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 ${\displaystyle h}$.

See also

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

References

1. Exact solutions in general relativity
2. Vacuum solution (general relativity)
3. Kerr metric
4. Solutions of the Einstein field equations
5. Schwarzschild metric
6. Einstein field equations
7. Two-body problem in general relativity
8. Mathematics of general relativity
9. Congruence (general relativity)
10. General relativity
11. Alternatives to general relativity
12. Pp-wave spacetime
13. History of general relativity
14. Weyl–Lewis–Papapetrou coordinates
15. Electrovacuum solution
16. Birkhoff's theorem (relativity)
17. Metric tensor (general relativity)
18. Special relativity
19. Kerr–Newman metric
20. Anti-de Sitter space