Conclusions and Outlook

We have established a two-parameter class of solutions of the Einstein gravitational field equation, corresponding to the presence of static fluid matter with the equation of state $P=\omega\rho$, for all values of $\omega$ in the relevant interval $(0,1/3]$. There are in fact three parameters in play, the equation of state parameter $\omega$, the Schwarzschild radius $r_{M}$ and the radius $r_{S}$ of the sphere containing essentially all the matter, where we could put our arbitrary radial reference point $r_{0}$, but the solutions depend only on $\omega$ and on a ratio such as $r_{M}/r_{S}$. The fluid matter is strongly localized, and the Schwarzschild solution is indeed the $r\to\infty$ asymptotic limit of the solutions we found here.

Two main physical facts can be abstracted from the results. The first is that the central space in the interior of a dense conglomeration of matter is necessarily shrunk, with actual radial lengths that are smaller, and in some circumstances much smaller, than the corresponding variations of the coordinate $r$. It is even possible that these lengths go all the way to zero in the limit in which a black hole with an event horizon would form. In any case, this reduces the available volume of space under the matter, and therefore compresses the matter even further. This seems to indicate that the apparent volume of a very dense star or black hole is therefore much larger than the actual physical one.

The second is that it is possible to have a static solution with a localized conglomeration of pure energy, that is, of matter with the pure radiation equation of state $\rho=3P$, matter that consists, therefore, of pure relativistic radiation. Of course this is, in part, a consequence of the fact that we are ignoring the outward radiation of energy from the surface of the fluid matter to asymptotic infinity. However, close to the limit in which a black hole would form the loss of energy by this mechanism is strongly contained by the red shift effect upon outgoing radiation originating from close to the position of the Schwarzschild radius, thus justifying the static hypothesis as an approximation. Therefore, we can only claim that this type of static or quasi-static solution with pure radiation exists in the case of distributions of fluid matter which are very close to a black-hole configuration.

In the limit in which we make $\pi(\xi_{i})\to\infty$, where $\xi _{i}$ is the position of the inflection point of $\gamma (\xi )$, we seem to have that $r_{S}\to r_{M}$, where $r_{S}>r_{M}$ is the radial position of the sphere that contains essentially all the fluid matter, and $r_{M}$ is the Schwarzschild radius of the corresponding mass $M$. At the same time we seem to have that $\gamma(\xi)\to-\infty$ for almost all $\xi<\xi_{i}$, so that all radial lengths within the matter distribution are going toward zero. In this limit one does seem to obtain a black hole, in the usual sense given to that term, or at least something that is indistinguishable from it when viewed from the outside. This is not, however, a ``naked'' black hole as is usually thought to be the case, but a ``dressed'' one, with plenty of matter and a definite, if unexpected, geometry within it.

It is very interesting to observe that this situation establishes an unexpected connection with G. 't Hooft's ideas about quantum mechanics and black holes [#!GTHooftNoInterior!#]. In describing his studies involving quantum entanglement around black holes, that author has used the figure of speech that the interior of the black hole ``is not really there'', motivated by quantum correlations between antipodal points of the surface of the black hole. Well, one way to have this situation realized is to have the internal volume of the black hole be zero, and the distances between these diametrically opposed points also be zero, just like all other radial lengths within the region containing the fluid matter. It is interesting that while that author's conclusions come from a quantum theory involving black holes, our conclusions here emerge from a theory which is, in so far as one can currently see, entirely classical.

We end with a few words about the possible routes for the continuation of this line of work. It is to be noted that this is a very simple model, in so far as the hypotheses about the matter are concerned, and one should not expect it to have immediate applicability in all realistic physical cases of interest. For realistic stars, which are known to have an internal structure consisting of layers, it would probably be necessary to extend the model to include equations of state that depend on $r$, that is, to exchange the constant $\omega$ for a function $\omega(r)$. Also with the intent of making the model more realistic, a possibly simple extension would be to the time-dependent case, still with spherical symmetry. This would concern only the solution strictly within the matter, since by the Jebsen-Birkhoff [#!JebsenTheorem!#,#!BirkhoffTheorem!#] theorem the Schwarzschild solution outside would not change in a significant way. It might be sufficient, though, to simply accommodate slowly changing solutions due to outward radiation from the surface of the fluid matter distribution.

It would certainly be interesting to extend the results obtained here to the case of the Kerr metric, so that the extended results would be applicable to rotating stars and black holes. However, this is likely to turn out to be quite a difficult enterprise. Another possible extension, as an alternative to having a fully non-homogeneous equation of state with a function $\omega(r)$, would be to fluid matter with a discretely non-homogeneous equation of state, possibly fluid matter with layered distributions, with different equations of state in each layer, each with a different constant $\omega$, tied up together by the imposition of appropriate boundary conditions at the interfaces between layers. Although it is not at all clear that this is actually possible, it seems like a possibility worthwhile examining. Instabilities in these configurations could possibly lead to fast transitions such as those which are characteristic of some astrophysical phenomena.

Since what we presented here is a previously unknown class of solutions of the Einstein gravitational field equation, it is likely that these new solutions will find applications in the study of stars, as well as of other denser astrophysical objects. Further study of the non-linear differential equation show in Equations (77) for the function $\beta(\xi)$, and in Equations (79) and (86) for the function $\gamma (\xi )$, may turn out to be useful to elucidate the actual physical structure of extremely dense objects such as neutron stars and black holes. In particular, further numerical studies aimed at clarifying the behavior of various aspects of the geometry in the limit in which $\pi(\xi_{i})\to\infty$, where $\xi _{i}$ is the position of the inflection point of $\gamma (\xi )$, a limit which we might loosely describe as the black-hole limit, would probably be quite interesting. This is currently being worked on, and will be included in the aforementioned forthcoming paper.