The previous questions are interesting but difficult to settle down rigorously. Indeed the very essence of an approximation is to cope with our ignorance of the higher-order terms in some expansion, but the higher-order terms are precisely the ones which would be needed for a satisfying answer to these problems. So we shall be able to give only some educated guesses and/or plausible answers, that we cannot justify rigorously, but which seem very likely from the standard point of view on the post-Newtonian theory, in particular that the successive orders of approximation get smaller and smaller as they should (in average), with maybe only few accidents occuring at high orders where a particular approximation would be abnormally large with respect to the lower-order ones. Admittedly, in addition, our faith in the estimation we shall give regarding the accuracy of the 3PN order for instance, comes from the historical perspective, thanks to the many successes achieved in the past by the post-Newtonian approximation when confronting the theory and observations. It is indeed beyond question, from our past experience, that the post-Newtonian method does work.
Establishing the post-Newtonian expansion rigorously has been the subject of numerous mathematical oriented works, see, e.g., [187, 188, 189]. In the present section we shall simply look (much more modestly) at what can be said by inspection of the explicit post-Newtonian coefficients which have been computed so far. Basically, the point we would like to emphasize35 is that the post-Newtonian approximation, in standard form (without using the resummation techniques advocated in Refs. [92, 60, 61]), is able to located the ICO of two black holes, in the case of comparable masses (), with a very good accuracy. At first sight this statement is rather surprising, because the dynamics of two black holes at the point of the ICO is so relativistic. Indeed one sometimes hears about the “bad convergence”, or the “fundamental breakdown”, of the post-Newtonian series in the regime of the ICO. However our estimates do show that the 3PN approximation is good in this regime, for comparable masses, and we have already confirmed this by the remarkable agreement with the numerical calculations, as detailed in Section 9.5.
Let us center our discussion on the post-Newtonian expression of the circular-orbit energy (194), developed to the 3PN order, which is of the form[139, 140, 95, 97, 37, 38, 103]: [37, 38] and is equivalent to the parameter of Refs. [139, 140]. We already gave in Equation (133) the relation linking them,
Before its actual computation in general relativity, it has been argued in Ref.  that the numerical value of could be , because for such a value some different resummation techniques, when they are implemented at the 3PN order, give approximately the same result for the ICO. Even more, it was suggested  that might be precisely equal to , with and harmonic-coordinate approaches , and the independent computation of Refs. [133, 132], have settled the value of this parameter in general relativity to be [92, 93, 94] and effective-one-body methods [60, 61, 94], which are designed to “accelerate” the convergence of the post-Newtonian series, do not in fact converge toward the same exact solution (or, at least, not as fast as expected).
In the limiting case , the expression (203, 204) reduces to the 3PN approximation of the energy for a test particle in the Schwarzschild background,stable circular orbit or ISCO, i.e. it corresponds to a point of dynamical unstability. Another important feature of Equation (208) is the singularity at the value which corresponds to the famous circular orbit of photons in the Schwarzschild metric (“light-ring” singularity). This orbit can also be viewed as the last unstable circular orbit. We can check that the post-Newtonian coefficients corresponding to Equation (208) are given by
Let us now discuss a few order-of-magnitude estimates. At the location of the ICO we have found (see Figure 1 in Section 9.5) that the frequency-related parameter defined by Equation (192) is approximately of the order of for equal masses. Therefore, we might a priori expect that the contribution of the 1PN approximation to the energy at the ICO should be of that order. For the present discussion we take the pessimistic view that the order of magnitude of an approximation represents also the order of magnitude of the higher-order terms which are neglected. We see that the 1PN approximation should yield a rather poor estimate of the “exact” result, but this is quite normal at this very relativistic point where the orbital velocity is . By the same argument we infer that the 2PN approximation should do much better, with fractional errors of the order of , while 3PN will be even better, with the accuracy .
Now the previous estimate makes sense only if the numerical values of the post-Newtonian coefficients in Equations (204) stay roughly of the order of one. If this is not the case, and if the coefficients increase dangerously with the post-Newtonian order , one sees that the post-Newtonian approximation might in fact be very bad. It has often been emphasized in the litterature (see, e.g., Refs. [77, 183, 92]) that in the test-mass limit the post-Newtonian series converges slowly, so the post-Newtonian approximation is not very good in the regime of the ICO. Indeed we have seen that when the radius of convergence of the series is (not so far from ), and that accordingly the post-Newtonian coefficients increase by a factor at each order. So it is perfectly correct to say that in the case of test particles in the Schwarzschild background the post-Newtonian approximation is to be carried out to a high order in order to locate the turning point of the ICO.
