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In the very well written article by C. Will, On the Unreasonable Effectiveness of the post-Newtonian Approximation in Gravitational Physics, he states:

The one question that remains open is the nature of the post-Newtonian sequence; we still do not know if it converges, diverges or is asymptotic. Despite this, it has proven to be remarkably effective.

What is the current status of the convergence of the post-Newtonian sequence? Are there indications that it is possibly asymptotic?

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  • $\begingroup$ arxiv.org/abs/1403.7377 is an article from 2014, a few years after the other one. $\endgroup$ Nov 20, 2021 at 1:51
  • $\begingroup$ @JohnHunter That paper is about the experimental confrontation with GR theory. It's a good paper, but doesn't address what I'm asking about. $\endgroup$ Nov 20, 2021 at 8:48

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The question about PN sequence is raiser mathematical one, nor physical or philosophical. It has a good answer in the paper "Newtonian and post-Newtonian approximations are asymptotic to general relativity" by T. Futamase and Bernard, F. Schutz, Phys. Rev. D 28, 2363 – Published 15 November 1983. The Abstract to this paper is clearly stated that the PN approximations "are genuine asymptotic approximations to general relativity". Also this question discussed in many papers such as Blanchet, L., “Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries”, Living Rev. Relativity, 9, lrr-2006-4, (2006). URL (cited on 3 August 2006): http://www.livingreviews.org/lrr-2006-4. 2.2, 3

"The Post-Newtonian Approximation for Relativistic Compact Binaries", by Toshifumi Futamase&Yousuke Itoh, Living Reviews in Relativity, 12 Mar 2007, 10(1):2

"Accuracy of the post-Newtonian approximation: Optimal asymptotic expansion for quasicircular, extreme-mass ratio inspirals" by Nicolás Yunes and Emanuele Berti, Phys. Rev. D 77, 124006 – Published 5 June 2008; Erratum Phys. Rev. D 83, 109901 (2011).

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    $\begingroup$ @DaddyKropotkin: Futamase & Schutz prove their result under assumptions of small velocities and near flatness, this is the domain of “reasonable effectiveness” of PN approximation. Will talks about how “unreasonably” well PN approximation works in strong gravity situations such as late stage inspirals. $\endgroup$
    – A.V.S.
    Nov 28, 2021 at 13:27
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    $\begingroup$ @DaddyKropotkin: I am aware of Wigner paper and hoped that you would understand my meaning: the result of Futamase and Schutz is inapplicable to the kind of situations C.Will is talking about. $\endgroup$
    – A.V.S.
    Nov 28, 2021 at 15:29
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    $\begingroup$ @DaddyKropotkin I think that Clifford M. Will and other doing a nice job to extend PN approximations to what they called as post-Newtonian theory. From the other hand Futamase & Schutz also did a nice job to prove that PN approximations are in fact asymptotic approximations to the sequence of fully relativistic solutions. It is not clear why Clifford M. Will does not mentioned Futamase & Schutz in his review. $\endgroup$ Nov 29, 2021 at 4:47
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    $\begingroup$ @A.V.S. I don't understand your comments. Could you post your thinking about Futamase & Schutz paper with some explanation? $\endgroup$ Nov 29, 2021 at 4:50
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    $\begingroup$ @DaddyKropotkin: The F&Sch result is applicable in situations with vanishing self-gravity, i.e. it could not be applied to things like neutron stars and black holes. Note that for compact binaries individual terms of PN expansions of various quantities are generically divergent (at high enough orders) both in UV and IR. While renormalization allows extraction of meaningful results from those expressions, blanket statement “PN expansion is asymptotic” without additional qualifiers is misleading. $\endgroup$
    – A.V.S.
    Dec 2, 2021 at 13:42

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