A microscopic analysis of shear acceleration in the relativistic regime

Relativistic shear flows, as encountered in jets of powerful astrophysical sources, offer promising sites of particle acceleration up to the highest energies. This stochastic acceleration process is usually described with Fokker-Planck transport equations, which assume small momentum gains per scattering. The present paper introduces a covariant microscopic description of relativistic shear acceleration, building on the formalism of generalized Fermi acceleration and formulating angular scattering in the locally comoving frame in terms of stochastic differential equations. This allows us to model relativistic shear profiles and to study the transition from gradual to non-gradual shear acceleration, where the particle mean free path becomes comparable to the shear length scale. Assuming an infinite shear layer, linear in the flow four-velocity, we observe that the shear rate peaks where the flow is mildly relativistic, whereas Lorentz aberration suppresses the transverse velocity of particles where the flow is relativistic. Consequently, particles that leave the central layer experience long waiting times with a heavy-tailed, power-law distribution. The momentum evolution becomes subdiffusive and the acceleration time can substantially exceed local estimates. We successfully reproduce the Monte Carlo energy distributions using a two-state continuous-time random walk with Lévy-distributed waiting times. We also show how a momentum-dependent mean-free path can introduce breaks in the spectrum and extend our discussion to finite jet profiles. These effects should be accounted for in models of relativistic jets as sources of high-energy particles.