5.4 Stability

An important problem is the question of the stability of spherically-symmetric steady states. At present, there are almost no theoretical results on the stability of the steady states of the Einstein–Vlasov system. Wolansky [195] has applied the energy-Casimir method and obtained some insights, but the theory is much less developed than in the Vlasov–Poisson case and the stability problem is essentially open. The situation is very different for the Vlasov–Poisson system, and we refer to [141] for a review on the results in this case.

However, there are numerical studies [21Jump To The Next Citation Point, 100Jump To The Next Citation Point, 171Jump To The Next Citation Point] on the stability of spherically-symmetric steady states for the Einstein–Vlasov system. The latter two studies concern isotropic steady states, whereas the first, in addition, treats anisotropic steady states. Here we present the conclusions of [21Jump To The Next Citation Point], emphasizing that these agree with the conclusions in [171Jump To The Next Citation Point, 100] for isotropic states.

To allow for trapped surfaces, maximal-areal coordinates are used, i.e., the metric is written in the following form in [21Jump To The Next Citation Point]

2 2 2 2 2 2 2 2 2( 2 2 2) ds = − (α − a β )dt + 2a βdtdr + a dr + r d𝜃 + sin 𝜃dĪ• .

Here the metric coefficients α, β, and a depend on t ∈ ℝ and r ≥ 0, α and a are positive, and the polar angles 𝜃 ∈ [0,π] and Ī• ∈ [0,2π ] parameterize the unit sphere. Thus, the radial coordinate r is the area radius. A maximal gauge condition is then imposed, which means that each hypersurface of constant t has vanishing mean curvature. The boundary conditions, which guarantee asymptotic flatness and a regular center, are given by

a(t,0 ) = a (t,∞ ) = α (t,∞ ) = 1. (66 )
Steady states are numerically constructed, and these are then perturbed in order to investigate the stability. More precisely, to construct the steady states the polytropic ansatz is used, cf. Section 5.1,
k l f (r,w, L ) = Φ (E, L) = (E0 − E )+(L − L0)+. (67 )
By specifying values on E0,L0 and α (0 ) steady states are numerically constructed. The distribution function f s of the steady state is then multiplied by an amplitude A, so that a new, perturbed distribution function is obtained. This is then used as initial datum in the evolution code. We remark that also other types of perturbations are analyzed in [21Jump To The Next Citation Point].

For k and l fixed each steady state is characterized by its central red shift Zc and its fractional binding energy Eb, which are defined by

--1-- -eb- Zc = α (0) − 1,Eb = M0 , where eb = M0 − M.


∫ ∞ ∫ ∞ ∫ ∞ 2 M0 = 4π 0 −∞ 0 a(t,r)f(t,r,w,L )dLdwdr

is the total number of particles, which, since all particles have rest mass one, equals the rest mass of the system. M is the ADM mass given by

∫ ∞ ( 3 ) M = 4π ρ(t,r) + -κ2(t,r) r2dr, 0 2

where κ = β∕rα. Both M0 and M are conserved quantities. The central redshift is the redshift of a photon emitted from the center and received at infinity, and the binding energy e b is the difference of the rest mass and the ADM mass. In Figure 5View Image and Figure 6View Image the relation between the fractional binding energy and the central redshift is given for two different cases.

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Figure 5: k = 0,l = 0,L0 = 0.1
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Figure 6: k = 0,l = 3∕2,L0 = 0.1

The relevance of these concepts for the stability properties of steady states was first discussed by Zel’dovich and Podurets [197], who argued that it should be possible to diagnose the stability from binding energy considerations. Zel’dovich and Novikov [196] then conjectured that the binding energy maximum along a steady state sequence signals the onset of instability.

The picture that arises from the simulations in [21Jump To The Next Citation Point] is summarized in Table 1. Varying the parameters k,l and L0 give rise to essentially the same tables, cf. [21Jump To The Next Citation Point].

Table 1: k = 0 and l = 1∕2.
Zc Eb A < 1 A > 1
0.21 0.032 stable stable
0.34 0.040 stable stable
0.39 0.040 stable stable
0.42 0.041 stable unstable
0.46 0.040 stable unstable
0.56 0.036 stable unstable
0.65 0.029 stable unstable
0.82 0.008 stable unstable
0.95 –0.015 unstable unstable
1.20 –0.078 unstable unstable

If we first consider perturbations with A > 1, it is found that steady states with small values on Zc (less than approximately 0.40 in this case) are stable, i.e., the perturbed solutions stay in a neighbourhood of the static solution. A careful investigation of the perturbed solutions indicates that they oscillate in a periodic way. For larger values of Zc the evolution leads to the formation of trapped surfaces and collapse to black holes. Hence, for perturbations with A > 1 the value of Zc alone seems to determine the stability features of the steady states. Plotting Eb versus Zc with higher resolution, cf. [21Jump To The Next Citation Point], gives support to the conjecture by Novikov and Zel’dovich mentioned above that the maximum of Eb along a sequence of steady states signals the onset of instability.

The situation is quite different for perturbations with A < 1. The crucial quantity in this case is the fractional binding energy Eb. Consider a steady state with Eb > 0 and a perturbation with A < 1 but close to 1 so that the fractional binding energy remains positive. The perturbed solution then drifts outwards, turns back and reimplodes, and comes close to its initial state, and then continues to expand and reimplode and thus oscillates, cf. Figure 7View Image.

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Figure 7: Zc = 0.47, Eb = 0.04, A = 0.99, T = 90.0

In [171] it is stated (without proof) that if Eb > 0 the solution must ultimately reimplode and the simulations in [21Jump To The Next Citation Point] support that it is true. For negative values of Eb, the solutions with A < 1 disperse to infinity.

A simple analytic argument is given in [21], which relates the question, whether a solution disperses or not. It is shown that if a shell solution has an expanding vacuum region of radius R (t) at the center with R (t) → ∞ for t → ∞, i.e., the solution disperses in a strong sense, then necessarily M0 ≤ M, i.e., Eb ≤ 0.

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