Relativity and gravitational theory Special relativity: kinematics Proposition

Length contraction

Builds on Lorentz transformation Lorentz factor

Statement

Let $S$ and $S'$ be inertial frames with coordinates $(t, x, y, z)$ and $(t', x', y', z')$ . Let $S'$ move with velocity $v$ along the $x$ axis of $S$ , where $\abs{v} < c$ . Let a rod be at rest in $S'$ , lying along the $x'$ axis, with length $L_0$ in $S'$ (its proper length). Define its length $L$ in $S$ as the distance between the positions of its two ends at one and the same time $t$ of $S$ . Then $L = \frac{L_0}{\gamma}, \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}},$ where $\gamma$ is the Lorentz factor. Lengths perpendicular to the direction of motion are the same in both frames.

Remarks

Proof idea: the Lorentz transformation gives $x' = \gamma (x - v t)$ . For the two ends at the same time $t$ , subtracting gives $L_0 = \Delta x' = \gamma \Delta x = \gamma L$ . Since $y' = y$ and $z' = z$ , transverse lengths do not change. Example: a rod of proper length $1$ metre moving at $v = 0.6c$ has length $0.8$ metre in $S$ . The measurement at equal times in $S$ is essential: the two measuring events are not simultaneous in $S'$ . This is why each observer can find the other's rods contracted without contradiction.

Going deeper

For a student

Length contraction and time dilation are two faces of one Lorentz transformation, and in many problems either one gives the answer. For a muon flying down through the atmosphere, the Earth observer uses time dilation, while the muon uses length contraction of the atmosphere. Paradoxes such as the ladder paradox are resolved by keeping track of which events are simultaneous in which frame.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Where this leads

1 page builds on this one.

Bell's spaceship paradox

Tag	`4A5MG1MJAZ`
URI	`m2://4A5MG1MJAZ`
Kind	statement, proposition
Writing	W0 stub
MSC	83A05
Status	This copy of the site has no build data. Run `m2 status 4A5MG1MJAZ`.
About	Lorentz transformation
Uses	Lorentz transformation, Lorentz factor
Same as	Wikidata Q836595 Wikipedia Length contraction
Changed	3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by

Bell's spaceship paradox