A Sampling of Numerical Results

Let us now show and briefly discuss some sample results obtained by means of a numerical computer program written to calculate the solution for $\gamma (\xi )$, so as to confirm the existence of solutions, as well as to confirm and display some of the main properties of the solutions, in order to illustrate their general behavior in a qualitative and visual way. A full description and discussion of the numerical solution, as well as of the properties of the solutions, will be given in a separate paper, which is forthcoming. This will include a detailed analysis of the $\pi(\xi_{i})\to\infty$ black-hole limit.

In the graphs shown in Figures 4 to 6 one can see results for $\gamma (\xi )$ (solid line), $\pi (\xi )$ (dashed line) and $\pi '(\xi )$ (dotted line) for several runs of a program written to solve Equation (86). The program used to plot these graphs employed the Runge-Kutta fourth-order integration algorithm, running in quadruple-precision mode, that is, in double-precision mode on a $64$-bit machine, with a self-adjustable increment for $\xi$, and about $1000$ plotting points. The most relevant physical data in each case is reported within each graph, such as the value of $\gamma(0)$ and the value of $\xi _{M}$. The graph in Figure 4 exemplifies the low energy-density regime, the one in Figure 5 the middle energy-density regime, and that in Figure 6 the high energy-density regime.

Figure 4: Sample solution for the function $\gamma (\xi )$ (solid line), showing also the the corresponding derivative function $\pi (\xi )$ (dashed line), as well as the second derivative function $\pi '(\xi )$ (dotted line). In this case we used $\omega =0.001$ and $\pi (\xi _{i})=0.1$ at the inflection point, so that we are in the low energy-density regime.

In two of these graphs, those in Figures 5 and 6, a rendering of the identity function is included (dash-dotted line), for comparison with $\gamma (\xi )$. Since the difference $[\xi-\gamma(\xi)]$ appears in the denominator of the factor $\exp[2\lambda(r)]$ that multiplies the term $dr^{2}$ in the invariant interval, the proximity between the two corresponding curves tells us how close we are to having a singularity in that factor. The critical point $\xi _{c}$ is the point of greatest proximity between $\xi$ and $\gamma (\xi )$ and thus indicates approximately where the corresponding event horizon would form, if and when that was the case. The critical point is also the point of minimum of the difference $[\xi-\gamma(\xi)]$, which seems to be always positive, as one would expect.

Figure 5: Sample solution for the function $\gamma (\xi )$ (solid line), showing also the identity function (dash-dotted line) and normalized versions of the corresponding derivative function $\pi (\xi )$ (dashed line) and second derivative function $\pi '(\xi )$ (dotted line), so that their maximum amplitudes become $1$. In this case we used $\omega =0.01$ and $\pi (\xi _{i})=2.0$ at the inflection point, so that we are in the middle energy-density regime.

Due to the rather large values either set or obtained for some of these quantities, in the graphs shown in Figures 5 and 6 the curves for $\pi (\xi )$ and $\pi '(\xi )$ are normalized so that the maximum value of their amplitudes become $1$. For the same reason, in the graph in Figure 6 the negative part of the data for $\gamma (\xi )$ is given in a logarithmic scale, using base-$10$ logs. What is actually plotted is the quantity $-\log_{10}[1-\gamma(\xi)]$. This means that the negative part of the curve for $\gamma (\xi )$ in this graph has much larger absolute values than is immediately apparent. All the other parts of the curves shown are on a linear scale, including the positive part of the curve for $\gamma (\xi )$.

Figure 6: Sample solution for the function $\gamma (\xi )$ (solid line), showing also the identity function (dash-dotted line) and normalized versions of the corresponding derivative function $\pi (\xi )$ (dashed line) and second derivative function $\pi '(\xi )$ (dotted line), so that their maximum amplitudes become $1$. In this case we used $\omega =0.1$ and $\pi (\xi _{i})=40.0$ at the inflection point, so that we are in the high energy-density regime. In this case the negative portion of $\gamma (\xi )$ is shown in a base-$10$ logarithmic scale.

The basic qualitative behavior of both the function $\gamma (\xi )$ and the function $\pi (\xi )$ is thus confirmed by the numerical analysis. The limits both for $\xi\to 0$ and for $\xi\to\infty$ behave just as was predicted by our previous analysis. In particular, the Schwarzschild solution is indeed the $r\to\infty$ asymptotic limit of our solutions here, thus indicating that the matter is indeed localized, and in most cases quite strongly so, with an apparent exponential decay of the energy density for large values of $\xi$. The fact that $\gamma (\xi )$ and hence $\beta (r)$ can become negative and large has important consequences. The fact that they become negative seems to be a general feature of all the solutions worked out so far. Once $\gamma (\xi )$ and hence $\beta (r)$ become negative, the metric factor $\exp[2\lambda(r)]$ becomes

$$\,{\rm e}^{2\lambda(r)} = \frac{1}{1+\vert\gamma(\xi)\vert/\xi},$$ (106)

which goes towards zero as $\vert\gamma(\xi)\vert/\xi$ increases. Note that, once $\gamma (\xi )$ becomes negative, there is no longer any possibility of this factor having a singularity like the one at $r_{M}$ in the Schwarzschild solution. When $\vert\gamma(\xi)\vert/\xi\gg 1$, we have for the physical element of length $d\ell$ in the radial direction

$$d\ell = \,{\rm e}^{\lambda(r)}dr$$

$$= \frac{dr}{\sqrt{1+\vert\gamma(\xi)\vert/\xi}}$$

$$\ll dr.$$ (107)

