Solving The Einstein Field Equation
Solving the Einstein field equation requires an iterative process. The solution is represented in the metric tensor
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Typically there is an initial guess for the tensor. The guess is used to calculate Christoffel symbols, which are used to calculate the curvature. If the Einstein field equation is not satisfied, the process is repeated.
Solutions occur in two forms, vacuum solutions and non-vacuum solutions. A vacuum solution is one in which the stress-energy tensor is zero. The relevant vacuum solution for circular orbits is the Schwarzschild metric. There are also a number of exact solutions that are non-vacuum solutions, solutions in which the stress tensor is non-zero.
Read more about this topic: Theoretical Motivation For General Relativity
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