Move fast enough and lengths themselves shrink along the direction of travel — not by illusion or by squashing, but because simultaneity, and with it the very act of measuring a moving object’s ends, is frame-dependent. This length contraction calculator takes a rest length and a speed (as a fraction of light speed) and returns the contracted length and the Lorentz factor γ that governs it, along with the everyday speeds where the effect is gloriously negligible.
The Formula
L = L₀ √(1 − v²/c²) = L₀ ÷ γ, with the Lorentz factor γ = 1 ÷ √(1 − β²) and β the speed as a fraction of c. The numbers everyone meets first: β = 0 gives γ = 1 (no contraction, the classical world); β = 0.6 gives γ = 1.25; β = 0.8 gives γ ≈ 1.667, so this page’s default 100 m craft measures 60.0 m from the frame it streaks past; β = 0.99 gives γ ≈ 7.09; β = 0.999 gives γ ≈ 22.4. The γ curve is the whole personality of relativity: flat and innocent below about 0.3c, then rearing toward infinity as β → 1, which is the mathematics refusing, on principle, to let anything massive arrive at light speed.
Measured, Not Squeezed
Length contraction is not a material compression — the traveller’s ship, rulers and body feel nothing, and in the ship’s own frame everything keeps its rest length exactly. The contraction lives in the measurement made from another frame: to measure a moving object’s length you must mark both ends simultaneously, and simultaneity itself is frame-relative (the other frame’s “same moment” is a different pair of events). Each frame therefore measures the other’s lengths contracted, symmetrically, and both are right — the resolution that dissolves the famous paradoxes (the pole and the barn fit or do not fit depending on which frame’s simultaneity referees, and both accounts agree on every physical outcome, like whether a door shut).
Evidence: Muons That Should Not Arrive
Cosmic-ray muons are created 10+ km up with a rest lifetime of 2.2 microseconds — at nearly c they should decay after ~660 m, yet they flood sea-level detectors. Two frame stories reconcile it, and length contraction is one of them: in the muon’s frame, the atmosphere’s 10 km is contracted by γ ≈ 20+ to under 500 m, comfortably crossable in its short life. (In Earth’s frame the same rescue is told as time dilation of the muon’s clock.) The atmosphere’s thickness literally measures differently for the particle crossing it — this calculator’s arithmetic, verified by particle counters on every mountain and campus since the 1940s.
Everyday Speeds, Honest Magnitudes
At a jet’s 250 m/s, β ≈ 8 × 10⁻⁷ and γ − 1 ≈ 3 × 10⁻¹³: a 60 m airliner contracts by about twenty picometres, a few dozen hydrogen atoms’ worth. At Earth’s orbital 29.8 km/s the effect is parts in 10⁹. GPS satellites live where it is small but not negligible (their story mixes this with gravitational time effects and nets microseconds a day, which engineers absolutely must correct). The effect becomes engineering only for particle accelerators — where it is not a correction but the terrain — and for the interstellar-flight literature, where γ is simultaneously the promise (distances shrink!) and the bill (energy per kilogram grows with the same γ).
Perpendicular Lengths and Other Fine Print
Only the dimension along the motion contracts; transverse lengths are invariant — a speeding sphere measures as an oblate spheroid from the side frame, never a pancake in all axes. The rest length L₀ must be measured in the object’s own rest frame — feeding the calculator a length you already measured from a moving frame double-counts the effect. And nothing here lets information outrun light: contraction is a relation between frame descriptions of the same object, not a motion, so causality sleeps soundly.
Common Mistakes
Expecting the traveller to experience the shrinkage (their frame is the rest frame — nothing contracts at home). Applying contraction to heights and widths. Using γ where 1/γ belongs (contracted means shorter: divide). Feeding β ≥ 1 — the widget refuses, as nature does. And picturing the contracted object as optically appearing squashed: Terrell rotation shows a fast object visually appears rotated as much as contracted, because the light reaching your eye left different parts at different times — measurement and appearance part company exactly where relativity gets interesting.
Related Tools
The relativity shelf continues in the Physics Calculators hub — the E = mc² Calculator prices rest energy, and the Kepler’s Third Law Calculator covers the classical orbits relativity later corrected.
Frequently Asked Questions
How do you calculate length contraction?
L = L0 × √(1 − v²/c²), equivalently L0 ÷ γ. A 100 m craft at 0.8c (γ ≈ 1.667) measures 60.0 m from the frame it passes. Only the dimension along the motion contracts.
What is the Lorentz factor?
γ = 1 ÷ √(1 − β²). It is 1 at rest, 1.25 at 0.6c, about 1.667 at 0.8c, 7.09 at 0.99c, and diverges as speed approaches c — the same factor that governs time dilation and relativistic energy.
Does the traveller feel their ship contract?
No. In the ship’s own frame it keeps its rest length exactly; contraction describes how other frames measure it, arising from the relativity of simultaneity rather than any physical squeezing.
At what speed does contraction become noticeable?
Roughly above 0.3c for percent-level effects (γ ≈ 1.05). At jet speeds the contraction of an airliner is picometres; the effect is daily engineering only in particle physics and speculative interstellar design.
Why can muons reach the ground if they live 2.2 microseconds?
In the muon’s frame the atmosphere is length-contracted (by γ of 20 or more) to a few hundred metres — crossable within its lifetime. In Earth’s frame the same fact is told as time dilation; both frames agree the muon arrives.
Do widths and heights contract too?
No — only the dimension parallel to the motion. A moving sphere measures as an oblate spheroid (flattened along travel), with transverse dimensions unchanged.