Magnetohydrodynamics in the extreme relativistic limit

We present two new formulations of magnetohydrodynamics (MHD), in the limit where the inertia of the charge carriers can be neglected. The first employs Lagrangian coordinates and generalizes Newcomb's formalism to allow for a variable time slicing. It contains an extremely simple prescription for generalizing the action of a relativistic Nambu-Goto string to four dimensions. It is also related by a duality transformation to the action presented by Achterberg. This transformation causes the perturbed and unperturbed Lagrangian coordinates to exchange roles as dynamical fields and background spacetime. Our second formulation introduces massless electrically charged fermions as the current carrying modes, and considers long wavelength perturbations with ${\ensuremath{\omega}}^{2}{,k}_{\ensuremath{\perp}}^{2}\ensuremath{\ll}\mathrm{eB}.$ Because the Fermi zero mode can be bosonized separately on each magnetic flux line, the current density may be written in terms of a four-dimensional axion field that acts as a Lagrange multiplier to enforce the MHD condition. The fundamental modes of the magnetofluid in this limit comprise two oppositely directed Alfv\'en modes and the fast mode, all of which propagate at the speed of light. We calculate the nonlinear interaction between two Alfv\'en modes, and show that in both formulations it satisfies the same simple expression. This provides the first exact treatment of the effects of compressibility on nonlinear interactions between MHD waves. We then summarize the interactions between Alfv\'en modes, between Alfv\'en modes and fast modes, and between fast modes in terms of a simplified Lagrangian. The three-mode interaction between fast modes is a magnetohydrodynamic analogue of the QED process of photon splitting, but occurs in background magnetic fields of arbitrary strength. The scaling behavior of an Alfv\'en wave cascade in a box is derived, paying close attention to boundary conditions. This result also applies to nonrelativistic MHD media and differs from those obtained by previous authors in the nonrelativistic regime. Finally, we briefly outline the physical processes which determine the inner scale of such a cascade in neutron star magnetospheres, black hole accretion disks, and \ensuremath{\gamma}-ray burst sources. At low charge density, the waves at the inner scale may become charge starved; whereas Compton drag is the dominant dissipative mechanism at large optical depth to electron scattering. A turbulent cascade leads to effective dissipation even in optically thick media, and in particular can significantly raise the entropy-baryon ratio in the relativistic outflows that power cosmological \ensuremath{\gamma}-ray bursts.