the Coulomb gauge
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The freedom to redefine the potentials without changing the fields lets you impose one extra condition to tidy them up. The Coulomb gauge makes the tidiest possible choice for the vector potential: it demands that A have zero divergence.
The gauge condition is div A = 0. Its immediate payoff is that Gauss's law collapses to Poisson's equation for the scalar potential, laplacian V = -rho/epsilon_0, the very same equation as in electrostatics — so V is determined by the instantaneous charge distribution, computed as if nothing propagated. Because the transverse (divergence-free) part of A is what survives, the Coulomb gauge is also called the transverse or radiation gauge, and the vector potential then obeys a wave equation driven only by the transverse part of the current.
The Coulomb gauge is prized in bound-state and non-relativistic problems — it is the natural gauge for atomic physics and for quantizing radiation, since the physical, transverse photon degrees of freedom are laid bare. Its notorious oddity is that V responds instantaneously to a change in charge anywhere, seemingly faster than light; this is not a paradox, because V is not measurable on its own — the acausal-looking scalar piece is exactly cancelled by a matching piece of dA/dt, so the physical field E still propagates at c.
In the Coulomb gauge the scalar potential of any charge distribution is the plain electrostatic V = (1/4 pi epsilon_0) times the integral of rho/|r - r'|, evaluated at the SAME instant — no retardation. All the causal, light-speed delay is carried instead by the transverse vector potential A.
div A = 0 turns Gauss's law into the instantaneous Poisson equation laplacian V = -rho/epsilon_0.
The instantaneous Coulomb potential looks acausal but is not observable by itself; the transverse A restores light-speed propagation of the physical field E. The Coulomb gauge is not Lorentz-invariant — the condition div A = 0 singles out a frame — so it is awkward for relativistic calculations, where the Lorenz gauge is preferred.