Gauge condition equivalent to condition that coordinate functions satisfy wave equation to first order

Question

Let $\eta_{ab}$ be the metric of special relativity and let $x^\mu$ be global inertial coordinates of $\eta_{ab}$. Let $\gamma_{ab}$ be a small perturbation of $\eta_{ab}$. How do I see that the gauge condition$$0 = \partial^a \overline{\gamma}_{ab} = \partial^a \gamma_{ab} - {1\over2}\partial_b \gamma$$is equivalent to the condition that the coordinate functions $x^\mu$ satisfy the wave equation $\nabla^a \nabla_a x^\mu = 0$ to first order in $\gamma_{ab}$, where $\nabla_a$ is the derivative operator associated with $g_{ab} = \eta_{ab} + \gamma_{ab}$?

Answer 1

It is important to first note that $x^\mu$ is a scalar function, not a vector, so that we must use the scalar wave operator. We will raise and lower indices with the background metric, $\eta_{ab}$. Working in the $x^\mu$ coordinate system, we then have\begin{align*} g^{ab} \nabla_a \nabla_b x^\mu & = g^{ab} \partial_a \partial_b x^\mu - g^{ab} {\Gamma^c}_{ab} \partial_c x^\mu \\ & = \sum_{\alpha, \beta, \gamma} \left\{g^{\alpha\beta}\partial_\alpha\partial_\beta x^\mu - g^{\alpha\beta} {\Gamma^\gamma}_{\alpha\beta} \partial_\gamma x^\mu\right\} \\ & = -\sum_{\alpha, \beta} g^{\alpha\beta}{\Gamma^\mu}_{\alpha\beta} \\ & = -\sum_{\alpha, \beta, \rho} {1\over2}g^{\alpha\beta} g^{\mu\rho}(\partial_{\alpha}g_{\beta\rho} + \partial_\beta g_{\alpha\rho} - \partial_\rho g_{\alpha\beta}) \\ & \approx -\sum_{\alpha, \beta, \rho} {1\over2}(\eta^{\alpha\beta} - \gamma^{\alpha\beta})(\eta^{\mu\rho} - \gamma^{\mu\rho})(\partial_\alpha \gamma_{\beta\rho} + \partial_\beta \gamma_{\alpha\rho} - \partial_\rho \gamma_{\alpha\beta}) \\ & \approx -\sum_{\alpha, \beta, \rho} {1\over2} \eta^{\alpha\beta} \eta^{\mu\rho}(\partial_\alpha \gamma_{\beta\rho} + \partial_\beta \gamma_{\alpha\rho} - \partial_\rho \gamma_{\alpha\beta}) \\ & = -\sum_\alpha \left\{\partial^\alpha {\gamma_\alpha}^\mu - {1\over2} \partial^\mu \gamma\right\}. \end{align*}On the third line, we used the fact that $\partial_\alpha x^\beta = {\delta^\beta}_\alpha$. On the fifth line, we used the fact that, to first order in $\gamma_{ab}$, the inverse metric is $g^{ab} = \eta^{ab} - \gamma^{ab}$.