Comparing gauge invariant Bardeen variables with Cartan variables for plane-waves and applications to modified gravity theories

We apply the gauge invariant Bardeen formalism to plane waves to first calculate the Bardeen variables by applying an appropriate gauge and then computing all the curvature tensors in Einstein's gravity. For plane-waves we plan to compare the gauge invariant Bardeen quantities with the Cartan invariants that encode all information about a spacetime and determine if they are in fact the same geometric quantities. If they are, we can develop a frame-based formalism for linear approximations of gravitational solutions, and this has applications to modified gravitational theories. If they are not, we can explicitly point out what information about the plane-waves that the Bardeen variables misses.