Solutions of the Einstein field equations

Solutions of the Einstein field equations are metrics of spacetimes that result from solving the Einstein field equations (EFE) of general relativity. Solving the field equations gives a Lorentz manifold. Solutions are broadly classed as exact or non-exact.

The Einstein field equations are

${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }\,=\kappa T_{\mu \nu },}$

where ${\displaystyle G_{\mu \nu }}$ is the Einstein tensor, ${\displaystyle \Lambda }$ is the cosmological constant (sometimes taken to be zero for simplicity), ${\displaystyle g_{\mu \nu }}$ is the metric tensor, ${\displaystyle \kappa }$ is a constant, and ${\displaystyle T_{\mu \nu }}$ is the stress–energy tensor.

The Einstein field equations relate the Einstein tensor to the stress–energy tensor, which represents the distribution of energy, momentum and stress in the spacetime manifold. The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, given the stress–energy tensor, the Einstein field equations are a system of ten partial differential equations in which the metric tensor can be solved for.

Where appropriate, this article will use the abstract index notation.