2 Global Existence Theorems for 1 Introduction to Kinetic Theory1.2 The Vlasov-Maxwell and Vlasov-Poisson

1.3 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 [76Jump To The Next Citation Point In The Article]. We also refer to Ehlers [27] and Stewart [85] for more background on kinetic theory in general relativity.

Let M be a four-dimensional manifold and let tex2html_wrap_inline1484 be a metric with Lorentz signature (-,+,+,+) so that tex2html_wrap_inline1488 is a spacetime. We use the abstract index notation, which means that tex2html_wrap_inline1484 is a geometric object and not the components of a tensor. See [89] 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 tex2html_wrap_inline1494 to this curve is the four-velocity of the particle. The four-momentum tex2html_wrap_inline1344 is given by tex2html_wrap_inline1498 . 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 TM, 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 tex2html_wrap_inline1516 constant are spacelike so that t is a time coordinate and tex2html_wrap_inline1520, 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 tex2html_wrap_inline1530 together with the spatial components of the four-momentum tex2html_wrap_inline1344 in these coordinates. The Vlasov equation then reads


Here a, b =0,1,2,3 and j =1,2,3, and tex2html_wrap_inline1538 are the Christoffel symbols. It is understood that tex2html_wrap_inline1540 is expressed in terms of tex2html_wrap_inline1542 and the metric tex2html_wrap_inline1484 using the relation tex2html_wrap_inline1546 (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,



In terms of initial data tex2html_wrap_inline1552 the solution to the Vlasov equation can be written as


where tex2html_wrap_inline1554 and tex2html_wrap_inline1556 solve (23Popup Equation) and (24Popup Equation), and where tex2html_wrap_inline1558 and tex2html_wrap_inline1560 .

In order to write down the Einstein-Vlasov system we need to define the energy-momentum tensor tex2html_wrap_inline1562 in terms of f and tex2html_wrap_inline1484 . In the coordinates tex2html_wrap_inline1568 on P we define


where as usual tex2html_wrap_inline1572 and | g | denotes the absolute value of the determinant of g . Equation (22Popup Equation) together with Einstein's equations


then form the Einstein-Vlasov system. Here tex2html_wrap_inline1578 is the Einstein tensor, tex2html_wrap_inline1580 the Ricci tensor and R is the scalar curvature. We also define the particle current density


Using normal coordinates based at a given point and assuming that f is compactly supported it is not hard to see that tex2html_wrap_inline1562 is divergence-free which is a necessary compatability condition since tex2html_wrap_inline1578 is divergence-free by the Bianchi identities. A computation in normal coordinates also shows that tex2html_wrap_inline1590 is divergence-free, which expresses the fact that the number of particles is conserved. The definitions of tex2html_wrap_inline1562 and tex2html_wrap_inline1590 immediately give us a number of inequalities. If tex2html_wrap_inline1596 is a future directed timelike or null vector then we have tex2html_wrap_inline1598 with equality if and only if f =0 at the given point. Hence tex2html_wrap_inline1590 is always future directed timelike if there are particles at that point. Moreover, if tex2html_wrap_inline1596 and tex2html_wrap_inline1606 are future directed timelike vectors then tex2html_wrap_inline1608, which is the dominant energy condition. If tex2html_wrap_inline1610 is a spacelike vector then tex2html_wrap_inline1612 . This is called the non-negative pressure condition. These last two conditions together with the Einstein equations imply that tex2html_wrap_inline1614 for any timelike vector tex2html_wrap_inline1596, 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 tex2html_wrap_inline1618 on the initial hypersurface S, the second fundamental form tex2html_wrap_inline1622 of S and matter data tex2html_wrap_inline1552 . The relations between a given initial data set tex2html_wrap_inline1628 on a three-dimensional manifold S and the metric tex2html_wrap_inline1618 on the spacetime manifold is that there exists an embedding tex2html_wrap_inline1634 of S into the spacetime such that the induced metric and second fundamental form of tex2html_wrap_inline1638 coincide with the result of transporting tex2html_wrap_inline1628 with tex2html_wrap_inline1634 . For the relation of the distribution functions f and tex2html_wrap_inline1552 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 tex2html_wrap_inline1638 should be equal to tex2html_wrap_inline1654, where tex2html_wrap_inline1446 sends each point of the mass shell over tex2html_wrap_inline1638 to its orthogonal projection onto the tangent space to tex2html_wrap_inline1638 . An initial data set for the Einstein-Vlasov system must satisfy the constraint equations, which read



Here tex2html_wrap_inline1662 and tex2html_wrap_inline1664, where tex2html_wrap_inline1666 is the future directed unit normal vector to the initial hypersurface and tex2html_wrap_inline1668 is the orthogonal projection onto the tangent space to the initial hypersurface. In terms of tex2html_wrap_inline1552 we can express tex2html_wrap_inline1440 and tex2html_wrap_inline1674 by (tex2html_wrap_inline1676 satisfies tex2html_wrap_inline1678 so it can naturally be identified with a vector intrinsic to S)


Here tex2html_wrap_inline1682 is the determinant of the induced Riemannian metric on S . We can now state the local existence theorem by Choquet-Bruhat [17] for the Einstein-Vlasov system.

Theorem 1    Let S be a 3-dimensional manifold, tex2html_wrap_inline1618 a smooth Riemannian metric on S, tex2html_wrap_inline1622 a smooth symmetric tensor on S and tex2html_wrap_inline1552 a smooth non-negative function of compact support on the tangent bundle TS of S . Suppose that these objects satisfy the constraint equations (26Popup Equation) and (27Popup Equation). Then there exists a smooth spacetime tex2html_wrap_inline1488, a smooth distribution function f on the mass shell of this spacetime, and a smooth embedding tex2html_wrap_inline1634 of S into M which induces the given initial data on S such that tex2html_wrap_inline1484 and f satisfy the Einstein-Vlasov system and tex2html_wrap_inline1638 is a Cauchy surface. Moreover, given any other spacetime tex2html_wrap_inline1720, distribution function f ' and embedding tex2html_wrap_inline1724 satisfying these conditions, there exists a diffeomorphism tex2html_wrap_inline1726 from an open neighbourhood of tex2html_wrap_inline1638 in M to an open neighbourhood of tex2html_wrap_inline1732 in M ' which satisfies tex2html_wrap_inline1736 and carries tex2html_wrap_inline1484 and f to tex2html_wrap_inline1742 and f ', respectively.

In this context we also mention that local existence has been proved for the Einstein-Maxwell-Boltzmann system [9] and for the Yang-Mills-Vlasov system [18].

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, [21]). 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 that well-understood. 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.

2 Global Existence Theorems for 1 Introduction to Kinetic Theory1.2 The Vlasov-Maxwell and Vlasov-Poisson

image The Einstein-Vlasov System/Kinetic Theory
Håkan Andréasson
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de