invariance

A quantity or law is invariant under a transformation if applying the transformation leaves it unchanged. Invariance is the precise technical counterpart of symmetry, and the two words are often used interchangeably in physics. Saying "the Lagrangian is invariant under time translation" is the same as saying "the system has time-translation symmetry".

Different invariances underpin different theories. Galilean invariance — that the laws of mechanics look the same in any uniformly moving frame — is a foundational assumption of Newtonian physics. Lorentz invariance — the relativistic strengthening of Galilean invariance to include the speed of light as a universal constant — is the foundational assumption of special relativity. Diffeomorphism invariance — that the laws are independent of any particular coordinate choice — is the core of general relativity. Gauge invariances — local phase freedoms — are the defining feature of the Standard Model.

Noether's theorem ties continuous invariance to conservation laws, making invariance not just a formal property but the origin of everything conserved in physics. Looking at a theory's invariances is how physicists work out what it predicts, and breaking a previously assumed invariance is how they discover new physics.