The Relativity of Time — Question 1

Question 1

A transverse light clock derives the dilation factor geometrically

A clock consists of parallel mirrors separated transversely by h in their rest frame. One tick is a complete light round trip. The apparatus moves along x at speed v, perpendicular to the mirror separation. Assume the transverse separation remains h.

1. Use one half-tick triangle to derive the laboratory tick interval.

2. Find the ratio of laboratory to proper tick intervals at $v=0.80c$.

Question 1 - Solution

A transverse light clock derives the dilation factor geometrically

(a) Method and calculation

$$(c\Delta t/2)^2=h^2+(v\Delta t/2)^2.$$$$\boxed{\Delta t=\frac{2h/c}{\sqrt{1-v^2/c^2}}},\qquad\Delta\tau=2h/c.$$

(b) Interpretation and further calculation

$$\frac{\Delta t}{\Delta\tau}=\gamma=\frac1{\sqrt{1-0.80^2}}.$$$$\boxed{\Delta t/\Delta\tau=5/3}.$$

Physical check

Both endpoints of a complete tick lie on the same clock; its rest-frame interval is proper time.