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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, \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, $$\eta_{\mu\nu}$$:

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

and dropping all terms which are of second or higher order in h.[1]

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

References

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.

Physics Encyclopedia