I have been studying gravitation waves radiated by a binary source. I have linearised Einstein's field equation and approximated the source to a Quadrupole moment to get the power radiated by the source. Now what is post-Newtonian approximation? I have read in Wikipedia that weak field limit doesn't work for binary stars. So, one has to stick on to Post-Newtonian approximation. What is post Newtonian approximation and why do we need to study for the strong fields? Please explain.

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The Post-Newtonian expansion allows us to describe the gravitational dynamics of the given situation in terms of Newtonian forces acting on a "Galilean" i.e. non-relativistic bodies. However, this means that the special-relativistic effects are embedded in the gravitational force.

The usual description of the post-Newtonian expansion will tell you that it is an expansion of this force in orders of $v/c$. Thus, it would seem the approximation will be valid even in strong fields - but it is not. The expansion starts with completely Newtonian equations and finds the appropriate corrections by successive iterations which find the next order correction in terms of $v/c$ - but this does not insure the rate of convergence of such iterations or convergence at all. Especially in very strong gravitational fields, e.g. for objects at a distance of few of their Schwarzschild radii, the iteration procedure will fail to give any reasonable results.

However, the main advantage of the post-Newtonian methods is that it seems to be most accurate (as compared with explicit relativistic simulations) and practical for a large range of astrophysical considerations. The two (or more) objects in gravitational interaction can be considered on an equal footing, there does not have to be any test-particle vs. dominating background, and the motions are in 3D with a single time parameter - which is useful.

If you want to see the screws and nuts of post-Newtonian theory, I recommend the review "The Post-Newtonian Approximation for Relativistic Compact Binaries" by Toshifumi Futamase and Yousuke Itoh.


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