Interactive · Length Contraction

The longest a ruler ever is is at home.

Each frame carries its own one-metre rod. Whichever frame you are in, the rod in your hand is a full metre. The rod in the other frame, blowing past you, is measured to be only $L_{0} / γ$ long along the direction of motion.

Setup

Two identical metre-sticks, one in Frame A and one in Frame B, each at rest in its own frame. They’re aligned with the direction of relative motion. Slide β to change the relative speed, and toggle who is "at rest" to feel the symmetry.

Relative velocity v / c β = 0.60 γ = 1.250

Observer:

Live readouts

γ: 1.250
Proper length L₀ (in rod’s rest frame): 1.00 m
Observed length L = L₀/γ (in the other frame): 0.800 m
Contraction (L₀ − L): 0.200 m

The rulers

Frame A — your rod (at rest)

measures full L₀

Frame B — their rod (moving)

contracted to L₀/γ = 0.800 m

Where it comes from

To measure the length of a moving rod, you have to mark its endpoints at the same time in your frame. That "at the same time" is the catch — different frames disagree about simultaneity.

Place a rod of proper length $L_{0}$ at rest in $S^{'}$ , with ends at $x_{1}^{'} = 0$ and $x_{2}^{'} = L_{0}$ . In $S$ (in which the rod moves at $v$ ) you measure both ends at the same time $t_{1} = t_{2} = t$ . Use the inverse Lorentz boost $x^{'} = γ (x - v t)$ :

x_{1}^{'} x_{2}^{'} = γ (x_{1} - v t) = γ (x_{2} - v t)

Subtracting:

L_{0} = x_{2}^{'} - x_{1}^{'} = γ (x_{2} - x_{1}) = γ L .

Solve for the length you measure in your frame:

L = \frac{L _{0}}{γ}

At your current $v = 0.60 c$ , $γ = 1.250$ , so a one-metre rod measures $L = 0.800 m$ .

Read the puzzle carefully

A common trap: "if A sees B’s rod contracted, and B sees A’s rod contracted, who is right?" Both. Each statement is a measurement made in that observer’s frame. The contraction is not an "appearance" or a "shrinkage of the atoms"; it is a consequence of how simultaneity is sliced through spacetime, which differs frame to frame.

Lengths perpendicular to the motion don’t contract — only the component along the boost direction does. So a moving cube becomes a flattened slab, not a smaller cube.