame where the truck is steady and the tunnel is moving, and using the principle of relativity, you'd expect that now the tunnel has shrunk instead of the truck, so that it can't fit.

• Hmm, hmm. By similar arguments people try to proof that length contraction is wrong (ladder in the shed, etc.). I grant there is a difference, because for length contraction, we need to measure in the direction of movement, while here it is orthogonal to it, so the velocity in the direction of measurement is zero. Yet: the axiom you require up-front is that movement (velocity) is symmetric, relative. Is this needed, or can we do without? – Harald Apr 15 at 12:26
• @Harald Thank you for raising the point. I had considered adding to my answer exactly your objection and its confutation, then refrained for sake of brevity. The fallacy in the ladder-in-the-shed paradox and the like is that relativity of simultaneity (of doors' opening and closure) isn't taken in due account. In present case no such events are involved. – Elio Fabri Apr 15 at 13:17
• @Harald As to symmetry of movement, surely the following is required: If frame B is moving wrt A with velocity $v$, then A is moving wrt B with velocity $-v$. This is no independent axiom and is frequently taken for obvious. It derives from space and time homogeneity and space isotropy - axioms anyway necessary for founding SR. The proof however isn't exactly trivial, AFAIK. – Elio Fabri Apr 15 at 13:18