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 ϕ describes the gravitational field in the sense given below. The Nordström–Vlasov system reads
∫ ∂2ϕ − △ ϕ = − e4ϕ ∘--𝔣dp---, (22 ) t x ℝ3 1 + |p|2 [ 2 −1∕2 ] ∂t𝔣 + p^⋅ ∇x 𝔣 − (∂tϕ + p^⋅ ∇x ϕ) p + (1 + |p| ) ∇x ϕ ⋅ ∇p 𝔣 = 0. (23 )
^p = ∘----p----, 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,ϕ) of this system is interpreted as follows. The spacetime is a Lorentzian manifold with a conformally-flat metric

gμν = e2ϕdiag (− 1,1,1, 1).

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

ϕ f (t,x,p ) = 𝔣(t,x,e p). (24 )
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.

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