Analysis of the Numerical Results

The graph seen in Figure 1 shows most of the functions involved in the solution, for a typical mid-range set of the dimensionless parameters, namely $n=3/2$, $C=0.05$, and $\pi _{e}=1.0$. The crucial function, from which everything else stems, is the function $\gamma (\xi )$, which in all cases has the same qualitative behavior shown in that graph, for any values of the parameters for which a solution exists. It is a very simple function, that slopes up monotonically from a constant negative value to a constant positive value. These two constant values determine $\xi_{\mu}$ and $\xi_{M}$ respectively. Its derivative $\pi (\xi )$ is also a simple function, with a single well-defined point of maximum. It is closely related to the matter energy density $\rho (\xi )$, which is also shown in the graph. The second derivative $\pi '(\xi )$ gives the numerical propagation function. For the value $n=3/2$, which is used in this case, it clearly marks the positions of the interface points, where its graph hits the $\xi$ axis at right angles.

The general behavior of the solution is that almost always there are two interface points and thus both an inner vacuum region and an outer vacuum region. However, the parameters can be judiciously adjusted so as to decrease the width of the inner vacuum region to zero, in which case the shell becomes a filled sphere. In this case one gets $\xi_{1}=0$, so that the inner interface tends to the origin, and then one gets $\xi_{\mu}=0$ as well, so that we have that $\gamma (\xi )$ is zero at the origin, that is, the Tooper boundary condition $\gamma(0)=0$ holds, as is the case in most treatments, such as in [#!LandauClassicalFields!#,#!WeinbergGC!#,#!Wald!#,#!MisnerThorneWheeler!#]. Therefore, this takes us back to the solutions found by Tooper [#!tooper!#].

For each given value of $C$ there is a minimum positive and non-zero value of $\pi_{e}$ that is allowed, which is the one that gives the Tooper solution. Below that minimum value of $\pi_{e}$ a hard singularity of $\pi (\xi )$ is generated, corresponding to an infinite matter energy density, and then there is no acceptable solution to the problem. On the other hand, the outer interface point seems to always exist, for all allowed values of the parameters of the system. The only limitation on the values of the parameters due to existence conditions for the solutions are those related to the inner interface.

The graph in Figure 2 shows the functions $\nu (\xi )$ and $\lambda (\xi )$ that describe the geometry of the solution for this same set of input parameters. Outside the outer interface these are the just the functions of the exterior Schwarzschild solution, for which we have that $\nu(\xi)=-\lambda(\xi)$, with $\lambda(\xi)>0$ and $\nu(\xi)<0$. Somewhere within the matter region there is a crossing of the graphs of $\nu (\xi )$ and $\lambda (\xi )$, so that for sufficiently small $\xi$ these signs are reversed, and we then have that $\lambda(\xi)<0$ and $\nu(\xi)>0$. Inside the inner interface these are the functions of the exact solution for the inner vacuum. Since these two functions have singularities at the origin, the graphs are limited to a region within which the graphs stay below certain maximum absolute values of $\nu (\xi )$ and $\lambda (\xi )$. It is worth emphasizing that these are the only two functions involved that diverge somewhere in their domains, and that such divergences only happen at the origin.

The graph in Figure 3 shows the quantity $\log(\pi)$ as a function of $\log(\xi-\xi_{1})$, for this first set of values of the parameters. This graph depicts the behavior of the function $\pi (\xi )$ in the approach of the inner interface at $\xi_{1}$ from within the matter region, which proceeds from the right to the left in the graph. In this limit $\pi (\xi )$ behaves indeed as $\left(\xi-\xi_{1}\right)^{3/2}$, as one can see from the fact that the slope of the straight line shown in the graphs is exactly $3/2$, within the numerical precision level in use. This straight line was obtained by linear regression from the numerical data. This result confirm the analysis made in Section 3, and it also shows that $\pi (\xi )$ in fact hits zero at $\xi_{1}$, since otherwise this log-log graph could not turn out to be a straight line.

In the graph shown in Figure 4 we display a high-density case, in which we use the value $n=3$, a smaller value of the parameter $C$, with $C=0.01$, and a much larger value of the parameter $\pi_{e}$, with $\pi _{e}=100.0$. In this case the inner vacuum is very wide in the radial direction, and all the matter is concentrated within a relatively thin shell located closer to the outer interface than to the inner one.

As one can see in the graph in Figure 5, in this case the geometry becomes much more extreme. The constant value of $\gamma (\xi )$ within the inner vacuum regions becomes large and negative. For large but finite values of $\pi_{e}$ this significantly decreases the physical volume of the inner vacuum region, as compared to its apparent volume. Note that within the inner vacuum region, since the radial lengths shrink, while the angular ones remain invariant, the geometry of a two-dimensional spacial section through the origin, of this four-dimensional geometry, is not embeddable in a flat three-dimensional space, in the way that the exterior Schwarzschild geometry is.

The data for the third set of parameters, shown in the graphs is Figures 6 to 8, a low-density example with $n=3/2$, $C=1/3$, and $\pi _{e}=10^{-3}$, shows how our solution approaches the behavior that is expected by our classical Newtonian intuition, along most of the matter distribution. This example is well along the limit leading to the Tooper solutions, as indicated by the very small value of $\xi_{\mu}$ given in the caption of Figure 6. In this case the inner vacuum region and the singularity that it contains become reduced to a very small region near the origin. The smaller the value of $\pi_{e}$, the smaller this region becomes, and in the $\pi_{e}\to 0$ limit it is reduced to a single point. Under appropriate conditions, with small but non-zero $\pi_{e}$, this small singular region may then be washed away by the thermal fluctuations of the system, since they violate spherical symmetry at a sufficiently small length scale.

In this case the geometry becomes almost flat almost everywhere, since both $\nu (\xi )$ and $\lambda (\xi )$ become very small, as can be seen in the graph shown in Figure 7. What curvature there is becomes quite smooth, and embeddable in three dimensions, all the way down to the point where $\lambda (\xi )$ becomes zero. This only fails to be the case within a very small region near the origin. In Figure 8 one can see the matter energy density, which has a maximum quite close to the origin, and is monotonically decreasing outward from there, just as is hypothesized in the case of the Tooper solutions. In the $\pi_{e}\to 0$ limit the point of maximum of the matter energy density tends to the origin.

Note that while in the low-density case the inner vacuum region that contains the singularity becomes ever smaller as $\pi_{e}$ decreases, in the case of the data for the second set of parameters, with a large value of $\pi_{e}$, the inner vacuum region and the effects of the central singularity spread throughout most of the interior of the object, and thus cannot be ignored. Therefore, while for low-density objects our solution does not differ significantly from the Tooper solutions, for dense objects it is rather dramatically different from it.