### 1.3 The Nordström–Vlasov system

Before turning to the main theme of this review, i.e., the Einstein–Vlasov system, we briefly review the
results on the Nordström–Vlasov system. Nordström gravity [120] is an alternative theory of gravity
introduced in 1913. By coupling this model to a kinetic description of matter the Nordström–Vlasov
system results. In Nordström gravity the scalar field describes the gravitational field in the sense given
below. The Nordström–Vlasov system reads
Here
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

The first mathematical study of this system was initiated by Calogero in [43], where the existence of static
solutions is established. The stability of the static solutions was then investigated in [52]. 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 [49] 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 [45]. Another interesting result for the Nordström–Vlasov system
is given in [36], where a radiation formula, similar to the dipole formula in electrodynamics, is rigorously
derived.