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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?

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    $\begingroup$ > "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. $\endgroup$
    – Ján Lalinský
    Commented Jun 11, 2022 at 17:34

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Following @Shikhar-Singh comment, while the potential solutions under the coulomb gauge seem to violate causality (are not retarded), changes in the fields derived from those potential would propagate at the speed of light. It is mainly a matter of mathematical convenience that we use the Lorenz gauge to compute the potentials since it is easier to derive the physical (retarded) fields from them. Needles to say that the choice of gauge fixing does not effect the physics

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Only the fields need to be retarded, not the potentials

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