Post-Newtonian expansion

Post-Newtonian expansions in general relativity are used for finding an approximate solution of the Einstein field equations for the metric tensor.

Expansion in 1/c²

The post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of matter, forming the gravitational field, to the speed of light, which in this case is better called the speed of gravity.

In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to Newton's law of gravity.

Expansion in h

Another approach is to expand the equations of general relativity in a power series in the deviation of the metric from its value in the absence of gravity

$h_{\alpha \beta} = g_{\alpha \beta} - \eta_{\alpha \beta} \,.$

To this end, one must choose a coordinate system in which the eigenvalues of $h_{\alpha \beta} \eta^{\beta \gamma} \,$ all have absolute values less than 1.

For example, if one goes one step beyond linearized gravity to get the expansion to the second order in h:

$g^{\mu \nu} \approx \eta^{\mu \nu} - \eta^{\mu \alpha} h_{\alpha \beta} \eta^{\beta \nu} + \eta^{\mu \alpha} h_{\alpha \beta} \eta^{\beta \gamma} h_{\gamma \delta} \eta^{\delta \nu} \,.$

$\sqrt{- g} \approx 1 + \tfrac12 h_{\alpha \beta} \eta^{\beta \alpha} + \tfrac18 h_{\alpha \beta} \eta^{\beta \alpha} h_{\gamma \delta} \eta^{\delta \gamma} - \tfrac14 h_{\alpha \beta} \eta^{\beta \gamma} h_{\gamma \delta} \eta^{\delta \alpha} \,.$

Hybrid expansion

Sometimes, as with the Parameterized post-Newtonian formalism, a hybrid approach is used in which both the reciprocal of the speed of gravity and masses are assumed to be small.

See also

• Parameterized post-Newtonian formalism
• Einstein–Infeld–Hoffmann equations
• Linearized gravity
• Coordinate conditions

External links

• "On the Motion of Particles in General Relativity Theory" by A.Einstein and L.Infeld