# Meaning of Field Equation Solutions

I have been trying to understand General Relativity theory better. From what I have gathered, Einstein's Field Equations are the tools by which the geometry of space-time can be mathematically defined. In my adventures on the internet trying to better understand this concept, I inevitably came upon wikipedia's article which said:

"The solutions of the Einstein field equations are metrics of spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as post-Newtonian approximations. Even so, there are numerous cases where the field equations have been solved completely, and those are called exact solutions."

Just so that I am clear- When we say that no "solution" has been found for the EFE given a system of a particular set of physical conditions are we saying that we have not found a way to mathematically define the space-time in that system? If this is the case, are there any physical systems for which we can not, in principle, mathematically define the space-time characteristics; or do we believe that all physical systems' space-times are, in principle, capable mathematical determination?

I also read that some "solutions" have been found for physical systems in which certain assumptions are made so as to make the calculation possible. Are all solutions of this form- that is, do all solutions to the EFE involve assumptions which are not necessarily true; or have some been found in which no such assumptions are made?

These questions remind me of another issue which I wised to bring up: Does anyone here know if Einstein's geometrical treatment of gravitation has had implications for the "Three-Body Problem"? I know that this problem is one in which no general solution has been found but that only a small number of special cases in which the behavior of the three-body system can be defined has been found. Does the theory of relativity enable us to at least increase the number of cases in which we can define the behavior of a three body system?

Thank you.