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In the coulomb gauge, the equations that describe the dynamics of $\Phi$ and $\vec{A}$ simplify to: $$ \Delta \Phi = - \frac{\rho}{\epsilon_0} \\ \Delta \vec{A} - \frac{\partial_t^2}{c^2} \vec{A} = - \mu_0 \vec{j} + \frac{1}{c^2}\nabla \partial_t \Phi $$ While the second equation looks like kind of a wave equation, the first one, describing $\Phi$, doesn't depend on $\vec{A}(\vec{x},t)$, and seems to be nonlocal. If I change $\rho$, then $\Phi$ will immediately change everywhere. How is this consistent with the requirement for electrodynamics to be a local theory?

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Although the scalar potential changes instantaneously with $\rho(\vec x , t)$, you have to keep in mind that the real physical quantities are the electric and magnetic fields. These are the quantities that directly interact with charged particles through the Lorentz force. Now the dependence of the electric and magnetic fields on the charge and current distribution is explicitly stated in Jefimenko's equations, which are the extension of the Coulomb and Biot-Savart law for time dependent cases. These equations are independent of the gauge you're using for your potentials, including the Coulomb gauge, because the electromagnetic fields are invariant under gauge transformations. You can in fact see that the fields depend on the charge and current distribution through a retarded time, which explicitly shows the locality you are talking about.

See the following Wikipedia page for further details on Jefimenko's equations:

https://en.wikipedia.org/wiki/Jefimenko%27s_equations

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