Gravitational waves in the post-Newtonian expansion

I have spent some time trying to understand gravitational waves in the context of the post-Newtonian expansion.

As far as I understand it, the general relativistic equation of motion can be approximated in the general form

$$\frac{\text{d}^2x^i}{\text{d}t^2}=\mathcal{O}(1)+\mathcal{O}\Bigl(\frac{v^2}{c^2}\Bigr)+\mathcal{O}\Bigl(\frac{v^4}{c^4}\Bigr)+ ...$$

where the first term is the Newtonian approximation, while the term of $$\mathcal{O}(v^2/c^2)$$ is the first post-Newtonian approximation (1PN), the term of $$\mathcal{O}(v^4/c^4)$$ is the 2PN-approximation, and so on. For example in Weinbergs "Gravitation and Cosmology" the (first) Post-Newtonian expansion starts with the expansion of the geodesic equation in Powers of (v/c) and the result is an equation of motion, where the (first) post-Newtonian correction is of $$\mathcal{O}(v^2/c^2)$$ compared to the Newtonian equation of motion (see chapter 9.1/p212 at least in my version of the book).

Now the first approximation for gravitational waves is the following: One expands the equation of motion to the first term, which gives Newtonian results. Then one can describe the motion of bodies in the source with Newtonian laws in the near region, while in the far region one can describe the gravitational waves produced by such motion. Furthermore one can calculate the energy carried away by those waves and by an energy-conversation argument derive that this energy has to be lost in the near region of the source. For a binary star system this gives a formula for the period decrease:

$$$$\dot{P}_b=-\frac{192\pi G^\frac{5}{3}}{5c^5}\Biggl(\frac{P_b}{2\pi}\Biggr)^{-\frac{5}{3}}\frac{m_1m_2}{(m_1+m_2)^\frac{1}{3}}\frac{1+\frac{73}{24}e^2+\frac{37}{96}e^4}{(1-e^2)^\frac{7}{2}}$$$$

This is effectively a modification of the motion in the near region of $$\mathcal{O}(v^5/c^5)$$, which corresponds to a 2.5PN correction. This is of course interesting since one starts with Newtonian motion describing the bodies where of course there is no decrease of the period, but in the end, one can calculate an effect that is far beyond the Newtonian approximation.

Now I have read, that this result is recovered, when one does the Post-Newtonian expansion properly (meaning one considers the effect the far field has on the near feld i.e. the radiation reaction) up to 2.5PN-Order. Then one can derive the formula above directly from the equation of motion. For example Maggiore writes in his book Gravitational Waves and Cosmology "[...] as we discussed in Section 5.3.5, this result can also be obtained directly from the post-Newtonian equations of motion of the binary system" referring to the formula written above. I have, however, not found a derivation of this. In the mentioned section there is some talk about the energy and what not, but not in any mathematical detail and sometimes sentences in textbooks are a little ambivalent or even not totally true, so I wanted to be safe and maybe somebody here is very knowledgeable in this topic and able to help.

Question 1: Is this correct? Can somebody point me to a book or a paper in which the formula is derived from the 2.5PN equation of motion?

In the 1PN-Approximation one expands the equation of motion up to $$\mathcal{O}(v^2/c^2)$$ and the result for a system of point masses are the Einstein-Infeld-Hoffmann-equations. Here my question is, whether it is possible, to do the same again. What I mean is: Solve the Einstein-Infeld-Hoffmann-equation either numerically or in some other way, describe the motion of bodies in the near region, calculate the gravitational waves (and their energy) that follow from that motion and derive some effect on the period-decrease of the system from that energy loss which would be of $$\mathcal{O}(v^7/c^7)$$? So in short:

Question 2 Can the procedure be repeated in 1PN?

• I don't have time for a proper answer, but this review may answer many of your questions: doi.org/10.12942/lrr-2014-2 Commented Sep 1, 2021 at 13:02
• Thank you. Unfortunately I am not familiar with blanchets formalism of higher post-newtonian physics and it is very difficult for me, to really grasp what is explained in that quit long review. But thanks anyway :) Commented Sep 2, 2021 at 9:17
• The leading order expression for the power is the Peters-Mathews formula, originally derived here: journals.aps.org/pr/abstract/10.1103/PhysRev.131.435. It should be the same thing as working with the leading order PN formulas. Commented Sep 2, 2021 at 18:52
• In the post-Newtonian regime, there exists a timescale hierarchy between the orbital, precession, and radiation reaction timescales: $t_{\rm orb} \ll t_{\rm pre} \ll t_{\rm rr}$. This hierarchy is typically exploited. This is useful because the precession effects become important at least at 1.5 PN order for a given binary separation, while radiation reaction shrinks the magnitude of the binary separation on a longer timescale. You might find details on this in the seminal textbook by Will and Poisson called Gravity, which has lots of derivations. Sorry I cant be more helpful. Commented Sep 2, 2021 at 19:21