The Einstein–Vlasov system

1.4 The Einstein–Vlasov system

In this section we will consider a self-gravitating collisionless gas and we will write down the Einstein–Vlasov system and describe its general mathematical features. Our presentation follows to a large extent the one by Rendall in [101

]. We also refer to Ehlers [38 ] and Stewart [109 ] for more background on kinetic theory in general relativity.

Let $M$ be a four-dimensional manifold and let $gab$ be a metric with Lorentz signature $(− ,+, +, + )$ so that $(M, gab)$ is a spacetime. We use the abstract index notation, which means that $gab$ is a geometric object and not the components of a tensor. See [117 ] for a discussion on this notation. The metric is assumed to be time-orientable so that there is a distinction between future and past directed vectors. The worldline of a particle with non-zero rest mass $m$ is a timelike curve and the unit future-directed tangent vector $a v$ to this curve is the four-velocity of the particle. The four-momentum $a p$ is given by $a mv$ . We assume that all particles have equal rest mass $m$ and we normalize so that $m = 1$ . One can also consider massless particles but we will rarely discuss this case. The possible values of the four-momentum are all future-directed unit timelike vectors and they constitute a hypersurface $P$ in the tangent bundle $T M$ , which is called the mass shell. The distribution function $f$ that we introduced in the previous sections is a non-negative function on $P$ . Since we are considering a collisionless gas, the particles travel along geodesics in spacetime. The Vlasov equation is an equation for $f$ that exactly expresses this fact. To get an explicit expression for this equation we introduce local coordinates on the mass shell. We choose local coordinates on $M$ such that the hypersurfaces $0 t = x =$ constant are spacelike so that $t$ is a time coordinate and $j x$ , $j = 1,2,3$ , are spatial coordinates (letters in the beginning of the alphabet always take values $0,1,2,3$ and letters in the middle take $1,2,3$ ). A timelike vector is future directed if and only if its zero component is positive. Local coordinates on $P$ can then be taken as $xa$ together with the spatial components of the four-momentum $a p$ in these coordinates. The Vlasov equation then reads

Here $a, b = 0,1,2,3$ and $j = 1, 2,3$ , and $Γ j ab$ are the Christoffel symbols. It is understood that $p0$ is expressed in terms of $j p$ and the metric $gab$ using the relation $a b gabp p = − 1$ (recall that $m = 1$ ).

In a fixed spacetime the Vlasov equation is a linear hyperbolic equation for $f$ and we can solve it by solving the characteristic system,

i i dX--= P--, (26 ) ds P 0 dP i P aP b ----= − Γ iab--0-. (27 ) ds P

In terms of initial data $f0$ the solution to the Vlasov equation can be written as

where $Xi(s,xa, pi)$ and $Pi(s,xa,pi)$ solve Equations (26

, 27

), and where $Xi (t,xa, pi) = xi$ and $P i(t,xa, pi) = pi$ .

In order to write down the Einstein–Vlasov system we need to define the energy-momentum tensor $Tab$ in terms of $f$ and $gab$ . In the coordinates $a a (x ,p )$ on $P$ we define

∫ 1∕2dp1dp2dp3 Tab = − f papb|g| ----------, ℝ3 p0

where as usual $pa = gabpb$ , and $|g |$ denotes the absolute value of the determinant of $g$ . Equation (25 ) together with Einstein’s equations,

G := R − 1-Rg = 8πT + Λg , ab ab 2 ab ab ab

then form the Einstein–Vlasov system. Here $Gab$ is the Einstein tensor, $Rab$ the Ricci tensor, $R$ is the scalar curvature and $Λ$ is the cosmological constant. In most of this review we will assume that $Λ = 0$ , but Section 2.3 is devoted to the case of non-vanishing cosmological constant (where also the case of adding a scalar field is discussed). We now define the particle current density

∫ a a 1∕2dp1-dp2-dp3 N = − ℝ3 f p |g| p0 .

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 compatability condition since $Gab$ 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 $a V$ is a future directed timelike or null vector then we have $a NaV ≤ 0$ with equality if and only if $f = 0$ at the given point. Hence $N a$ is always future directed timelike if there are particles at that point. Moreover, if $V a$ and $W a$ are future directed timelike vectors then $TabV aW b ≥ 0$ , which is the dominant energy condition. If $Xa$ is a spacelike vector then $a b TabX X ≥ 0$ . This is called the non-negative pressure condition. These last two conditions together with the Einstein equations imply that $a b RabV V ≥ 0$ for any timelike vector $a V$ , which is the strong energy condition. 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 for the Einstein–Vlasov system that we will state below. Before stating that theorem we will first discuss the initial conditions imposed.

The data in the Cauchy problem for the Einstein–Vlasov system consist of the induced Riemannian metric $gij$ on the initial hypersurface $S$ , the second fundamental form $kij$ of $S$ and matter data $f0$ . The relations between a given initial data set $(gij,kij)$ on a three-dimensional manifold $S$ and the metric $gij$ 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 $(gij,kij)$ 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

Here $ρ = Tabnanb$ and $ja = − habTbcnc$ , where $na$ is the future directed unit normal vector to the initial hypersurface and $hab = gab + nanb$ is the orthogonal projection onto the tangent space to the initial hypersurface. In terms of $f0$ we can express $ρ$ and $l j$ by ( $a j$ satisfies $a naj = 0$ so it can naturally be identified with a vector intrinsic to $S$ )

∫ a (3) 1∕2dp1dp2 dp3 ρ = f p pa| g| --------j-, ∫ℝ3 1 + pjp j = f p |(3)g |1∕2dp1 dp2 dp3. l ℝ3 l

Here $(3) | g|$ is the determinant of the induced Riemannian metric on $S$ . We can now state the local existence theorem by Choquet-Bruhat [25 ] 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 (29 , 30 ). 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 neighbourhood of $ψ (S )$ in $M$ to an open neighbourhood of $ψ′(S)$ in $M ′$ which satisfies $χ ∘ ψ = ψ ′$ and carries $gab$ and $f$ to $g′ab$ and $f ′$ , respectively.

In this context we also mention that local existence has been proved for the Yang–Mills–Vlasov system in [26 ], and that this problem for the Einstein–Maxwell–Boltzmann system is treated in [11 ]. This result is however not complete, the non-negativity of $f$ is left unanswered. Also, the hypotheses on the scattering kernel in this work leave some room for further investigation. This problem concerning physically reasonable assumptions on the scattering kernel seems not well understood in the context of the Einstein–Boltzmann system, and a careful study of this issue would be desirable.

A main theme in the following sections is to discuss special cases for which the local existence theorem can be extended to a global one. There are interesting situations when this can be achieved, and such global existence theorems are not known for Einstein’s equations coupled to other forms of phenomenological matter models, i.e. fluid models (see, however, [30 ]). In this context it should be stressed that the results in the previous sections show that the mathematical understanding of kinetic equations on a flat background space is well-developed. On the other hand the mathematical understanding of fluid equations on a flat background space (also in the absence of a Newtonian gravitational field) is not satisfying. It would be desirable to have a better mathematical understanding of these equations in the absence of gravity before coupling them to Einstein’s equations. This suggests that the Vlasov equation is natural as matter model in mathematical general relativity.