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

[math]\displaystyle{ g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} +\mathcal{O}(h^2) }[/math],

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

## See also

* Exact solutions in general relativity
* Linearized gravity
* Post-Newtonian expansion
* Parameterized post-Newtonian formalism
* Numerical relativity

## References

1. ↑ Sean M. Carroll (2004). Spacetime and Geometry: An Introduction to General Relativity. Addison-Wesley Longman, Incorporated. pp. 274–279. ISBN 978-0-8053-8732-2. https://books.google.com/books?id=1SKFQgAACAAJ.

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