Why retardation showed up for Lorenz gauge but not coulomb gauge?

There's two common gauge used for EM, Lorenz gauge and Coulomb gauge

However, if you look at the solution of those gauge, the retardation only showed up for Lorenz gauge, but not for coulomb gauge.

Especially, even though the electric potential solution might be understood ambiguously, the reasoning for the solution of vector potentials, $$\nabla^2A_C-\frac{1}{c^2}\frac{\partial^2 A_C}{\partial t^2}= -\mu_0 j_\bot$$ and $$\nabla^2A_L-\frac{1}{c^2}\frac{\partial^2 A_L}{\partial t^2}= -\mu_0 j,$$ did not appear to be so obvious.

Why retardation showed up for Lorenz gauge but not coulomb gauge?

• > "if you look at the solution of those gauge" -- These equations do not have unique solution. You can find retarded, advanced, or any mix of these. Usually retarded fields are considered as "physical", and then the vector potential does manifest retardation, even in the Coulomb gauge. Jun 11 at 17:34