Lorentzinvariant

Lorentzinvariant refers to a property of physical quantities or equations that remain unchanged under Lorentz transformations, the symmetry operations of special relativity that mix space and time coordinates through boosts and rotations. A Lorentzinvariant quantity is effectively frame-independent: its value is the same in all inertial reference frames.

Mathematically, a quantity is Lorentzinvariant if it is a scalar under the Lorentz group SO(1,3). The spacetime interval s^2 = c^2 t^2 - x^2 - y^2 - z^2 is a canonical example, remaining the same for all observers related by Lorentz transformations. In relativistic dynamics, the invariant mass m is another key example, since the squared four-momentum p_mu p^mu equals m^2 c^2 for a particle at rest. Other invariants include the inner product of four-vectors and certain constructed scalars from fields, such as the Lorentz-invariant Lagrangian densities used in field theories.

In quantum field theory and high-energy physics, Lorentzinvariance is a guiding principle: laws and equations are formulated to be Lorentzinvariant so that predictions do not depend on the observer’s inertial frame. This invariance underpins the classification of particles by mass and spin through representations of the Lorentz group and constrains interaction terms in Lagrangians.

It is important to distinguish Lorentzinvariance from Lorentz covariance. Invariance refers to the unchanged value of a quantity, while covariance means the form of equations preserves structure under Lorentz transformations; the quantities themselves may transform.

See also: Lorentz transformation, Lorentz group, Minkowski space, four-vectors, spacetime interval, Noether’s theorem, quantum field theory.