Entropy covector field and macroscopic observables for rotating and non-rotating relativistic kinetic gases around a Schwarzschild black hole
In this article, we derive the components of the entropy covector field for a relativistic kinetic gas composed of collisionless, spinless, massive, and uncharged particles following bound orbits in a curved spacetime background. By assuming a dependence on the inclination angle of the particle orbits, we consider two distinct models that describe a rotating and a non-rotating relativistic kinetic gas around a Schwarzschild black hole. We analyze the behavior of key macroscopic observables (including the anisotropy parameter and the kinetic temperature) which are constructed from the particle density, energy density, and principal pressures. We aim to characterize and compare the morphology of the resulting configurations, thereby extending and complementing a previous work. The results reveal significant differences between the rotating and non-rotating cases, particularly in the asymptotic behavior of the anisotropy parameter, kinetic temperature, and average pressure, highlighting the role of angular momentum in shaping the macroscopic properties of collisionless gases in strong gravitational fields.