Conclusions

In this paper we have given the complete static solution of the Einstein field equations for the case of a spherically symmetric distribution of gaseous matter satisfying the equation of state of a polytrope. We have arrived at some of the same important and rather unexpected conclusions that we had already come across in a previous paper [#!LiquidShells!#] on the solution for shells of liquid matter.

One new fact, which is different from that previous case, is that in this case the inner and outer boundaries of the matter, that define a spherical shell, are not imposed by hand, but arise as inevitable consequences of the dynamics of the system, as determined by the Einstein field equations. This makes it impossible to ignore these solutions, for there is no arbitrary choice of a geometrical character involved. All the free parameters of the system are related only to the physical characteristics of the matter, and all geometrical characteristics that arise are inevitable consequences of the equations.

The new solutions converge back to the known Tooper [#!tooper!#] family of solutions in certain limits of the free parameters of the physical system, in which the shell becomes a filled sphere. While for each value of the index $n$ that appears in the polytropic equation of state the Tooper family of solutions is described by a two-dimensional parameter space, the family that we present here is described by a three-dimensional parameter space.

One of the conclusions that is just like in the previous case involving liquid matter is that all solutions of this type 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. This field can be characterized as being repulsive with respect to the origin, which explains why we do not see an infinite concentration of matter at that point.

Unlike the problem examined in the previous paper [#!LiquidShells!#], which can be seen as having a somewhat academic character, the problem we examine here can have direct applications to astrophysical objects. Not only we can use the isentropic case $n=3/2$ to represent the external convective layer of a star, and the case $n=3$ to represent its internal radiative layer, but we might also consider using both, tied up to one another by means of appropriate interface boundary conditions at an intermediary point, in order to represent a star in a more complete fashion, including the two main layers that are known to exist.