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

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