Past exam of the general relativity and cosmology course of the University of Oxford / 2017 / 3 / a / Solution

Work consistently to first order in the metric perturbation, with ${\partial }_{\mu }{\xi }^{\nu }$ of the same small order. Under the passive change $x^{\prime \mu }=x^{\mu }+{\xi }^{\mu }(x)$,Applying this to the full metric tensor and writing $g_{\mu \nu }^{\prime }={\eta }_{\mu \nu }+h_{\mu \nu }^{\prime }$ with the same fixed numerical Minkowski metric givesIndices on $\xi$ are lowered with $\eta$. This is the gauge transformation of the perturbation relative to a fixed background. If the background were transformed as a tensor as well, the difference of those two fully transformed metrics would instead obey the homogeneous tensor transformation law. The inhomogeneous terms arise from retaining the fixed-background identification used in linearized gravity.

The trace changes by $h^{\prime }=h-2{\partial }_{\alpha }{\xi }^{\alpha }$, so the trace-reversed metric perturbation changes asLet $F_{\nu }={\partial }^{\mu }{\bar{h}}_{\mu \nu }$. Commuting partial derivatives givesThus choose a solution of the inhomogeneous wave equationSuch a solution exists locally with suitable initial data, and can be selected with appropriate wave-equation boundary conditions. This imposes the Lorenz gauge in linearized gravity, ${\partial }^{\mu }{\bar{h}}_{\mu \nu }^{\prime }=0$. All the divergence terms in the supplied vacuum equation then vanish, leavingThe remaining freedom $\square {\xi }_{\nu }=0$ is the residual gauge symmetry of linearized gravity.

Using the old coordinates in these derivatives is consistent at the retained order. IndeedActing on a first-order perturbation, the correction is a product of two first-order quantities and is discarded. Similarly replacing the argument $x^{\prime }$ by $x$ changes $h^{\prime }$ by $O(h^2)$ under the same perturbative ordering. This does not assert an exact equality of the two coordinate differential operators; it is the linear approximation about the fixed flat background.