Introduction

The exterior Schwarzschild solution [#!Schwarzschild!#,#!Wald!#] of the Einstein field equations has played a major role in General Relativity. It describes the effects of gravitation in the vacuum outside a time-independent spherically symmetric distribution of matter. One of the reasons for its importance is its generality -- it only depends on the spherical symmetry and on the total energy of the matter distribution. Jebsen and Birkhoff [#!JebsenTheorem!#,#!BirkhoffTheorem!#] have shown that this solution is still valid even in time-dependent situations, provided that the spherical symmetry is preserved. Another reason for its popularity is the association of the coordinate singularity of this solution, which occurs for a certain value of the radial coordinate, with the presence of an event horizon, thus leading to the concept of black holes.

Less known -- even absent in many standard textbooks on General Relativity -- is the interior Schwarzschild solution [#!SchwarzschildInternal!#,#!Wald!#]. It gives the metric of the space inside a spherically symmetric matter distribution with an energy density which is constant with the radial coordinate. This other solution can be continuously joined with the Schwarzschild vacuum solution that is valid outside the matter distribution. It is less general in that it only describes matter distributions with energy densities that do not depend on the radial coordinate $r$. In addition, it does not contain any singularities. This point is emphasized in many texts, for example in [#!Wald!#,#!MisnerThorneWheeler!#]. Basically, in order to avoid singularities at the center of the matter distribution a certain integration constant is set equal to zero.

For a spherical matter shell characterized by an inner radius $r_{1}$, an outer radius $r_{2}$ and an energy density constant with $r$ the situation is more involved. In the inner vacuum region, where $r<r_{1}$, the solution of the Einstein equations leads to an integration constant, heretofore denoted by $r_{\mu}$, which determines the singularities in the entire inner vacuum region. There are no singularities only if $r_{\mu}=0$. In analogy with what is done for the interior Schwarzschild solution one may feel tempted to set $r_{\mu}=0$ by hand and eliminate all singularities. However, as we are going to show in this paper, the correct approach is to start in the outer vacuum region ($r>r_{2}$), where the exterior Schwarzschild solution holds, and use the continuity of the solution in the two boundaries of the three regions to determine the constant $r_{\mu}$. The rather surprising result is that the imposition of the surface boundary conditions implies that $r_{\mu}>0$, so that the solutions do contain a singularity at the origin. In addition, one can prove that this condition has to be satisfied in order to produce solutions with non-negative pressure inside the matter shell.

It is remarkable that the boundary conditions on matter interfaces for the Einstein field equations seem to play a smaller than expected role in the literature. A rare example in which the role of these boundary conditions is emphasized can be found in [#!XiaochuMei!#], although the author of that paper only obtained solutions containing a negative pressure region inside the matter shell. By analyzing these negative pressure solutions the author concluded that matter cannot collapse towards the center of black holes in general relativity. We are going to show in this paper that it is possible to obtain physically reasonable matter shell solutions of the Einstein equations with non-negative and finite pressure inside the shell. It is important to emphasize that the singularity at the origin in the inner vacuum region does not lead to any divergence of the matter quantities, and in fact stabilizes the matter shell structure. This is so because the gravitational field within the inner vacuum region turns out to be repulsive with respect to the origin. Our solutions for matter shells are expressed in terms of a single integral which for some values of the physical parameters can be written in terms of elementary functions and constitute a new class of exact solutions of the Einstein field equations.

Results similar to the ones we present here were obtained numerically for the case of neutron stars, with a Chandrasekhar-style equation of state [#!WeinbergGC!#], by Ni [#!NiNeutrStars!#], including the presence of inner and outer matter-vacuum interfaces. However, the crucial consideration of the interface boundary conditions was missing from that analysis, thus leading to incomplete results. The discussion of the interface boundary conditions was subsequently introduced by Neslušan [#!NeslusanNeutrStars!#], thus completing the analysis of the case of the neutron stars. Just as in the present work, the discussion of the interface boundary conditions led, also in that case, to an inner vacuum region containing a singularity at the origin and a gravitational field pointing away from the origin, that is, repulsive with respect to the origin. The present work can be considered as an exactly solvable laboratory model that illustrates some of the properties of that numerical solution. It also shows that the properties of the inner vacuum region are not artifacts of that particular problem or of that particular type of equation of state.

This paper is organized as follows. In Section 2 we state and solve the problem; in Section 3 we derive the main physical properties of the solution; in Section 4 we present a two-parameter family of explicit solutions and a few numerical examples; and in Section 5 we present our conclusions.