### 1.3 The Nordström–Vlasov system

Before turning to the main theme of this review, i.e. the Einstein–Vlasov system, we introduce a system
which has many mathematical features in common with the Vlasov–Maxwell system and which recently has
been mathematically studied for the first time. It is based on an alternative theory of gravity
which was given by Nordström [70] in 1913. By coupling this model to a kinetic description
of matter the Nordström–Vlasov system results. In Nordström gravity the scalar field
contains the gravitational effects as described 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 which, in the
coordinates , takes the form

The particle distribution defined on the mass shell in this metric is given by

The first mathematical study of this system was undertaken by Calogero in [19], 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 [21] 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 [23] where
weak solutions are obtained in analogy with [35], in [22] where a continuation criterion for existence of
classical solutions is established in analogy with [46] (an alternative approach is given in [76]),
and in [62] where global existence in the case of two space dimensions is proved in analogy
with [44]. The spherically symmetric case is studied in [6] 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 [6] 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.