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 which, in the coordinates , takes the form
The particle distribution defined on the mass shell in this metric is given by, where static solutions are analyzed and where also more details on the derivation of the system are given. Although the Nordström–Vlasov model of gravity does not describe physics correctly (in the classical limit the Vlasov–Poisson system of Newtonian gravity  is however obtained) it can nevertheless serve as a platform for developing new mathematical methods. The system has some common features with the Einstein–Vlasov system (see next Section 1.4) but seems in many respects to be less involved. The closest relationship from a mathematical point of view is the Vlasov–Maxwell system; this is evident in  where weak solutions are obtained in analogy with , in  where a continuation criterion for existence of classical solutions is established in analogy with  (an alternative approach is given in ), and in  where global existence in the case of two space dimensions is proved in analogy with . The spherically symmetric case is studied in  and cannot be directly compared to the spherically symmetric Vlasov–Maxwell system (i.e. the Vlasov–Poisson system) since the hyperbolic nature of the equations is still present in the former system while it is lost in the latter case. In  it is shown that classical solutions exist globally in time for compactly supported initial data under the additional condition that there is a lower bound on the modulus of the angular momentum of the initial particle system. It should be noted that this is not a smallness condition and that the result holds for arbitrary large initial data satisfying this hypothesis.
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