In the nonlinear terms we can simply use

Expanding and organizing Equation (287) with the linear terms given explicitly and the quadratic terms collected in the expression for , we obtain the long expression with all terms functions of either or :
Using Equations (269) and (271), the and in Equation (288) are replaced by . On the righthand side all the variables, e.g., ., are functions of ‘’; their functional forms are the same as when they were functions of and the linear terms are again explicitly given and the quadratic terms are collected in the
To proceed, we use the complex centerofmass condition, namely, , and solve for . This is accomplished by first reversing the calculation via

and then, before extracting the harmonic component, replacing the by , via the inverse of Equation (271),

using Equation 280
In this process several of the quadratic terms cancel out and new ones arise.The final expression for , given in terms of the complex world line expressed as a function of , then becomes our basic equation:
This, which becomes the analogue of the Newtonian dipole expression , is our central relationship. Almost all of our results in the following sections follow directly from it.We emphasize that prior to this discussion/derivation, the and the were independent quantities but in the final expression the is now a function of the .
Note that the linear term

coincides with the earlier results in the stationary case, Equation (259). From

we have
We will see shortly that there is a great deal of physical content to be found in the nonlinear terms in Equation (292).http://www.livingreviews.org/lrr20096 
This work is licensed under a Creative Commons License. Problems/comments to 