How do I use Einstein's field equations to solve for particle locations?

[Petyab]

Can somebody explain a little bit about how to actually use Einstein's field equations to solve for particle locations?

Relevant information:
parentheses are sub-scripts
R(uv)-1/2guvR+guv(cosmological constant sign)=(8piG/c^4)T(uv)

where R is the Einstein Tensor

R is described by wikipedia as the same as the Ricci tensor

R(uv) is the Ricci Tensor

The Ricci tensor is described by wikipedia as "represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space." (wikipedia, October 16th, 2011) But what if it doesn't deviate? Should a value of near zero be used?

g(uv) is the inverse metric tensor which seems to be an important part that deals with the causal mathematical discription of curvature, placement, and so forth.


G is Newton's gravitational constant

Hey, I kind of get this...take the value...use it.

T(uv) is the stress energy tensor

This is connected with the flux of energy against and amongst objects.

I know it's hard mathematics and there's a lot involved but it seems that two of the big things should be near zero and so I'm wondering how to get the other parts to make more sense...help?

 

[jfy4]

For practical use of GR, the Linearized field equations are fine. These are just like Maxwell's equations and are more familiar. To calculate the trajectories of particles you really just want the geodesic equation, which is not strictly related to the field equations.
Linearized Gravity:
$$\Box g_{\alpha\beta}=\frac{16\pi G}{c^4}T_{\alpha\beta}$$
geodesic equation
$$\frac{\partial u^{\alpha}}{\partial x^{\beta}}+\Gamma^{\alpha}_{\beta\gamma}u^{\gamma}=0$$