Lorentz covariance

If a differential operator $L$ is a linear map and has nonzero kernel of a linear map, the equation $LA=J$ determines $A$ only up to a homogeneous solution. Lorentz covariance of $L$ and $J$ therefore does not force independently selected solutions in different inertial frames to obey a particular tensor transformation law. A covariant Green function and consistently transformed homogeneous data do provide such a choice. For the four-potential in Lorenz gauge, $\square A^{\mu }=-{\mu }_0{J}^{\mu }$ still permits $A^{\mu }\mapsto A^{\mu }+{\partial }^{\mu }\chi$ with $\square \chi =0$, a residual Lorenz gauge symmetry. Constant spatial potentials for zero sources are a concrete example.

A physical quantity is Lorentz invariant when its value is unchanged under Lorentz transformations. Tensor equations can be Lorentz covariant while their components transform. Fixed-vector gauge choices need not make covariance manifest, although physical gauge-invariant predictions retain the symmetry.

Dynamical equations whose solutions transform into solutions under a Lorentz transformation, holding fixed the theory's physical background data. A scalar matter equation constructed from the Minkowski metric has this property; introducing a preferred timelike vector field into the matter equation can destroy it.