Non-linear scattering of plane gravitational and electromagnetic waves

Fully non-linear, plane-symmetric exact solutions of the Einstein equations describing the scattering of gravitational and electromagnetic waves have existed for many years. For these closed-form solutions to be found, idealized wave profiles such as the Dirac delta and Heaviside theta functions must be assumed. Although pathological in that future curvature singularities generically occur, these exact solutions give useful insights into the non-linear features of the scattering process. Only a limited number of exact solutions exist and this leaves many other physically-motivated scattering situations without a non-linear description. The aim of this paper is to shed light on these unexplored cases. This is achieved through numerical solutions of the Friedrich-Nagy initial boundary value problem for the Einstein equations coupled to the source-free Maxwell equations in plane symmetry. Interesting non-linear scattering effects are presented for a variety of electromagnetic and gravitational wave profiles not currently described by an exact solution. Implications for electromagnetic wave observations are investigated through analyzing the time-delay and frequency shift imparted on the electromagnetic wave through the non-linear scattering with a gravitational wave of a given strain. Simple arguments supported by the numerical solutions suggest dramatic effects on the radiation may be observable.