Length Contraction

Length Contraction

Special relativity does not only affect time — it also affects space. A moving object is shorter along its direction of motion as measured by a stationary observer.

The Formula

If an object has rest length $L_0$ (measured in its own rest frame), then in a frame where the object moves at velocity $v$, its measured length is:

$L = \frac{L_0}{\gamma}$

Because $\gamma \geq 1$, the observed length $L \leq L_0$. At $v = 0$: $L = L_0$. As $v \to c$: $L \to 0$.

Only the dimension along the direction of motion contracts. Perpendicular dimensions are completely unchanged.

Recovering the Rest Length

Inverting the relation:

$L_0 = L\,\gamma$

If you know the contracted length and the speed, you can recover the rest length.

Symmetry

Contraction is reciprocal: if frame $A$ sees frame $B$'s objects as shortened, then frame $B$ equally sees frame $A$'s objects as shortened by the same factor. Neither frame is privileged. The effect is real but not an optical illusion — it reflects the geometry of spacetime.

Example: A Spaceship

A spaceship of rest length $L_0 = 100$ m traveling at $v = 0.8c$ (where $\gamma = 5/3$) appears only $60$ m long to an observer at rest. To the astronauts on board, the ship is still $100$ m.

Your Task

Implement:

• length_contraction(L0, v) — returns contracted length $L = L_0 / \gamma$
• rest_length(L, v) — returns rest length $L_0 = L \times \gamma$

Use $c = 299792458.0$ m/s, defined inside each function.