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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?

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  • $\begingroup$ Making this a comment, since it's a guess, but $\nabla_i A^i=0$. $\endgroup$ – Sean E. Lake Jun 26 '18 at 16:11
  • $\begingroup$ Would it still imply that $A_0=0$ ? $\endgroup$ – N.E. Jun 26 '18 at 16:18
  • $\begingroup$ 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. $\endgroup$ – Michael Seifert Jun 26 '18 at 17:25
  • $\begingroup$ $A_0=0$ is the Weyl gauge. You can have $A_0=0$ in the Coulomb gauge, but that's coincidental to $\rho=0$ everywhere. $\endgroup$ – Sean E. Lake Jun 26 '18 at 17:37
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    $\begingroup$ The Coulomb gauge is impossible to specify geometrically in curved space without a distinguished temporal vector field, or something equivalent. $\endgroup$ – Qmechanic Jun 26 '18 at 18:34

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