2 The Einstein–Vlasov System

In this section we consider a self-gravitating collision-less gas in the framework of general relativity and we present the Einstein–Vlasov system. It is most often the case in the mathematics literature that the speed of light c and the gravitational constant G are normalized to one, but we keep these constants in the formulas in this section since in some problems they do play an important role. However, in most of the problems discussed in the forthcoming sections these constants will be normalized to one.

Let M be a four-dimensional manifold and let gab be a metric with Lorentz signature (− ,+, +, + ) so that (M, gab) is a spacetime. The metric is assumed to be time-orientable so that there is a distinction between future and past directed vectors.

The possible values of the four-momentum pa of a particle with rest mass m belong to the mass shell Pm ⊂ T M, defined by

Pm := {(xa,pa) ∈ TM : gab(xa)papb = − m2c2, pa is future directed}. (25 )
Hence, if m > 0, a Pm (x ) is the set of all future-directed time-like vectors with length cm, and if m = 0 it is the set of all future-directed null vectors. On Pm we take (xa, pj),a = 0,1,2,3 and j = 1,2, 3 (letters in the beginning of the alphabet always take values 0,1,2,3 and letters in the middle take 1,2,3) as local coordinates, and p0 is expressed in terms of pj and the metric in view of Equation (25View Equation). Thus, the density function fm is a non-negative function on Pm. Below we drop the index m on fm and simply write f.

Since we are considering a collisionless gas, the particles follow the geodesics in spacetime. The geodesics are projections onto spacetime of the curves in Pm defined in local coordinates by

dxa ----= pa, ds dpj- j b c ds = − Γ bcp p .
Here a Γ bc are the Christoffel symbols. Along a geodesic the density function a j f = f (x ,p ) is invariant so that
d a j ---f(x (s),p (s)) = 0, ds

which implies that

∂f ∂f pa----− Γ jabpapb---= 0. (26 ) ∂xa ∂pj
This is accordingly the Vlasov equation. We point out that sometimes the density function is considered as a function on the entire tangent bundle TM rather than on the mass shell P ⊂ T M m. The Vlasov equation for a a f = f (x ,p ) then takes the form
∂f ∂f pa---a − Γ abcpbpc-a-= 0. (27 ) ∂x ∂p
This equation follows from (26View Equation) if we take the mass shell condition gabpapb = − m2c2 into account. Indeed, by abuse of notation, we have
0 ∂f--= ∂f--+ -∂f-∂p--, ∂xa ∂xa ∂p0 ∂xa ∂f ∂f ∂f ∂p0 --j-= --j-+ ---0--j. ∂p ∂p ∂p ∂p
Here f is considered as a function on Pm in the left-hand side, and on TM in the right-hand side. From the mass shell condition gabpapb = − m2c2 we derive
0 ∂p-- = − -1pbpcΓ c, ∂xa p0 ab ∂p0 pj --j-= − --. ∂p p0
Inserting these relations into (26View Equation) we obtain (27View Equation). If we let t = x0, and divide the Vlasov equation (26View Equation) by p0 we obtain the most common form in the literature of the Vlasov equation
∂f- pj∂f-- 1--j a b-∂f- ∂t + p0∂xj − p0Γabp p ∂pj = 0. (28 )
In a fixed spacetime the Vlasov equation (28View Equation) is a linear hyperbolic equation for f and we can solve it by solving the characteristic system,
dXi P i ----= --0, (29 ) ds i P a b dP-- i P-P--- ds = − Γ ab P 0 . (30 )
In terms of initial data f 0 the solution of the Vlasov equation can be written as
f (xa, pi) = f0(Xi (0,xa,pi),Pi(0,xa,pi)), (31 )
where i a i X (s,x ,p) and i a i P (s,x ,p ) solve Equations (29View Equation, 30View Equation), and where
i a i i i a i i X (t,x ,p ) = x and P (t,x ,p ) = p .

In order to write down the Einstein–Vlasov system we need to know the energy-momentum tensor T mab in terms of f and gab. We define

∘ ----∫ dp1dp2dp3 Tamb = c |gab| fpapb----------, (32 ) ℝ3 − p0
where, as usual, p = g pb a ab, and |g | ab denotes the absolute value of the determinant of g ab. We remark that the measure
∘ ---- |gab| μ := ------dp1dp2dp3, − p0

is the induced metric of the submanifold a Pm (x ) ⊂ TxaM, and that μ is invariant under Lorentz transformations of the tangent space, and it is often the case in the literature that Tmab is written as

∫ Tm = c f p p μ. ab Pm (xa) a b

Let us now consider a collisionless gas consisting of particles with different rest masses
m ,m ,...,m 1 2 N, described by N density functions f ,j = 1,...,N mj. Then the Vlasov equations for the different density functions fmj, together with the Einstein equations,

∑N R − 1Rg + Λg = 8πG-- Tmk , ab 2 ab ab c4 ab k=1

form the Einstein–Vlasov system for the collision-less gas. Here Rab is the Ricci tensor, R is the scalar curvature and Λ is the cosmological constant.

Henceforth, we always assume that there is only one species of particles in the gas and we write Tab for its energy momentum tensor. Moreover, in what follows, we normalize the rest mass m of the particles, the speed of light c, and the gravitational constant G, to one, if not otherwise explicitly stated that this is not the case.

