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.