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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 φ contains the gravitational effects as described below. The Nordström–Vlasov system reads
∫ ∂2tφ − ā–³x φ = − e4φ š”£āˆ˜--dp----, (22 ) 1 + |p|2 [ 2 −1āˆ•2 ] ∂tš”£ + p^⋅ ∇x š”£ − (∂tφ + p^⋅ ∇x φ) p + (1 + |p| ) ∇x φ ⋅ ∇p š”£ = 0. (23 )
Here
p ^p = āˆ˜--------- 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 (š”£,φ) 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 = e2φdiag (− 1,1,1, 1). μν

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

φ f (t,x,p ) = š”£(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.
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