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
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
].
![]() |
![]() |
![]() |
![]() |
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.,
.
http://www.livingreviews.org/lrr-2011-4 |
Living Rev. Relativity 14, (2011), 4
![]() This work is licensed under a Creative Commons License. E-mail us: |