Force -free magnetohydrodynamic waves: Non-linear interactions and the effects of strong gravity

The non-linear dynamics of a magnetic field is studied in the presence of a tenuous, conducting plasma and a relativistically strong gravitational field. In particular, we study the propagation and non-linear interactions of magnetohydrodynamic waves in the regime 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 non-linearities induced by a gravitational field. We treat the perturbed electromagnetic field as a fluid of infinite conductivity. We calculate the scattering of a torsional (Alfven) wave by a gravitational potential, and demonstrate a nonlinear coupling with a compressive (fast) wave which is second order in the amplitude of the Alfven wave. In a cylindrically symmetric spacetime with slow rotation, the coupling is second order in gt&phis; and first order in the amplitude of the wave. We also give a fresh analysis of the non-linear 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 non-relativistic 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 Alfven waves. We show how spacetime curvature modifies the collision between two torsional waves, in both the weak- and strong-field regimes. Finally, we show that the scattering of a torsional Alfven wave by a Schwarzschild spacetime will lead to the emission of a fast mode with a logarithmically divergent amplitude.