Length contraction

Time dilation has a geometric twin: length contraction. Define the proper length L₀ of an object (or a distance between two points) as its length measured in the frame where it's at rest. Any observer moving relative to that frame, measuring along the direction of relative motion, gets a shorter number — and only along that direction; lengths perpendicular to the motion are unaffected, exactly as we assumed for the light clock's mirror spacing in the previous lesson.

Here's the cleanest way to see why, using time dilation you already have. Say a fixed distance L₀ separates Earth from a distant star, both at rest in Earth's frame — that's the proper length, since Earth and the star don't move relative to each other. A spaceship covers that distance at speed v; Earth-based clocks (synchronized between Earth and the star) time the trip at t = L₀/v. But the ship's own onboard clock is present at both departure and arrival, so it reads the smaller, dilated proper time t_ship = t/γ. Now switch to the ship's point of view: it considers itself at rest and sees Earth and the star rushing toward it at speed v, separated by whatever distance L it measures. Since the ship's clock is again present at both events, the time it reads for the star to arrive is simply t_ship = L/v. Setting the two expressions for t_ship equal, v cancels and L = L₀/γ falls straight out.

Because γ ≥ 1 always, L ≤ L₀ — objects and distances are always measured shorter in a frame where they're moving, never longer, and the two effects share the exact same factor γ. It's tempting to picture the ship physically crumpling, but nothing local happens to the ship at all; length, like simultaneity, is a relationship between an object and the frame doing the measuring, not a property of the object alone.