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} .
See also Exact solutions in general relativity Linearized gravity Post-Newtonian expansion Parameterized post-Newtonian formalism Numerical relativity
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