Gravitational Radiation in the Limit of High Frequency. I. The Linear Approximation and Geometrical Optics

A formalism is developed for obtaining approximate gravitational wave solutions to the vacuum Einstein equations of general relativity in situations where the gravitational fields of interest are quite strong. To accomplish this we assume the wave to be of high frequency and expand the vacuum field equations in powers of the correspondingly small wavelength, getting an approximation scheme valid for all orders of $\frac{1}{r}$, for arbitrary velocities up to that of light, and for all intensities of the gravitational field. To lowest order in the wavelength, we obtain a gauge-invariant linearized equation for gravitational waves which is just a covariant generalization of that for massless spin-2 fields in a flat background space. This wave equation is solved in the WKB approximation to show that gravitational waves travel on null geodesics of the curved background geometry with their amplitude, frequency, and polarization modified by the curvature of space-time in exact analogy to light waves.