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, $g$, as a background space-time, $\backslash gamma$, (which is usually an exact solution) plus some small perturbation, $h$. Then one is able to solve the Einstein field equations as a series in $h$, 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, $\backslash eta\_\{\backslash mu\backslash nu\}$:

::$g\_\{\backslash mu\backslash nu\}\; =\; \backslash eta\_\{\backslash mu\backslash nu\}\; +\; h\_\{\backslash mu\backslash nu\}\; +\backslash mathcal\{O\}(h^2)$,

and dropping all terms which are of second or higher order in $h$.{{cite book|author=Sean M. Carroll|title=Spacetime and Geometry: An Introduction to General Relativity|url=https://books.google.com/books?id=1SKFQgAACAAJ|year=2004|publisher=Addison-Wesley Longman, Incorporated|isbn=978-0-8053-8732-2|pages=274–279}}