Relativistic dynamical friction in stellar systems

We extend the classical formulation of the dynamical friction effect on a test star by Chandrasekhar to the case of relativistic velocities and velocity distributions also accounting for post-Newtonian corrections to the gravitational force. The original kinetic framework is revised and used to construct a special-relativistic dynamical friction formula where the relative velocities changes in subsequent encounters are added up with Lorentz transformation and the velocity distribution of the field stars accounts for relativistic velocities. Furthermore, a simple expression is obtained for systems where the post-Newtonian correction on the gravitational forces become relevant even at non-relativistic particle velocities. Finally, using a linearized Lagrangian we derive another expression for the dynamical friction expression in a more compact form than that of Lee (1969). Comparing our formulation with the classical one, we observe that a given test particle suffers a slightly stronger drag when moving through a distribution of field stars with relativistic velocity distribution. Vice versa, a purely classical treatment of a system where post-Newtonian (PN) corrections should be included, over estimates the effect of dynamical friction at low test particle velocity, regardless of the form of velocity distribution. Finally, a first order PN dynamical friction covariant formulation is less strong than its classical counterpart at small velocities but much higher for large velocities over a broad range of mass ratios