Go to previous page Go up Go to next page

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 f contains the gravitational effects as described below. The Nordström-Vlasov system reads
integral @2tf - /_\ xf = - e4f f V~ -dp----, (22) 1 + |p|2 [ 2 -1/2 ] @tf + p . \~/ xf - (@tf + p . \~/ xf) p + (1 + |p| ) \~/ xf . \~/ pf = 0. (23)
Here
p p = V~ -------- 1 + |p|2

denotes the relativistic velocity of a particle with momentum p . The mass of each particle, the gravitational constant, and the speed of light are all normalized to one. A solution (f,f) of this system is interpreted as follows: The spacetime is a Lorentzian manifold with a conformally flat metric which, in the coordinates (t,x), takes the form

g = e2fdiag(- 1,1,1, 1). mn

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

f f (t,x,p) = f(t,x,e p). (24)
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  [46Jump To The Next Citation Point] (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  [6Jump To The Next Citation Point] 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.
Go to previous page Go up Go to next page