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

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

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

See also

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

References

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