What do I do once the Field Equations have been assembled

[Frogeyedpeas]

I was working on a problem the other day involving the Tensor versions of the Einstein Field Equations where I defined a metric (minkowski) and then defined a Stress-Energy-Momentum Tensor and solved for the corresponding Ricci curvature Tensor, now that I have all of this solved what do I do with it? How does it assemble into a PDE?

If you want me to I can post the exact "hypothetical equation" but it appears to be a lot of work so I'll restrain from posting it unless its necessary.

 

[elfmotat]

You can't define a metric and the stress-energy at the same time. You can define a metric and figure out what the stress-energy would need to be to produce it, or you can define a particular stress-energy and figure out what its corresponding metric is.

The field equations themselves are PDE's - the Ricci curvature tensor is related to the metric by two partial derivatives.

 

[Frogeyedpeas]

So if that was the case then, how are the equations solved... Suppose I supply it a stress-energy tensor. Then how would I be able to figure out the curvature and the metric from just the stress energy? And also I am right now only having access to the tensor form of the equations (I do not know how the tensors themselves are inter-related as you hinted to)

 

[PAllen]

There are two main ways solutions of the field equations are derived. Well, there is a third, nonphysical way elfmotat mentioned: start with an arbitrary metric, compute the Einstein tensor = stress energy tensor from it, see if it is plausible (most likely not). The more physical approaches are:

Decide on a class of symmetries you seek in esp. a vacuum solution, specify the metric form that expresses these symmetries, then solve Einstein tensor = 0 for this general form. This is how you get the classic vacuum solutions. [Edit: you can also specify boundary conditions, including overall topology].

The other physical approach is the ADM initial value problem. This is much more complex. You express physics on an initial spacelike 3-surface, meeting constraints, and numerically evolve the field equations from there. This method is discussed, for example, in:

http://relativity.livingreviews.org/Articles/lrr-2000-5/
and
http://relativity.livingreviews.org/Articles/lrr-2012-2/

 

[PAllen]

I should note that, mathematically, there is an additional approach. Arbitrarily specify a stress energy tensor on the whole manifold meeting chosen energy conditions. Note that GR is then not predicting anything about motion of matter and energy because you have specified all of that in your arbitrary tensor T. Then, you can solve for the metric from the G = T, (G the Einstein tensor, constants ignored). Generally, there should, in principle, be solutions because you have 10 PDEs for 10 metric components. Given the metric, you can then examine the behavior of test particles.

Obviously, the fact that you specify all behavior a priori makes this approach undesirable, and I've never read a book or paper that actually went through such an exercise. I just mention it for mathematical completeness.