Length Contraction Calculator

Length contraction is a phenomenon predicted by Einstein's special theory of relativity. An object moving at a significant fraction of the speed of light appears shorter along its direction of motion when measured by a stationary observer, compared to its length measured in its own rest frame. The effect is also known as Lorentz contraction.

The formula is L = L₀ × √(1 − v²/c²), where L is the observed (contracted) length, L₀ is the proper length (rest length), v is the relative velocity of the object, and c is the speed of light (~299,792,458 m/s). This can also be written as L = L₀ / γ, where γ is the Lorentz factor.

The Lorentz factor, denoted γ (gamma), is defined as γ = 1 / √(1 − v²/c²). It quantifies how much time, length, and relativistic mass change at a given velocity. At low speeds γ ≈ 1 and effects are negligible; as velocity approaches c, γ approaches infinity.

Proper length is the length of an object measured in the reference frame where the object is at rest. It is the maximum possible length of the object — any observer moving relative to the object will measure a shorter length due to relativistic contraction.

Length contraction is only significant at speeds that are a substantial fraction of the speed of light (roughly above 10% of c, or ~30,000 km/s). At everyday speeds — even the fastest spacecraft — the contraction is so tiny it is immeasurable. For example, at 90% of c, an object contracts to about 44% of its rest length.

No. According to special relativity, no object with mass can reach or exceed the speed of light (c ≈ 299,792 km/s). As velocity approaches c, the energy required becomes infinite. If v = c were entered, the contracted length would be zero and the Lorentz factor would be undefined (infinite), which is physically impossible for a massive object.

The ladder paradox is a classic thought experiment in special relativity. A ladder moving at relativistic speed appears short enough to fit inside a barn that would be too small for it at rest — yet from the ladder's reference frame, the barn appears contracted instead. This apparent paradox is resolved by the relativity of simultaneity: the two ends of the ladder don't pass through the barn doors at the same time in both frames.

No. Length contraction only occurs along the axis of motion (the direction of travel). The dimensions perpendicular to the direction of motion remain unchanged. So a sphere moving at relativistic speed would appear as a flattened disk when measured by a stationary observer.