Let us now investigate the features of the energy momentum tensor for Vlasov matter. We define the particle current density

∫ a a∘ ----dp1dp2dp3- N = − 3 f p |gab| p . ℝ 0

Using normal coordinates based at a given point and assuming that f is compactly supported, it is not hard to see that Tab is divergence-free, which is a necessary compatibility condition since the left-hand side of (2) is divergence-free by the Bianchi identities. A computation in normal coordinates also shows that N a is divergence-free, which expresses the fact that the number of particles is conserved. The definitions of T ab and N a immediately give us a number of inequalities. If V a is a future-directed time-like or null vector then we have a NaV ≤ 0 with equality if and only if f = 0 at the given point. Hence, a N is always future-directed time-like, if there are particles at that point. Moreover, if V a and W a are future-directed time-like vectors then TabV aW b ≥ 0, which is the dominant energy condition. This also implies that the weak energy condition holds. If Xa is a space-like vector, then a b TabX X ≥ 0. This is called the non-negative pressure condition, and it implies that the strong energy condition holds as well. That the energy conditions hold for Vlasov matter is one reason that the Vlasov equation defines a well-behaved matter model in general relativity. Another reason is the well-posedness theorem by Choquet-Bruhat [55Jump To The Next Citation Point] for the Einstein–Vlasov system that we state below. Before stating that theorem we first discuss the conditions imposed on the initial data.

The initial data in the Cauchy problem for the Einstein–Vlasov system consist of a 3-dimensional manifold S, a Riemannian metric g ij on S, a symmetric tensor k ij on S, and a non-negative scalar function f0 on the tangent bundle TS of S.

The relationship between a given initial data set (gij,kij) on S and the metric gab on the spacetime manifold is, that there exists an embedding ψ of S into the spacetime such that the induced metric and second fundamental form of ψ (S) coincide with the result of transporting (g ,k ) ij ij with ψ. For the relation of the distribution functions f and f0 we have to note that f is defined on the mass shell. The initial condition imposed is that the restriction of f to the part of the mass shell over ψ (S) should be equal to f0 ∘ (ψ −1,d(ψ)− 1) ∘ ϕ, where ϕ sends each point of the mass shell over ψ(S ) to its orthogonal projection onto the tangent space to ψ (S). An initial data set for the Einstein–Vlasov system must satisfy the constraint equations, which read

ij 2 R − kijk + (trk ) = 16πρ, (33 ) ∇ikil − ∇l(trk ) = 8πjl. (34 )
Here a b ρ = Tabn n and a ab c j = − h Tbcn, where a n is the future directed unit normal vector to the initial hypersurface, and ab ab a b h = g + n n is the orthogonal projection onto the tangent space to the initial hypersurface. In terms of f0 we can express ρ and jl by (ja satisfies naja = 0, so it can naturally be identified with a vector intrinsic to S)
∫ a ∘ ----dp1dp2dp3 ρ = f0p pa |gij|-------j-, ∫ ℝ3 ∘ ---- 1 + pjp j = f p |g |dp1dp2dp3. l ℝ3 0 l ij
We can now state the local existence theorem by Choquet-Bruhat [55Jump To The Next Citation Point], for the Einstein–Vlasov system.

Theorem 1 Let S be a 3-dimensional manifold, gij a smooth Riemannian metric on S, kij a smooth symmetric tensor on S and f 0 a smooth non-negative function of compact support on the tangent bundle T S of S. Suppose that these objects satisfy the constraint equations (33View Equation, 34View Equation). Then there exists a smooth spacetime (M, gab), a smooth distribution function f on the mass shell of this spacetime, and a smooth embedding ψ of S into M, which induces the given initial data on S such that gab and f satisfy the Einstein–Vlasov system and ψ (S) is a Cauchy surface. Moreover, given any other spacetime ′ ′ (M ,gab), distribution function ′ f and embedding ′ ψ satisfying these conditions, there exists a diffeomorphism χ from an open neighborhood of ψ (S) in M to an open neighborhood of ψ ′(S ) in M ′, which satisfies χ ∘ ψ = ψ ′ and carries gab and f to g′ab and f′, respectively.

The above formulation is in the case of smooth initial data; for information on the regularity needed on the initial data we refer to [55] and [118]. In this context we also mention that local existence has been proven for the Yang–Mills–Vlasov system in [56], and that this problem for the Einstein–Maxwell–Boltzmann system is treated in [30Jump To The Next Citation Point]. However, this result is not complete, as the non-negativity of f is left unanswered. Also, the hypotheses on the scattering kernel in this work leave some room for further investigation. The local existence problem for physically reasonable assumptions on the scattering kernel does not seem well understood in the context of the Einstein–Boltzmann system, and a careful study of this problem would be desirable. The mathematical study of the Einstein–Boltzmann system has been very sparse in the last few decades, although there has been some activity in recent years. Since most questions on the global properties are completely open let us only very briefly mention some of these works. Mucha [117] has improved the regularity assumptions on the initial data assumed in [30]. Global existence for the homogeneous Einstein–Boltzmann system in Robertson–Walker spacetimes is proven in [125], and a generalization to Bianchi type I symmetry is established in [124].

In the following sections we present results on the global properties of solutions of the Einstein–Vlasov system, which have been obtained during the last two decades.

Before ending this section we mention a few other sources for more background on the Einstein–Vlasov system, cf. [156Jump To The Next Citation Point, 158, 73, 176].

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