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. The approximations are expanded in small parameters which express orders of deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher order terms can be added to increase accuracy, but for strong fields sometimes it is preferable to solve the complete equations numerically. This method is a common mark of effective field theories. In the limit, when the small parameters are equal to 0, the post-Newtonian expansion reduces to Newton's law of gravity.

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.

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

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

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