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.
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