Abstract
The post-Newtonian Jeans equation for stationary self-gravitating systems is derived from the post-Newtonian Boltzmann equation in spherical coordinates. The Jeans equation is coupled with the three Poisson equations from the post-Newtonian theory. The Poisson equations are functions of the energy-momentum tensor components which are determined from the post-Newtonian Maxwell–Jüttner distribution function. As an application, the effect of a central massive black hole on the velocity dispersion profile of the host galaxy is investigated and the influence of the post-Newtonian corrections are determined.
Highlights
The momentum density hydrodynamic equation for stationary self-gravitating systems, which follows from the collisionless Boltzmann equation, is denominated in astrophysics as the Jeans equation
The Jeans equation is a differential equation for the momentum density of stationary self-gravitating systems which is coupled with the Poisson equation for the Newtonian gravitational potential
We have determined the stationary radial component of the momentum density hydrodynamic equation from the post-Newtonian Boltzmann in spherical coordinates. It represents the post-Newtonian Jeans equation for stationary spherically symmetrical self-gravitating systems, which is coupled with the three Poisson equations of the first post-Newtonian theory
Summary
The post-Newtonian approximations are solutions of Einstein’s field equations in successive powers of the ratio v/c where v is a typical speed of the system and c the light speed. The aim of this paper is to determine the first post-Newtonian approximation of the momentum density hydrodynamic equation for stationary spherically symmetrical selfgravitating systems, which corresponds to the first post-Newtonian approximation to the Jeans equation. As an application of the theory developed here, the system of equations consisting of the stationary spherically symmetrical post-Newtonian Jeans equation and Poisson equations is solved numerically for the determination of the effect of a central massive black hole on the velocity dispersion profile of the host galaxy and the corresponding influence of the post-Newtonian corrections in the solution of this problem. The work is outlined as follows: In Section 2, we introduce the post-Newtonian Poisson equations and calculate the energy-momentum tensor components from the postNewtonian Maxwell–Jüttner equilibrium distribution function.
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