Non-exact solutions in general relativity

Non-exact solutions in general relativity are solutions of Albert Einstein's field equations of general relativity which hold only approximately. These solutions are typically found by treating the gravitational field, ${\displaystyle g}$, as a background space-time, ${\displaystyle \gamma }$, (which is usually an exact solution) plus some small perturbation, ${\displaystyle h}$. Then one is able to solve the Einstein field equations as a series in ${\displaystyle h}$, dropping higher order terms for simplicity.

A common example of this method results in the linearised Einstein field equations. In this case we expand the full space-time metric about the flat Minkowski metric, ${\displaystyle \eta _{\mu \nu }}$:

${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+h_{\mu \nu }+{\mathcal {O}}(h^{2})}$,

and dropping all terms which are of second or higher order in ${\displaystyle h}$.[1]