# Covariant version of the Coulomb gauge

In curved spacetime, it is possible to define the covariant version of the Lorenz gauge, going from $\partial_\mu A^\mu =0$ to $\nabla _\mu A^\mu =0$ in some curved spacetime $g_{\mu \nu}$. What is the equivalent to that for the Coulomb gauge?

• Making this a comment, since it's a guess, but $\nabla_i A^i=0$. – Sean E. Lake Jun 26 '18 at 16:11
• Would it still imply that $A_0=0$ ? – N.E. Jun 26 '18 at 16:18
• Coulomb gauge doesn't necessarily imply $A_0 = 0$ even in flat spacetime, since you can always apply a gauge transformation $A_\mu \to A_\mu + \partial_\mu \lambda$ with $\lambda$ a function of $t$ only. – Michael Seifert Jun 26 '18 at 17:25
• $A_0=0$ is the Weyl gauge. You can have $A_0=0$ in the Coulomb gauge, but that's coincidental to $\rho=0$ everywhere. – Sean E. Lake Jun 26 '18 at 17:37
• The Coulomb gauge is impossible to specify geometrically in curved space without a distinguished temporal vector field, or something equivalent. – Qmechanic Jun 26 '18 at 18:34