Introduction

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 $P=\omega\rho$ where $P$ is the pressure, $\rho$ is the energy density, and $\omega$ is a positive real number in the interval $(0,1/3]$. 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 $r_{S}$, for a sufficiently large value of $r_{S}$, satisfying the condition that $r_{S}\geq r_{M}$, where $r_{M}$ is the Schwarzschild radius associated to the total mass $M$. 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 $r_{M}$, what we will present here is a two-parameter family of solutions, parametrized by $r_{M}$ and $\omega$. 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.