While studying some topics on retarded potential, I came across the following statement, and I have some trouble understanding why this is true.

When we use the Coulumb gauge (instead of the Lorentz gauge), the equations for the scalar and vector potentials $\phi_C$ and $\vec{A}_C$ are no longer wave equations but can still be solved for the wave equation and Green function as,

$$\phi_C(\vec{r},t)=\int d^3r'\frac{\rho(\vec{r}',t)}{R} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vec{A_C}(\vec{r},t)=\int d^3r\frac{\vec{j}_\bot(\vec{r}',t-R/c)}{R}$$

where $R=|\vec{r}-\vec{r}|$ and the transverse current is $\vec{j}_\bot=\vec{j}-\nabla\frac{\partial}{\partial t}\phi$.

It is evident that the scalar potential $\phi$ isn't retarded, and the vector potential $\vec{A}$ seems to be retarded since it depends of $t-R/c$ but it isn't retarded. This is due to the properties of $\vec{j}_\bot$.

Why isn't the vector potential retarded despite having a clear dependence on retarded time? Does the transverse current cancel that in some way, or how does it prevent it from being retarded?


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