5.5 Existence of axisymmetric static solutions

As we have seen above, a broad variety of static solutions of the Einstein–Vlasov system has been established, all of which share the restriction that they are spherically symmetric. The recent investigation [18Jump To The Next Citation Point] removes this restriction and proves the existence of static solutions of the Einstein–Vlasov system, which are axially symmetric but not spherically symmetric. From the applications point of view this symmetry assumption is more “realistic” than spherical symmetry, and from the mathematics point of view the complexity of the Einstein field equations increases drastically if one gives up spherical symmetry. Before discussing this result, let us mention that similar results have been obtained for two other matter models. In the case of a perfect fluid, Heilig showed the existence of axisymmetric stationary solutions in [92Jump To The Next Citation Point]. These solutions have non-zero angular momentum since static solutions are necessarily spherically symmetric. In this respect the situation for elastic matter is more similar to Vlasov matter. The existence of static axisymmetric solutions of elastic matter, which are not spherically symmetric, was proven in [1Jump To The Next Citation Point]. Stationary solutions with rotation were then established in [2Jump To The Next Citation Point].

Let us now briefly discuss the method of proof in [18Jump To The Next Citation Point], which relies on an application of the implicit function theorem. Also, the proofs in [92, 1, 2] make use of the implicit function theorem, but apart from this fact the methods are quite different.

The set-up of the problem in [18Jump To The Next Citation Point] follows the work of Bardeen [31], where the metric is written in the form

2 2 ds2 = − c2e2ν∕cdt2 + e2μdρ2 + e2μdz2 + ρ2B2e −2ν∕c dφ2 (68 )
for functions ν,B, μ depending on ρ and z, where t ∈ ℝ,ρ ∈ [0,∞ [,z ∈ ℝ and φ ∈ [0,2π]. The Killing vector fields ∂t and ∂ ϕ correspond to the stationarity and axial symmetry of the spacetime. Solutions are obtained by perturbing off spherically symmetric steady states of the Vlasov–Poisson system via the implicit function theorem and the reason for writing ν∕c2 in the metric, instead of ν, is that ν converges to the Newtonian potential UN of the steady state in the limit c → ∞. Asymptotic flatness is expressed by the boundary conditions
|(ρl,zim)|→ ∞ ν(ρ,z) = |(ρl,zim)|→ ∞ μ(ρ,z) = 0, |(ρl,zi)m|→ ∞ B (ρ,z) = 1. (69 )
In addition the solutions are required to be locally flat at the axis of symmetry, which implies the condition
2 ν(0,z)∕c + μ(0,z) = lnB (0,z), z ∈ ℝ. (70 )
Let us now recall from Section 5.1 the strategy to construct static solutions by using an ansatz of the form
f(x,v ) = Φ (E, L),
where E and L are conserved quantities along characteristics. Due to the symmetries of the metric (68View Equation) the following quantities are constant along geodesics:
E := − g (∂ ∕∂t,pa) = c2e2ν∕c2p0 ∘ ---------------------------------------------- = c2eν∕c2 1 + c−2 (e2μ(p1)2 + e2μ(p2)2 + ρ2B2e −2ν∕c2(p3)2), (71 ) L := g(∂∕∂φ, pa) = ρ2B2e −2ν∕c2p3. (72 )
Here a p are the canonical momenta. E can be thought of as a local or particle energy and L is the angular momentum of a particle with respect to the axis of symmetry. For a sufficiently regular Φ the ansatz function f satisfies the Vlasov equation and upon insertion of this ansatz into the definition of the energy momentum tensor (32View Equation) the latter becomes a functional T = T (ν,B, μ) αβ αβ of the unknown metric functions ν,B, μ. It then remains to solve the Einstein equations with this energy momentum tensor as right-hand side. The Newtonian limit of the Einstein–Vlasov system is the Vlasov–Poisson system and the strategy in [18Jump To The Next Citation Point] is to perturb off spherically symmetric steady states of the Vlasov–Poisson system via the implicit function theorem to obtain axisymmetric solutions. Indeed, the main result of [18Jump To The Next Citation Point] specifies conditions on the ansatz function Φ such that a two parameter (γ and λ) family of axially-symmetric solutions of the Einstein–Vlasov system passes through the corresponding spherically symmetric, Newtonian steady state, whose ansatz function we denote by ϕ. The parameter γ = 1∕c2 turns on general relativity and the parameter λ turns on the dependence on L. Since L is not invariant under arbitrary rotations about the origin the solution is not spherically symmetric if f depends on L. It should also be mentioned that although γ is a priori small, which means that c is large, the scaling symmetry of the Einstein–Vlasov system can be used to obtain solutions corresponding to the physically correct value of c. The most striking condition on the ansatz function Φ, or rather on the ansatz function ϕ of the corresponding Vlasov–Poisson system, needed to carry out the proof is that it must satisfy
6 + 4πr2a (r) > 0, r ∈ [0,∞ [, N

where

∫ ′( 1- 2 ) aN(r) := 3 ϕ 2|v| + UN (r) dv. ℝ

An important argument in the proof is indeed to justify that there are steady states of the Vlasov–Poisson system satisfying this condition.

It is of course desirable to extend the result in [18Jump To The Next Citation Point] to stationary solutions with rotation. Moreover, the deviation from spherically symmetry of the solutions in [18] is small and an interesting open question is the existence of disk-like models of galaxies. In the Vlasov–Poisson case this has been shown in [74].


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