The creation of multiple images by a gravitational wave

We describe gravitational lensing by a gravitational wave, in the regime in which multiple images of a light source are created. We adapt the vector formalism employed for ordinary gravitational lenses to the case of a non–stationary spacetime, and we derive an approximate condition for multiple imaging. It is shown that certain astrophysical sources of gravitational waves satisfy this condition. To appear in Proceedings of the Pacific Conference on Gravitation and Cosmology, Seoul, Korea, 1–6 February 1996. We study gravitational lensing with a gravitational wave acting as the lens. It is well–known that exact gravitational waves deflect light, and light propagation through linearized (cosmological) gravitational waves outside the laboratory has also been studied [1], expecially in conjunction with their frequency shift effect on the photons of the cosmic microwave background. It has also been suggested that gravitational waves may create multiple images of distant light sources [2]. We concentrate on the latter aspect of lensing by gravitational waves, for two reasons: i) the possibility of multiple images is associated to strong amplifications of the light source, which makes easier to detect gravitational waves; ii) a previous analysis [3] concluded that the amplification is negligible. This conclusion was based on the Raychaudhuri’s equation, and it is not valid when multiple images are involved. 1 The vector formalism for lensing gravitational waves We consider a gravitational wave localized in a region of space between a light source and an observer. The spacetime metric is gμν = ημν + hμν , where ημν is the Minkowski metric and |hμν | ≪ 1. Let us consider a light ray whose unperturbed path is parallel to the z-axis, with 4-momentum p = pμ(0) + δp μ = (1 + δp, δp, δp, 1 + δp), where δp are small deflections. We work in the geometric optics approximation, which holds if λg.w. >> λe.m. and λe.m. > λg.w. /DL (where λg.w. and λe.m. are the wavelengths of the gravitational wave and of the photon, respectively, and DL is the observer-lens distance). We set the geometry as customary in gravitational lens theory, using lens and source planes with coordinates (x, y) and (sx, sy) respectively. The action of the lens is described by the plane-to-plane mapping x 7−→ s (A = 1, 2), with s given by the “lens equation” s = x + DLDLS DS δp(x) (1.1) (DLS and DS are the lens-source and the observer-source distances respectively). The map described by Eq. (1.1) has the Jacobian matrix