When $\xi\to 0$ we have that $\gamma (\xi )$ tends to some possibly large but finite negative value, so that we actually have that

$$\lim_{\xi\to 0} \vert\gamma(\xi)\vert/\xi = \infty,$$ (108)

which means that

$$\lim_{r\to 0} \,{\rm e}^{2\lambda(r)} = 0.$$ (109)

The numerical data seems to indicate that, in the situations in which $[\xi_{c}-\gamma(\xi_{c})]$ is close to zero at the critical point, the quantity $\vert\gamma(\xi)\vert$ seems to tend to become very large for smaller non-zero values of $\xi$, so that the radial lengths are significantly shrunk in most of the interior of the fluid matter distribution. The data seems to be consistent with the situation in which, in the limit in which we make $\pi(\xi_{i})\to\infty$, we have that $[\xi_{c}-\gamma(\xi_{c})]\to 0$, and also that $\gamma(0)\to -\infty$ linearly with $\pi(\xi_{i})$. In fact, it seems that we also have that

$$\lim_{\pi(\xi_{i})\to\infty} \vert\gamma(\xi)\vert = \infty,$$ (110)

for all values of $\xi$ smaller than $\xi _{r}$, so that in this limit the radial lengths shrink all the way to zero in most of the interior of the region containing the matter. In this way, the approach to a black-hole configuration seems to be tied up with a complete shrinkage of most of the volume of the region where the fluid matter is located. Note that, due to the shrinking of the radial lengths, below the value $\xi _{r}$ of $\xi$ there is no possibility of working out an illustrative isometric embedding such as the one we worked out for the Schwarzschild solution, shown in Figure 1.

In such circumstances the geometry within the region containing the fluid matter distribution does have some rather odd characteristics indeed, since the radial lengths shrink while the angular lengths at the same position do not change at all. In essence, this is what makes the embedding just mentioned impossible. Since it suffices for one of the lengths involved do decrease in order for the volume to decrease, this radial shrinkage does cause the volume to shrink under the fluid matter distribution. Note that, although the angular lengths do not change, they cease to be anything close to geodesics of the spatial geometry. One can always go from any point to any other point within the region containing the fluid matter distribution through only radial displacements, going from the first point to the center and from the center to the second point. With enough radial shrinkage, the distance traversed in this way will be smaller than that of a path between the two points that involves angular displacements. If the radial lengths shrink all the way to zero, then the geodesic distances between any two points in the shrunk region are zero. In effect, it is as if all the points in that region become effectively the same point.

According to the numerical data, the larger the value of $\pi(\xi_{i})$, where $\xi _{i}$ is the position of the inflection point of $\gamma (\xi )$, the closer to a singularity of $\exp[2\lambda(r)]$ at the critical point, where $[\xi_{c}-\gamma(\xi_{c})]\to 0$, we get. But so far indications are that one never actually gets to such a singular solution for finite values of $\pi(\xi_{i})$. The larger the value of $\pi(\xi_{i})$, the closer $r_{M}$ will be to the radial position $r_{S}$ of the sphere that contains essentially all the fluid matter, coming from within. In the cases which are closer to exhibiting a singularity of $\exp[2\lambda(r)]$, and therefore closer to the formation of a black hole with an event horizon, the remaining internal volume seems to be highly concentrated near the surface of the matter distribution, while the radial lengths seem to shrink to zero somewhat faster for smaller values of $\xi$. In fact, under these conditions the energy density seems to develop a large and narrow peak near that surface, as can be seen in some cases, if one recalls that $\pi(\xi)=\mathfrak{T}(\xi)$ is closely related to the energy density. All these issues will be described and examined in detail in the aforementioned forthcoming paper.

The case $\omega=1/3$ corresponds to ``fluid matter'' in the form of pure radiation, and it is interesting that even in this case, so long as $\pi(\xi_{i})$ is large enough, we do still obtain static solutions, in which the fluid matter is still strongly localized. Of course, at least in part this is being allowed by the fact that we are ignoring losses of energy by outward radiation from the surface of the matter distribution, a more detailed treatment of which would certainly lead to non-static solutions. However, if we are close to a configuration that has a singularity of $\exp[2\lambda(r)]$ at the critical point, where $[\xi_{c}-\gamma(\xi_{c})]\to 0$, then the strong red shift effect for any outward radiation from the vicinity of the spherical surface at that radial position will tend to make such energy losses very small, and then our static solution may still be a fair approximation of reality.