frameinvariant

Frame invariance refers to the property that a quantity remains unchanged under a change of reference frame. In physics and mathematics, a frame-invariant quantity is the same for all observers related by symmetry transformations, such as Galilean and Lorentz transformations, or by diffeomorphisms in curved spacetime.

Key examples in special relativity include the spacetime interval s^2 = c^2 t^2 - x^2 - y^2 - z^2, which is invariant under Lorentz transformations, and the rest mass m, which is a Lorentz scalar satisfying p^μ p_μ = m^2 c^2.

More generally, invariants include four-vector norms and scalar products that yield frame-independent numbers, such as p^μ p_μ and u^μ v_μ for appropriate four-vectors.

In general relativity, invariants are formed by contractions of tensor fields. Scalars such as the Ricci scalar R and the Kretschmann scalar R_{μνρσ} R^{μνρσ} remain unchanged under coordinate transformations and characterize intrinsic properties of spacetime.

Frame invariance is central to the formulation and interpretation of physical laws. Noether's theorem links continuous symmetries to conserved quantities, such as energy from time translation symmetry and momentum from spatial translation symmetry; angular momentum from rotational symmetry.

Not all quantities are frame-invariant. Velocities, momenta, and kinetic energy depend on the chosen reference frame, so comparing results may require constructing invariant combinations or using invariant formulations of the theory.