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.

Living Rev. Relativity 14, (2011), 4
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