Harmonic coordinate condition

Remarks

Equivalent forms are $\partial_\alpha \bigl( \sqrt{\abs{\det g}} \, g^{\alpha\mu} \bigr) = 0$ and $g^{\alpha\beta} \Gamma^\mu_{\alpha\beta} = 0$ for each $\mu$, where $\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols. This follows from $\square_g x^\mu = -g^{\alpha\beta} \Gamma^\mu_{\alpha\beta}$. In harmonic coordinates the Ricci tensor is $R_{\mu\nu} = -\tfrac12 g^{\alpha\beta} \partial_\alpha \partial_\beta g_{\mu\nu} + Q_{\mu\nu}(g, \partial g)$, with $Q_{\mu\nu}$ quadratic in the first derivatives. So the vacuum equations $R_{\mu\nu} = 0$ become a quasilinear hyperbolic system for $g_{\mu\nu}$. Example: inertial coordinates on Minkowski space are harmonic, since all Christoffel symbols vanish. Non-example: in Minkowski space with spatial spherical coordinates $(t, r, \theta, \varphi)$, the function $r$ is not harmonic, since $\square r = \Delta r = 2/r \ne 0$ for the flat metric of signature $(-,+,+,+)$. Harmonic coordinates exist near any point: one solves the wave equation $\square_g x^\mu = 0$ with suitable initial values.