Force-free magnetohydrodynamic waves: Nonlinear interactions and effects of strong gravity

The propagation and nonlinear interactions of magnetohydrodynamic waves are considered in the force-free limit, where the inertia of the conducting matter which enforces the MHD condition E·B=0 can be neglected in comparison with the inertia of the electromagnetic field. By extending the analysis beyond the WKB approximation, we are able to study the nonlinearities induced by a gravitational field. We treat the perturbed electromagnetic field as a fluid of infinite conductivity. We calculate the scattering of a torsional (Alfv\'en) wave by a gravitational potential, and demonstrate a nonlinear coupling with a compressive (fast) wave which is second-order in the amplitude of the Alfv\'en wave. In a cylindrically symmetric spacetime with slow rotation, the coupling is second-order in g_tφ and first order in the amplitude of the wave. We also give a fresh analysis of the nonlinear interactions between compressive and torsional waves in Minkowski space, with a focus on the relative strengths of their three- and four-mode interactions. In contrast with nonrelativistic magnetofluids, the effects of compression are always present. In the case of colliding fast waves, a net displacement of the field lines across (at least) one of the colliding wavepackets is shown to have a strong effect on the outgoing waveform, and to have a qualitatively different interpretation than was previously suggested for colliding Alfv\'en waves. Finally, we show how spacetime curvature modifies the collision between two torsional waves, in both the weak and strong field regimes.