5.1 Existence of spherically-symmetric static solutions

Let us assume that spacetime is static and spherically symmetric. Let the metric have the form
ds2 = − e2μ(r)dt2 + e2λ(r)dr2 + r2(d 𝜃2 + sin2𝜃d φ2),

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, r→∞ r→ ∞

and a regular center requires

λ(0) = 0.

Following the notation in Section 3.1, the time-independent Einstein–Vlasov system reads

μ−λ----v----- ∘ ------2-μ−λ x- e ∘ 1 + |v|2 ⋅ ∇xf − 1 + |v| e μr r ⋅ ∇vf = 0, (55 ) −2λ 2 e (2rλr − 1) + 1 = 8πr ρ, (56 ) e−2λ(2rμ + 1) − 1 = 8πr2p. (57 ) r
Recall that there is an additional Equation (39View Equation) of second order, which contains the tangential pressure pT, but we leave it out since it follows from the equations above. The matter quantities are defined as before:
∫ ∘ ------2- ρ (x ) = 3 1 + |v| f(x,v)dv, (58 ) ∫ℝ ( )2 p (x ) = x-⋅ v f(x,v) ∘--dv----. (59 ) ℝ3 r 1 + |v|2
The quantities
μ(r)∘ ------2- 2 2 2 2 E := e 1 + |v|, L = |x| |v | − (x ⋅ v) = |x × v |,

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), (60 )
for some function Φ, the Vlasov equation is automatically satisfied.

A common assumption in the literature is to restrict the form of Φ to

Φ(E, L ) = ϕ(E )(L − L0)l+, (61 )
where l > − 1∕2, L0 ≥ 0 and x+ = max {x, 0}. If we furthermore assume that
k ϕ (E ) = (E − E0)+,k > − 1,

for some positive constant E0, then we obtain the polytropic ansatz. The case of isotropic pressure is obtained by letting l = 0 so that Φ only depends on E.

In passing, we mention that for the Vlasov–Poisson system it has been shown [35] 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 whether or not this was also true for the Einstein–Vlasov system. This was settled in 1999 by Schaeffer [169], 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 on static solutions are based on this ansatz.

By inserting the ansatz f(x, v) = Φ(E, L) in the matter quantities ρ and p, a non-linear system for λ and μ is obtained, in which

− 2λ 2 e (2r λr − 1) + 1 = 8 πr G Φ(r,μ), e− 2λ(2r μr + 1 ) − 1 = 8 πr2H Φ(r,μ),
where
2π ∫ ∞ ∫ r2(𝜖2− 1) 𝜖2 G Φ(r,μ) = -2- Φ(eμ(r)𝜖,L )∘--------------dLd 𝜖, r 1 0 𝜖2 − 1 − L ∕r2 2π ∫ ∞ ∫ r2(𝜖2− 1) ∘ -------------- H Φ(r,μ) = --- Φ(eμ(r)𝜖,L ) 𝜖2 − 1 − L ∕r2dLd 𝜖. r2 1 0

Existence of solutions to this system was first proven in the case of isotropic pressure in [144Jump To The Next Citation Point], and extended to anisotropic pressure in [134Jump To The Next Citation Point]. The main difficulty is to show that the solutions have finite ADM mass and compact support. The argument in these works to obtain a solution of compact support is to perturb a steady state of the Vlasov–Poisson system, which is known to have compact support. Two different types of solutions are constructed, those with a regular centre [144Jump To The Next Citation Point, 134Jump To The Next Citation Point], and those with a Schwarzschild singularity in the centre [134Jump To The Next Citation Point].

In [145Jump To The Next Citation Point], Rein and Rendall go beyond the polytropic ansatz and obtain steady states with compact support and finite mass under the assumption that Φ satisfies

Φ (E, L) = c(E − E0 )k+Ll + O ((E0 − E )δ++k )Ll as E → E0,

where k > − 1, l > − 1∕2, k + l + 1∕2 > 0, k < l + 3∕2. This result is obtained in a more direct way and is not based on the perturbation argument used in [144, 134]. Their method is inspired by a work on stellar models by Makino [114], in which he considers steady states of the Euler–Einstein system. In [145] there is also a discussion about steady states that appear in the astrophysics literature, and it is shown that their result applies to most of these steady states. An alternative method to obtain steady states with finite radius and finite mass, which is based on a dynamical system analysis, is given in [76].


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