Relativistic transport theory for cosmic rays

The general relativistic form of Liouville's equation or the Boltzmann equation is used to derive relativistic equations describing the transport of cosmic rays in a magnetized plasma supporting a spectrum of hydromagnetic waves. A local Lorentz frame (the comoving frame) moving with the waves or turbulence scattering the cosmic rays is used to specify the individual particle momentum. Since the comoving frame is in general a noninertial frame in which the observer's (spatial) volume element is expanding (contracting) and shearing, geometric energy change terms appear in the cosmic-ray continuity equation which consist of the relativistic generalization of the adiabatic deceleration term obtained in previous analyses, and a further term involving the acceleration vector of the scatterers. Moment equations which are differential with respect to the magnitude of the particle momentum p' are obtained. When integrated over all momenta p', the moment equations lead to a hydrodynamical description of cosmic-ray transport. The diffusion approximation is used to close the moment equations, and to obtain a convection-diffusion form of the continuity equation. A brief discussion of the drift approximation is given, which in the lowest order approximation leads to a pitch-angle-dependent propagation equation for the cosmic rays. Possible astrophysical applications of the equations are discussed