where ,
,
. As before, asymptotic flatness is expressed by the boundary
conditions
and a regular center requires
Following the notation in Section 3.1, the time-independent Einstein–Vlasov system reads
Recall that there is an additional Equation (39
are conserved along characteristics. is the particle energy and
is the angular momentum squared. If
we let
A common assumption in the literature is to restrict the form of to
for some positive constant , then we obtain the polytropic ansatz. The case of isotropic pressure is
obtained by letting
so that
only depends on
.
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 . 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 in the matter quantities
and
, a non-linear system
for
and
is obtained, in which
Existence of solutions to this system was first proven in the case of isotropic pressure in [144], and
extended to anisotropic pressure in [134
]. 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 [144
, 134
], and those with a
Schwarzschild singularity in the centre [134
].
In [145], Rein and Rendall go beyond the polytropic ansatz and obtain steady states with compact
support and finite mass under the assumption that
satisfies
where ,
,
,
. 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].
http://www.livingreviews.org/lrr-2011-4 |
Living Rev. Relativity 14, (2011), 4
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