In general this would not be the case but in the linearised theory of gravity where the Riemann tensor is approximated to first order in the metric perturbation h. It can be shown directly from the transformations laws that each component of the Riemann tensor is Invariant to first order in the...
In the book general relativity by Hobson the gravitational wave of a binary merger is computed in the frame of the binary merger as well as the TT-gauge. I considered what components of the Riemann tensor along the x-axis in both gauges. The equation for the metric in the source and TT-gauge are...