Specify the Stress Energy Tensor and Calculate the Curvature

Question

I have a simple question about general relativity and the Einstein field equations, I wonder if you can specify the stress energy tensor, i.e. specify some mass distribution in space and then calculate the curvature to later find equations of motions etc, instead of starting out with how the geomerty would look. I am quite new to general relativity and so I am bound to have misconceptions.

Edit: 2013 December 19th I have found this article which at page 10, Chapter 5, section 5.2 does something simillar to what I meant, apperently there is a general from for the Stress energy tensor (for what is known as a perfect fluid(?)), and from it they derive something simillar to the second component in the normal schwarzschild metric i.e $$A(r)=(1-\frac{2U}{r})^{-1}$$ where $U$ is the energy. I do have one remaining question, the name of the general form of the stress energy tensor confuses me somewhat, "perfect fluid" is it just its name, and is it still fully capabable of describing the stress energy tensor in general relativity?

Answer 1

In principle, yes, you can specify the stress tensor and solve the resulting equations, but in practice, this is hard to do because the field equations are non-linear PDEs...darn.

The simplest possible example is the case in which the stress tensor vanishes; $T_{\mu\nu} = 0$ namely the vacuum equations. The field equations with vanishing cosmological constant then become \begin{align} R_{\mu\nu} = 0. \end{align} Manifolds with this property are called Ricci flat. Minkowski space $\mathbb R^{3,1}$ is such a manifold; it is Ricci flat everywhere, but so is, for example, the exterior of the Schwarzschild black hole.

I am far from an expert on this stuff, but in my experience solutions for a given stress tensor are usually obtained by constructing an ansatz containing some free parameters (usually exhibiting certain symmetries), and then determining if the ansatz solves the equations for certain values of the parameters.

One is often forced to consider numerical solutions, and there is a whole industry dedicated to this: numerical relativity.

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