Gauge invariant Bardeen variables for plane waves and comparison with Cartan invariants

Usually the Newman-Penrose (NP) formalism is used to investigate the polarization states of gravitational waves. However, the gauge invariant Bardeen formalism is much more suitable because the number of polarization states is always equal to or less than the number of Bardeen variables and the calculations are much simpler in this formalism. In this project we apply the Bardeen formalism to plane gravitational waves to first calculate the Bardeen variables by applying an appropriate gauge and then computing 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, then 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, then we can explicitly point out what information about the plane waves the Bardeen variables miss.