By prescribing the value , the equations can be solved, but the resulting solution will in general
not satisfy the boundary condition
, but it will have some finite limit. It is then possible to shift
both the cut-off energy
and the solution by this limit to obtain a solution, which satisfies
.
A convenient way to handle the problem that
and
cannot both be treated as free parameters is
to use the ansatz
as in [22]. This gives an equation for
, which can be rewritten in terms of the function
In this way the cut-off energy disappears as a free parameter of the problem and we thus have the four free
parameters and
. The structure of the static solutions obtained in [22
] is as
follows:
If the energy density can be strictly positive or vanish at
(depending on
) but it is always strictly positive sufficiently close to
. Hence, the support of the
matter is an interval
with
, and we call such states ball configurations. If
the support is in an interval
, and we call such steady states for
shells.
The value determines how compact or relativistic the steady state is, and the smaller values the
more relativistic. For large values, recall
, a pure shell or a pure ball configuration is obtained,
cf. Figure 1
for a pure shell. Note that we depict the behavior of
but we remark that the pressure
terms behave similarly but the amplitudes of
and
can be very different, i.e., the steady states can
be highly anisotropic.
For moderate values of the solutions have a distinct inner peak and a tail-like outer peak, and by
making
smaller more peaks appear, cf. Figure 2
for the case of ball configurations.
In the case of shells there is a similar structure but in this case the peaks can either be separated
by vacuum regions or by thin atmospheric regions as in the case of ball configurations. An
example with multi-peaks, where some of the peaks are separated by vacuum regions, is given in
Figure 3.
A different feature of the structure of static solutions is the issue of spirals. For a fixed
ansatz of the density function , there is a one-parameter family of static solutions, which are
parameterized by
. A natural question to ask is how the ADM mass
and the radius of the
support
change along such a family. By plotting for each
the resulting values for
and
a curve is obtained, which reflects how radius and mass are related along such a
one-parameter family of steady states. This curve has a spiral form, cf. Figure 4
. It is shown
in [22
] that in the isotropic case, where
the radius-mass curves always have a spiral
form.
http://www.livingreviews.org/lrr-2011-4 |
Living Rev. Relativity 14, (2011), 4
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