# 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.

## Answers and Replies

Matterwave
Science Advisor
Gold Member
The Einstein field equations tells us how the geometry of spacetime (encoded by the Einstein tensor) is related to the stress-energy present in the space time. It is a set of non-linear coupled partial differential equations, which means it is very difficult indeed to solve, generally speaking. However, us not knowing how to solve the equations for a given physical set up does not mean that no solution exists. For any physical system, a solution to the EFE's must exist, for if this weren't true, General Relativity would be very limited in scope indeed. (Sadly, I can not recall where to find a reference for showing the existence and uniqueness to solutions of the Einstein Field Equations, if anyone has a reference, that would be great) However, without any simplifying assumptions, the solution could simply be too hard to figure out.

Any solution to basically any physical problem of sufficient complexity will include simplifying assumptions. A completely general solution is usually out of reach. As you mentioned, even in Newtonian gravity, which is far simpler than General Relativity, no known analytic solution to the general 3-body problem exists.

As for your last point. If we are unable to solve even the 2-body problem in GR, I think it is pushing it to expect us to be able to solve a 3-body problem in GR.