4 AcknowledgementsThe Einstein-Vlasov System/Kinetic Theory2.2 Cosmological solutions

3 Stationary Solutions to the Einstein-Vlasov System

Equilibrium states in galactic dynamics can be described as stationary solutions of the Einstein-Vlasov system, or of the Vlasov-Poisson system in the Newtonian case. Here we will consider the former case for which only static, spherically symmetric solutions have been constructed, but we mention that in the latter case also, stationary axially symmetric solutions have been found by Rein [61].

In the static, spherically symmetric case, the problem can be formulated as follows. Let the spacetime metric have the form


where tex2html_wrap_inline1989 . As before, asymptotic flatness is expressed by the boundary conditions


and a regular centre requires


Following the notation in section  2.1, the time-independent Einstein-Vlasov system reads



The matter quantities are defined as before:


The quantities


are conserved along characteristics. E is the particle energy and L is the angular momentum squared. If we let


for some function tex2html_wrap_inline1995, the Vlasov equation is automatically satisfied. The form of tex2html_wrap_inline1995 is usually restricted to


where l >-1/2 and tex2html_wrap_inline2001 . If tex2html_wrap_inline2003, for some positive constant tex2html_wrap_inline2005, this is called the polytropic ansatz. The case of isotropic pressure is obtained by letting l =0 so that tex2html_wrap_inline1995 only depends on E . We refer to [57Jump To The Next Citation Point In The Article] for information on the role of tex2html_wrap_inline2013 .

In passing, we mention that for the Vlasov-Poisson system it has been shown [13] that every static spherically symmetric solution must have the form tex2html_wrap_inline2015 . This is referred to as Jeans' theorem. It was an open question for some time to decide whether or not this was also true for the Einstein-Vlasov system. This was settled in 1999 by Schaeffer [83], who found solutions that do not have this particular form globally on phase space, and consequently, Jeans' theorem is not valid in the relativistic case. However, almost all results in this field rest on this ansatz. By inserting the ansatz for f in the matter quantities tex2html_wrap_inline1440 and p, a nonlinear system for tex2html_wrap_inline2023 and tex2html_wrap_inline2025 is obtained, in which




Existence of solutions to this system was first proved in the case of isotropic pressure in [64Jump To The Next Citation Point In The Article] and then extended to the general case in [57Jump To The Next Citation Point In The Article]. The main problem is then to show that the resulting solutions have finite (ADM) mass and compact support. This is accomplished in [64Jump To The Next Citation Point In The Article] for a polytropic ansatz with isotropic pressure and in [57Jump To The Next Citation Point In The Article] for a polytropic ansatz with possible anisotropic pressure. They use a perturbation argument based on the fact that the Vlasov-Poisson system is the limit of the Einstein-Vlasov system as the speed of light tends to infinity [63]. Two types of solutions are constructed, those with a regular centre [64, 57Jump To The Next Citation Point In The Article], and those with a Schwarzschild singularity in the centre [57]. In [66Jump To The Next Citation Point In The Article] Rendall and Rein go beyond the polytropic ansatz and assume that tex2html_wrap_inline1995 satisfies


where tex2html_wrap_inline2029 . They show that this assumption is sufficient for obtaining steady states with finite mass and compact support. The result is obtained in a more direct way and is not based on the perturbation argument mentioned above. Their method is inspired by a work on stellar models by Makino [51], in which he considers steady states of the Euler-Einstein system. In [66] there is also an interesting discussion about steady states that appear in the astrophysics literature. They show that their result applies to most of these steady states, which proves that they have the desirable property of finite mass and compact support.

All solutions described so far have the property that the support of tex2html_wrap_inline1440 contains a ball about the centre. In [60] Rein shows that there exist steady states whose support is a finite, spherically symmetric shell, so that they have a vacuum region in the centre.

At present, there are almost no known results concerning the stability properties of the steady states to the Einstein-Vlasov system. In the Vlasov-Poisson case, however, the nonlinear stability of stationary solutions has been investigated by Guo and Rein [39] using the energy-Casimir method. In the Einstein-Vlasov case, Wolansky [90] has applied the energy-Casimir method and obtained some insights, but the theory in this case is much less developed than in the Vlasov-Poisson case and the stability problem is essentially open.

4 AcknowledgementsThe Einstein-Vlasov System/Kinetic Theory2.2 Cosmological solutions

image The Einstein-Vlasov System/Kinetic Theory
Håkan Andréasson
© Max-Planck-Gesellschaft. ISSN 1433-8351
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