Conclusions

In this paper we have given the complete and exact solution of the Einstein field equations for the case of a shell of liquid matter. Although this particular problem can be seen as having a somewhat academic nature, it does lead us to two important and unexpected conclusions. One of them is that all solutions for shells of liquid matter have a singularity at the origin, within the inner vacuum region, that does not, however, lead to any kind of pathological behavior involving the matter. The other is that, contrary to what is usually thought, a non-trivial gravitational field does exist within a spherically symmetric central cavity, namely the inner vacuum region.

The geometry within the cavity is associated with a spacetime that is contracted in the radial direction, rather than expanded. It is easy to verify that, unlike what happens in the outer vacuum region, the proper radial length, $\ell_{1}$, say from $r=0$ to $r=r_{1}$, is in fact smaller than the corresponding radial coordinate $r_{1}$. We have that $d\ell_{1}=\sqrt{g_{11}}\,dr$, and therefore

$$\ell_{1} = \int_{0}^{r_{1}}dr\, \sqrt{\frac{r}{r+r_{\mu}}}$$

$$< \int_{0}^{r_{1}}dr$$

$$= r_{1},$$ (89)

given that $r_{\mu}>0$. This illustrates the fact that the radial lengths within the inner vacuum region are contracted rather than expanded. The true physical volume of the inner vacuum region is therefore correspondingly smaller than the apparent coordinate volume. This renders this inner geometry not embeddable in the illustrative way that is usually employed in the case of the outer vacuum region.

The gravitational field associated to this geometry, inside the inner vacuum region, can be interpreted as a repulsive field with respect to the origin. This can be ascertained from an examination of the sign of the derivative of $\nu(r)$ in the inner and outer vacuum regions, and its interpretation in terms of the energy of a photon traveling in the radial direction. This sign is positive in the outer vacuum region, corresponding to an attractive field towards the origin, and negative in the inner vacuum region, corresponding to an repulsive field away from the origin. Of course, since $\nu'(r)$ is a continuous function, and since we enter the matter region from the outer vacuum region with a positive derivative, and exit it into the inner vacuum region with a negative derivative, there must be a point within the matter region where $\nu'(r)=0$, and where the derivative flips sign. This is clearly the point $r_{e}$ of minimum of $\nu(r)$, which is also the point of minimum of $z(r)$, and hence the point of maximum of the pressure $P(r)$, a point which already had a role to play in our arguments.

The arisal of a spherically symmetric region where the gravitational field is repulsive rather that attractive with respect to the origin may feel contrary to our classical intuition regarding gravity. However, this type of situation can arise even in the context of a Newtonian framework in flat spacetime, if we use a slightly modified potential. One can acquire an intuitive understanding of the unexpected situation in the inner vacuum region by considering the Newtonian argument for the determination of the gravitational force within a hollow spherically symmetric thin shell of matter, but with a potential that behaves as $1/r^{1+\epsilon}$ for some $\vert\epsilon\vert\ll 1$, thus leading to a force that behaves as $1/r^{2+\epsilon}$.

If one considers a test mass at a point in the interior of the hollow shell, at the position $\vec{r}$ with respect to the center, it is not difficult to use the usual Newtonian argument to show that, if $\epsilon>0$, then the resulting gravitational force at that point is oriented outward, in the direction of $\vec{r}$, towards the shell of matter. In other words, the attraction by the part of the shell that is closer to the point $\vec{r}$ outweighs the attraction from the opposite side, thus leading to a resulting force that repels particles away from the origin. Note that this argument involving a potential behaving in a way other than $1/r$ is the same that can be used to model the precession of the perihelion of orbits in General Relativity using this Newtonian framework. That precession is prograde precisely if $\epsilon>0$.

It is interesting to note that this configuration of the gravitational field tends to stabilize the shell of liquid matter, since any particle of matter that detaches from the liquid and wanders into one of the vacuum regions will be driven back to the bulk of the liquid by the gravitational field. This can be interpreted as a successful stability test satisfied by all the solutions. The general tendency of the gravitational field is therefore that of compressing the shell of fluid matter, from both sides. This suggests that the same interpretation should be valid in the case of a gaseous fluid.

The singularity at the origin is usually thought to be associated with an infinite concentration of matter there, and thus considered to be an evil that must be avoided at any cost. However, this argument only makes any sense at all if one thinks of that singularity as a point of gravitational attraction, rather than as a point of repulsion of matter. Here we do have the singularity, but not the infinite concentration of matter at the origin, due to the repulsive character of the gravitational field around the origin. In any case, the existence of the singularity is not a question of choice, of course, since it is required by the field equations and by the interface boundary conditions that follow from them. One is not at liberty to impose that $r_{\mu}=0$ in order to avoid this singularity.