Understanding Einstein's Field Equations Solutions

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In summary, the Einstein field equations are used to mathematically define the geometry of space-time in relation to the stress-energy present. They are non-linear equations and can only be solved with approximations. Some solutions have been found by making assumptions, but a complete solution for a 3-body system is still not possible. It is believed that a solution exists for any physical system, but it may be too difficult to solve without simplifying assumptions. The theory of relativity has not yet enabled us to solve the 3-body problem, and a completely general solution is usually not possible for any physical problem.
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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.
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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.

1. What are Einstein's field equations?

Einstein's field equations are a set of 10 non-linear partial differential equations that form the foundation of the general theory of relativity. They describe the relationship between the curvature of space-time and the distribution of matter and energy.

2. Why is it important to understand Einstein's field equations?

Understanding Einstein's field equations is crucial for understanding the fundamental laws of gravity and the behavior of space and time in the presence of matter and energy. They also allow us to make predictions about the behavior of massive objects, such as planets and stars, in the universe.

3. What are some common solutions to Einstein's field equations?

Some common solutions to Einstein's field equations include the Schwarzschild solution, which describes the gravitational field around a spherically symmetric mass, and the Friedmann-Lemaitre-Robertson-Walker solution, which describes the large-scale structure of the universe.

4. How do scientists use Einstein's field equations to study the universe?

Scientists use Einstein's field equations in conjunction with observational data to make predictions about the behavior of the universe. They also use them to study the effects of massive objects, such as black holes, on the surrounding space-time and to understand the dynamics of the early universe.

5. Are there any current challenges in understanding Einstein's field equations?

Yes, there are still many challenges in fully understanding Einstein's field equations. One of the main challenges is reconciling them with quantum mechanics, as the two theories describe the behavior of the universe at different scales. There is also ongoing research into finding new solutions to the equations that can explain phenomena such as dark matter and dark energy.

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