What happens when the two masses are comparable ()? It is clear that the accuracy of the post-Newtonian approximation depends crucially on how rapidly the post-Newtonian coefficients increase with . We have seen that in the case of the Schwarzschild metric the latter increase is in turn related to the existence of a light-ring orbit. For continuing the discussion we shall say that the relativistic interaction between two bodies of comparable masses is “Schwarzschild-like” if the post-Newtonian coefficients increase when . If this is the case this signals the existence of something like a light-ring singularity which could be interpreted as the deformation, when the mass ratio is “turned on”, of the Schwarzschild light-ring orbit. By analogy with Equation (210) we can estimate the location of this “pseudo-light-ring” orbit by
In Table 1 we present the values of the coefficients in the test-mass limit (see Equation (209) for their analytic expression), and in the equal-mass case when the ambiguity parameter takes the “uncorrect” value defined by Equation (206), and the correct one predicted by general relativity. When we clearly see the expected increase of the coefficients by roughly a factor 3 at each step. Now, when and we notice that the coefficients increase approximately in the same manner as in the test-mass case . This indicates that the gravitational interaction in the case of looks like that in a one-body problem. The associated pseudo-light-ring singularity is estimated using Equation (211) as
Now in the case but when the ambiguity parameter takes the correct value , we see that the 3PN coefficient is of the order of instead of being . This suggests, unless 3PN happens to be quite accidental, that the post-Newtonian coefficients in general relativity do not increase very much with . This is an interesting finding because it indicates that the actual two-body interaction in general relativity is not Schwarzschild-like. There does not seem to exist something like a light-ring orbit which would be a deformation of the Schwarzschild one. Applying Equation (211) we obtain as an estimate of the “light ring”,no notion of a light-ring orbit in the real two-body problem. Or, one might say (pictorially speaking) that the light-ring orbit gets hidden inside the horizon of the final black hole formed by coalescence. Furthermore, if we apply Equation (211) using the 2PN approximation instead of the 3PN one , we get the value instead of Equation (213). So at the 2PN order the metric seems to admit a light ring, while at the 3PN order it apparently does not admit any. This erratic behaviour reinforces our idea that it is meaningless (with our present 3PN-based knowledge, and until fuller information is available) to assume the existence of a light-ring singularity when the masses are equal.
It is impossible of course to be thoroughly confident about the validity of the previous statement because we know only the coefficients up to 3PN order. Any tentative conclusion based on 3PN can be “falsified” when we obtain the next 4PN order. Nevertheless, we feel that the mere fact that in Table 1 is sufficient to motivate our conclusion that the gravitational field generated by two bodies is more complicated than the Schwarzschild space-time. This appraisal should look cogent to relativists and is in accordance with the author’s respectfulness of the complexity of the Einstein field equations.
We want next to comment on a possible implication of our conclusion as regards the so-called post-Newtonian resummation techniques, i.e. Padé approximants [92, 93, 94], which aim at “boosting” the convergence of the post-Newtonian series in the pre-coalescence stage, and the effective-one-body (EOB) method [60, 61, 94], which attempts at describing the late stage of the coalescence of two black holes. These techniques are based on the idea that the gravitational two-body interaction is a “deformation” - with being the deformation parameter - of the Schwarzschild space-time. The Padé approximants are valuable tools for giving accurate representations of functions having some singularities. In the problem at hands they would be justified if the “exact” expression of the energy (whose 3PN expansion is given by Equations (203, 204)) would admit a singularity at some reasonable value of (e.g., ). In the Schwarzschild case, for which Equation (210) holds, the Padé series converges rapidly : The Padé constructed only from the 2PN approximation of the energy - keeping only and - already coincide with the exact result given by Equation (208). On the other hand, the EOB method maps the post-Newtonian two-body dynamics (at the 2PN or 3PN orders) on the geodesic motion on some effective metric which happens to be a -deformation of the Schwarzschild space-time. In the EOB method the effective metric looks like Schwarzschild by definition, and we might of course expect the two-body interaction to own the main Schwarzschild-like features.
Our comment is that the validity of these post-Newtonian resummation techniques does not seem to be compatible with the value , which suggests that the two-body interaction in general relativity is not Schwarzschild-like. This doubt is confirmed by the finding of Ref.  (already alluded to above) that in the case of the wrong ambiguity parameter the Padé approximants and the EOB method at the 3PN order give the same result for the ICO. From the previous discussion we see that this agreement is to be expected because a deformed light-ring singularity seems to exist with that value . By contrast, in the case of general relativity, , the Padé and EOB methods give quite different results (cf. the Figure 2 in ). Another confirmation comes from the light-ring singularity which is determined from the Padé approximants at the 2PN order (see Equation (3.22) in ) as
Finally we come to the good news that, if really the post-Newtonian coefficients when stay of the order of one (or minus one) as it seems to, this means that the standard post-Newtonian approach, based on the standard Taylor approximants, is probably very accurate. The post-Newtonian series is likely to “converge well”, with a “convergence radius” of the order of one36. Hence the order-of-magnitude estimate we proposed at the beginning of this section is probably correct. In particular the 3PN order should be close to the “exact” solution for comparable masses even in the regime of the ICO.
© Max Planck Society and the author(s)