Final answer:
The Coulomb gauge is not covariant because it does not include time as a variable, assuming instantaneous action for the electric field and violating the principle of causality from special relativity. On the other hand, the Lorenz gauge is Lorentz invariant and thus maintains consistency across different reference frames in special relativity.
Step-by-step explanation:
The reason we say the Coulomb gauge is not covariant, whereas the Lorenz gauge is, lies in their respective definitions and their alignment with the theory of special relativity. The Coulomb gauge condition is defined by ∇⋅α = 0, where α is the vector potential. This gauge condition does not make explicit allowance for changes in the reference frame of observers moving relative to one another (i.e., it lacks Lorentz invariance).
In contrast, the Lorenz gauge condition, given by ∇·A + (1/c²)∂V/∂t = 0, where A is the vector potential and V is the scalar potential, is explicitly Lorentz invariant. This means it remains consistent under Lorentz transformations, which are at the core of the theory of special relativity elucidating how measurements of space and time are related for observers in different inertial frames of reference.
The ultimate reason why the Coulomb gauge cannot be covariant is because it presupposes the instantaneous action at a distance for the electric field, violating the principle of causality central to special relativity. The electric field in the Coulomb gauge is determined by the scalar potential, which is instantaneous and does not explicitly include time as a variable, clashing with the relativistic concept that information cannot propagate faster than the speed of light.