Gauge-invariant formulation for the gravitational wave equations

A gauge-invariant formulation for the gravitational wave equations is presented. Using this approach, weak, plane wave solutions in a vacuum are derived in various theories. These include general relativity with two modes of polarization with helicity $\pm 2$, Yang's theory with three modes of polarization with helicity $\pm 2$ and $0$, and so-called "general metric theories" with six modes of polarization with helicity $\pm 2$, $\pm 1$, and two $0$'s. To identify the polarizations of gravitational waves, it is explicitly demonstrated how the gauge-invariant approach reproduces the earlier results.