Length contraction

Length contraction is the relativistic kinematic effect that an object of proper length L₀ measured in its own rest frame is measured by any inertial observer moving at velocity v along its long axis to have a contracted length L = L₀/γ, where γ = 1/(1 − β²)^(1/2) is the Lorentz factor and β = v/c. Dimensions perpendicular to the direction of motion are unchanged. The contraction is symmetric: A's metre stick is contracted as measured from B's frame, and B's metre stick is contracted as measured from A's frame. There is no contradiction because the two measurements use different simultaneity slices — to measure the length of a moving object requires marking the positions of its endpoints at the same time in the observer's frame, and "at the same time" is frame-dependent.

Length contraction was first proposed by George FitzGerald (1889) and Hendrik Lorentz (1892) as a physical contraction in the aether — a real shortening of material bodies caused by their motion through the medium, just enough to explain the Michelson-Morley null. Einstein's 1905 reformulation made it instead a geometric kinematic consequence of the Lorentz transformation, with no aether and no material compression: nothing physical happens to the object; its proper length in its own frame is unchanged; only the measured length in another frame contracts. Operationally, length contraction explains the §11.4 magnetism-as-relativistic-electrostatics result: in the rest frame of a current's drift electrons, the lattice of positive ions has its spacing contracted, producing a net positive line density and a Coulomb attraction that, in the lab frame, looks like a magnetic force.