The purpose of this paper is to establish the general solution of the
Einstein gravitational field equation, in the presence of localized
sources, under a certain set of symmetry conditions. We will assume that
we have static fluid matter with a spherically symmetric energy density
and a definite equation of state, given by where
is the
pressure,
is the energy density, and
is a positive real
number in the interval
. All the matter present will be assumed
to be localized, that is, to be essentially all contained within a certain
sphere whose surface is at the radial position
, for a sufficiently
large value of
, satisfying the condition that
,
where
is the Schwarzschild radius associated to the total mass
. In this paper we will use the terms ``mass'' and ``energy''
interchangeably, to refer to the total energy content of the fluid
matter.
While the Schwarzschild solution is in fact a one-parameter family of
solutions parametrized by the Schwarzschild radius , what we will
present here is a two-parameter family of solutions, parametrized by
and
. The solution presented will be given mostly
analytically, with the exception of a single dimensionless real function,
which displays a fairly simple qualitative behavior, but which at least
for now can only be obtained in detail numerically. However, we will see
that the main properties of this function can be ascertained analytically.