Length contraction is one of the two signature effects of Einstein's special relativity (alongside time dilation): objects moving relative to an observer measure shorter along their direction of motion than they do at rest. This calculator computes the contracted length, the Lorentz factor, and how contraction scales across a range of speeds.
How length contraction works
Length contraction follows directly from the postulate that the speed of light is constant for every observer. If two observers must agree on how fast light travels, they cannot agree on distances and time intervals in the same way classical mechanics assumes. The formula L = L₀√(1 − v²/c²) captures this: the contracted length L, seen by an observer the object is moving relative to, is always less than or equal to the proper length L₀, and the two are equal only when v = 0.
Crucially, contraction is relative and only affects the dimension parallel to motion. An observer riding along with the object always measures its full proper length — to them, nothing has contracted. It is only observers who see the object moving who measure the shorter length.
The Lorentz factor ties contraction and time dilation together
The Lorentz factor γ = 1/√(1 − v²/c²) is the single number that governs both major relativistic effects at a given velocity: lengths divide by γ (contraction), while time intervals multiply by γ (dilation). At everyday speeds γ is so close to 1 that both effects are unmeasurable without extremely precise instruments. Only as v approaches a significant fraction of c does γ climb noticeably — reaching 2 at about 86.6% of c, and diverging toward infinity as v approaches c itself.
Inputs, units, and limits
Proper length can be entered in any consistent length unit (meters, feet, etc.) — the contracted length is returned in the same unit. Velocity can be entered as a fraction of c (0 to just under 1) or in m/s, and the calculator converts between the two automatically when you switch units. The formula is undefined at or above the speed of light, since no object with mass can reach c; the calculator flags any velocity at or above c rather than returning a result. In everyday engineering contexts — vehicles, aircraft, even orbital spacecraft — v/c is so small that length contraction has no measurable effect; it only becomes significant for particle accelerators, cosmic rays, and thought experiments involving near-light-speed travel.