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 [21, 100, 171] 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 [21], emphasizing that these agree with the conclusions in [171, 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 [21]

Here the metric coefficients , and depend on and , and are positive, and the polar angles and parameterize the unit sphere. Thus, the radial coordinate is the area radius. A maximal gauge condition is then imposed, which means that each hypersurface of constant has vanishing mean curvature. The boundary conditions, which guarantee asymptotic flatness and a regular center, are given by

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,
By specifying values on and steady states are numerically constructed. The distribution function of the steady state is then multiplied by an amplitude , 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 [21].

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

Here

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

where . Both and are conserved quantities. The central redshift is the redshift of a photon emitted from the center and received at infinity, and the binding energy is the difference of the rest mass and the ADM mass. In Figure 5 and Figure 6 the relation between the fractional binding energy and the central redshift is given for two different cases.

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 [21] is summarized in Table 1. Varying the parameters and give rise to essentially the same tables, cf. [21].

Table 1: and .
 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 , it is found that steady states with small values on (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 the evolution leads to the formation of trapped surfaces and collapse to black holes. Hence, for perturbations with the value of alone seems to determine the stability features of the steady states. Plotting versus with higher resolution, cf. [21], gives support to the conjecture by Novikov and Zel’dovich mentioned above that the maximum of along a sequence of steady states signals the onset of instability.

The situation is quite different for perturbations with . The crucial quantity in this case is the fractional binding energy . Consider a steady state with and a perturbation with 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 7.

In [171] it is stated (without proof) that if the solution must ultimately reimplode and the simulations in [21] support that it is true. For negative values of , the solutions with 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 at the center with for , i.e., the solution disperses in a strong sense, then necessarily , i.e., .