Virtual laboratory/Physics/Relativity/Length contraction (Lorentz contraction)

Length contraction (Lorentz contraction)

An object moving at high speed contracts along the direction of motion: L=L₀/γ, where γ=1/√(1−v²/c²). The transverse dimension to the motion stays unchanged. Always L ≤ L₀ — the object is never longer than its proper length. Change the speed and watch the contraction.

Length contraction (Lorentz contraction)
Physics · Theory of relativity · Length contraction (FIZ.STW)
Speed vv (as a fraction of cc)0.400 ·c
The rocket moves at nearly the speed of light. The dimension transverse to the motion (its height):
speed v
0.400·c
Lorentz factor γ
1.09
contraction L/L₀
0.917
🎯 Challenge: hit the contraction
streak: 0

I will give you a target contraction L/L₀ — pick a speed that achieves it. L = L₀/γ, always L ≤ L₀.

What just happened?

The rocket accelerated to nearly the speed of light and the length you measured shrank — but only along the direction of motion. The height stayed identical.

Length contraction (Lorentz contraction):

L=L0γ,  γ=11−v2/c2L = \frac{L_0}{\gamma}, \; \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

— L0L_0 is the proper length (measured in the frame where the object is at rest), LL — the length measured by an observer relative to whom the object moves, γ≥1\gamma \ge 1 — the Lorentz factor.

In which direction does the object contract?

Only along the direction of motion. The transverse dimensions (height, width across) remain unchanged — that is why the rocket becomes "shorter" but not "lower".

For whom is the object shorter, and when is it visible?

For the observer relative to whom the object moves. In the rocket’s own frame the astronaut measures the full L0L_0 — nothing contracts. The effect is noticeable only at vv close to cc, because only then does γ\gamma clearly exceed 1. Always L≤L0L \le L_0 — the object is never longer than its proper length.

Why you need this for the exam:

You solve special-relativity problems with two formulas: L=L0/γL = L_0/\gamma (length contraction) and time dilation. The trick: decide who measures the proper length (the rest frame) and who measures the contracted one (the moving frame), and substitute γ\gamma.

Remember: an object in motion contracts only along the direction of motion — L = L₀/γ, always L ≤ L₀. The transverse dimension does not change.

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Theory & formulas

Lorentz contraction

An object moving at high speed is shorter along the direction of motion: L=L0γ,γ=11−v2c2L = \dfrac{L_0}{\gamma},\qquad \gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}} Since γ≥1\gamma \ge 1, always L≤L0L \le L_0.

Only along the motion

Only the dimension parallel to the velocity contracts. Dimensions transverse to the motion stay unchanged.

For whom is the object shorter

The contraction is seen by an observer relative to whom the object moves. In the object's own frame its length is the unchanged L0L_0 (the proper length). The effect is noticeable only at vv close to cc.

Notation: LL — the length measured by the observer, L0L_0 — the proper length, γ\gamma — the Lorentz factor, vv — the speed, cc — the speed of light.

Frequently asked questions

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In which direction does the object contract?

Only along the direction of motion. Dimensions perpendicular to the velocity stay unchanged.

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For whom is the object shorter?

For an observer relative to whom the object moves. In the object's own frame its length is the unchanged L₀.

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When is it visible?

At speeds close to c. At everyday speeds the contraction is unimaginably small.

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