On particle acceleration around shocks. I.

We derive a relativistically covariant (although not manifestly so) equation for the distribution function of particles accelerated at shocks, which applies also to extremely relativistic shocks and arbitrarily anisotropic particle distributions. The theory is formulated for arbitrary pitch-angle scattering and reduces to the well-known case for small-angle scatterings via a Fokker-Planck approximation. The boundary conditions for the problem are completely reformulated introducing a physically motivated Green's function; the new formulation allows derivation of the particle spectrum both close and far away from the injection energy in an exact way, while it can be shown to reduce to a power law at large particle energies. The particle spectral index is also recovered in a novel way. Contact is made with the Newtonian treatment.