denotes the relativistic velocity of a particle with momentum . The mass of each particle, the gravitational constant, and the speed of light are all normalized to one. A solution of this system is interpreted as follows. The spacetime is a Lorentzian manifold with a conformally-flat metric
The particle distribution defined on the mass shell in this metric is given by, where the existence of static solutions is established. The stability of the static solutions was then investigated in . Although the Nordström–Vlasov model of gravity does not describe physics correctly, the system approaches the Vlasov–Poisson system in the classical limit. Indeed, it is shown in  that solutions of the Nordström–Vlasov system tend to solutions of the Vlasov–Poisson system as the speed of light goes to infinity.
The Cauchy problem was studied by several authors [51, 50, 15, 108, 131] and the question of global existence of classical solutions for general initial data was open for some time. The problem was given an affirmative solution in 2006 by Calogero . Another interesting result for the Nordström–Vlasov system is given in , where a radiation formula, similar to the dipole formula in electrodynamics, is rigorously derived.
Living Rev. Relativity 14, (2011), 4
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