Stationary Solutions to the Einstein

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 [85 ].

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

2 2μ(r) 2 2λ(r) 2 2 2 2 2 ds = − e dt + e dr + r (dθ + sin θdϕ ),

where $r ≥ 0$ , $θ ∈ [0,π ]$ , $ϕ ∈ [0,2π]$ . As before, asymptotic flatness is expressed by the boundary conditions

lim λ(r) = lim μ (r) = 0, ∀t ≥ 0, r→ ∞ r→∞

and a regular centre requires

λ(0) = 0.

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

μ−λ v ∘ --------μ−λ x e ∘--------2 ⋅ ∇xf − 1 + |v|2e μr --⋅ ∇vf = 0, (47 ) 1 + |v| r e−2λ(2rλr − 1) + 1 = 8πr2 ρ, (48 ) −2λ 2 e (2rμr + 1) − 1 = 8πr p. (49 )

The matter quantities are defined as before:

∫ ∘ ------2- ρ (x ) = 3 1 + |v| f(x,v) dv, (50 ) ∫ℝ ( )2 p (x ) = x-⋅ v f(x,v) ∘--dv----. (51 ) ℝ3 r 1 + |v|2

The quantities

E := eμ(r)∘1 -+-|v|2, L = |x |2|v|2 − (x ⋅ v)2 = |x × v|2

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

f(x, v) = Φ(E, L)

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

where $l > − 1∕2$ and $L ≥ 0 0$ . If $φ(E ) = (E − E )k 0 +$ , $k > − 1$ , for some positive constant $E 0$ , this is called the polytropic ansatz. The case of isotropic pressure is obtained by letting $l = 0$ so that $Φ$ only depends on $E$ . We refer to [81

] for information on the role of $L0$ .

In passing, we mention that for the Vlasov–Poisson system it has been shown [15 ] that every static spherically symmetric solution must have the form $f = Φ(E, L )$ . 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 [107 ], 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 $ρ$ and $p$ , a nonlinear system for $λ$ and $μ$ is obtained, in which

where

2π ∫ ∞∫ r2(ε2−1) ε2 G Φ(r,μ) = -2- Φ (eμ(r)ε,L)∘---------------dL d ε, r 1 0 ε2 − 1 − L ∕r2 2π ∫ ∞∫ r2(ε2−1) ∘ -------------- H Φ(r,μ) = --- Φ (eμ(r)ε,L) ε2 − 1 − L∕r2 dL dε. r2 1 0

Existence of solutions to this system was first proved in the case of isotropic pressure in [90 ] and then extended to the general case in [81 ]. The main problem is then to show that the resulting solutions have finite (ADM) mass and compact support. This is accomplished in [90 ] for a polytropic ansatz with isotropic pressure and in [81 ] 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 [89 ]. Two types of solutions are constructed, those with a regular centre [90 , 81 ], and those with a Schwarzschild singularity in the centre [81 ]. In [91 ] Rendall and Rein go beyond the polytropic ansatz and assume that $Φ$ satisfies

Φ(E, L) = c(E − E )k Ll + O ((E − E )δ+k)Ll as E → E , 0 + 0 + 0

where $k > − 1$ , $l > − 1∕2$ , $k + l + 1∕2 > 0$ , $k < l + 3∕2$ . 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 [67 ], in which he considers steady states of the Euler–Einstein system. In [91 ] 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 $ρ$ contains a ball about the centre. In [84 ] 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 [51 ] using the energy-Casimir method. In the Einstein–Vlasov case, Wolansky [121 ] 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.