Gauge-invariant Bardeen variables for plane waves and their relation to Cartan and Killing invariants

The Newman-Penrose (NP) formalism is traditionally used to analyze the polarization content of gravitational waves, while the gauge-invariant Bardeen formalism provides a complementary, and often simpler, description based on the irreducible scalar, vector, and tensor perturbations of the metric. In this work we apply the Bardeen formalism to plane gravitational waves in Minkowski spacetime, computing all scalar, vector, and tensor gauge-invariant variables explicitly and demonstrating that only the two transverse-traceless tensor modes survive, as expected for vacuum waves in general relativity. We then compare these Bardeen variables with curvature-based invariants constructed using the linearized Cartan--Karlhede (CK) algorithm. We show that the CK invariants do not distinguish the $\oplus$ and $\times$ modes. Instead, we show that the Bardeen variables coincide with the Killing invariants obtained by projecting the metric perturbation onto the translational Killing vectors of the Minkowski background. Thus, in the plane-wave case, the physical gravitational-wave degrees of freedom are encoded in invariants generated from the Killing vector fields, rather than Cartan invariants.