Virial inequalities for steady states in relativistic galactic dynamics

TL;DR

This paper establishes virial inequalities for steady states in relativistic galactic models, extending known non-relativistic results to the Nordström-Vlasov and Einstein-Vlasov systems, with implications for energy and mass relations.

Contribution

It derives new virial inequalities for steady states in relativistic Vlasov systems, including bounds involving energy, rest mass, and redshift, using vector field multiplier techniques.

Findings

Energy of steady states bounded by rest mass in Nordström-Vlasov system

Inequality involving energy, rest mass, and redshift in Einstein-Vlasov system

Use of integral inequalities and vector field multipliers in proofs

Abstract

It is well known that steady states of the Vlasov-Poisson system, a widely used model in non-relativistic galactic dynamics, have negative energy. In this paper we derive the analogous property for two relativistic generalizations of the Vlasov-Poisson system: The Nordstr\"om-Vlasov system and the Einstein-Vlasov system. In the first case we show that the energy of steady states is bounded by their total rest mass; in the second case, where we also assume spherical symmetry, we prove an inequality which involves not only the energy and the rest mass, but also the central redshift. In both cases the proof makes use of integral inequalities satisfied by time depedent solutions and which are derived using the vector fields multipliers method.