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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$\begingroup$ Btw you can have follow-up questions on details and if you feel your question has been sufficiently answered you can accept it by clicking the check mark next to the answer. (You can do this also in the case of your three other questions.) $\endgroup$– VoidJan 27, 2015 at 9:28
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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.
