Galilean invariance
The principle that the laws of mechanics take the same form in all inertial frames related by Galilean transformations — uniform translation at constant velocity. The pre-relativistic statement of relativity, valid for low speeds.
Definition
Galilean invariance is the principle that the laws of classical mechanics — Newton's three laws, conservation of momentum, the equations of motion of a free particle — hold in identical form in every inertial frame related to another by a Galilean transformation x' = x − vt, t' = t. Galileo Galilei articulated the idea in 1632 in his Dialogue Concerning the Two Chief World Systems, with the famous "ship's cabin" argument: a person sealed below decks performing mechanics experiments cannot tell from those experiments alone whether the ship is at rest in the harbour or sailing at constant velocity in calm water. Newton's first law promotes this observational symmetry to a structural feature of the theory: an inertial frame is defined as one in which a force-free particle moves in a straight line at constant speed, and Newton's second law F = ma holds in identical form in any such frame.
The Galilean transformation preserves time as a universal scalar (t' = t in every frame), preserves Euclidean distances at any single instant, and adds velocities by simple vector addition: u' = u − v. It is the symmetry group of pre-relativistic mechanics. The trouble is that Maxwell's equations are not invariant under Galilean transformations — the wave equation derived from them yields a propagation speed c set by the constants ε₀ and μ₀, with no reference to a frame. Either Maxwell is wrong (no), or the Galilean transformation is wrong (yes, in the limit of high relative velocity), or there is a privileged "aether frame" in which Maxwell holds and other frames pick up corrections (the consensus 1860s–1900s, demolished by Michelson-Morley). The resolution is the Lorentz transformation, which reduces to the Galilean transformation in the limit β → 0.