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

g μ ν = η μ ν + h μ ν + O ( h 2 ) {\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  h {\displaystyle h} .[1]

## See also[edit]

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

## References[edit]

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.

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