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I am studying the Damour paper on the Post-Minkowskian approximation to the 2 bodies problem in General Relativity (paper) but can't really understand the explanation of the previous state of the art, namely the Post-Newtonian approximation: a way to "perturbe" the Newtonian flat space-time assuming small velocities.

I understand that, once is established that the two momentums $I_R$ and $I_\phi$ are the same in the real and effective scenario, one has to look for a map $f$ that connects the real and effective energies: $E^{eff}=f(E^{real})$ and this map will be expanded in a series of $\frac{1}{c^2}$. In the paper the $\alpha_i$ are the coefficients of this series. How does one calculate them?

Edit: if anyone is still interested, I found another interesting paper from 1998 by Buonanno and Damour, and chapters 2, 3 and 4 cover the subject, so question solved.

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