Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations (EFE) of general relativity. The solutions are Lorentz metrics. Solutions are usually classed as exact or non-exact.

The Einstein field equations are

$G\_\{ab\}\; \backslash ,\; =\; \backslash kappa\; T\_\{ab\}$

where $\backslash kappa$ is a constant, and the Einstein tensor on the left side of the equation is equated to the stress-energy tensor representing the energy and momentum present in the spacetime. The Einstein tensor is built up from the metric tensor and it's partial derivatives; thus, the EFE are a system of ten partial differential equations to be solved for the metric.

Exact solutions

Exact solutions are Lorentz metrics that are conformable to a physically realistic stress-energy tensor and which are obtained by solving the EFE exactly in closed form.

Non-exact solutions

Those solutions that are not exact are called non-exact solutions. Such solutions mainly arise due to the difficulty of solving the EFE in closed form and often take the form of approximations to ideal systems. Many non-exact solutions may be devoid of physical content, but serve as useful counterexamples to theoretical conjectures.

General relativity