Kinetic gases in static spherically symmetric modified dispersion relations

We study the dynamics of a collisionless kinetic gas in the most general static, spherically symmetric dispersion relation. For a static, spherically symmetric kinetic gas, we derive the most general solution to these dynamics, and find that any solution is given by a one-particle distribution function which depends on three variables. For two particular solutions, describing a shell of monoenergetic orbiting particles and a purely radial inflow, we calculate the particle density as a function of the radial coordinate. As a particular example, we study a $\kappa$-Poincar\'e modification of the Schwarzschild metric dispersion relation and derive its influence on the particle density. Our results provide a possible route towards quantum gravity phenomenology via the observation of matter dynamics in the vicinity of massive compact objects.