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 higherorder terms in some expansion, but the higherorder 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 postNewtonian 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 lowerorder 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 postNewtonian approximation when confronting the theory and observations. It is indeed beyond question, from our past experience, that the postNewtonian method does work.
Establishing the postNewtonian 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 postNewtonian coefficients which have been computed so far. Basically, the point we would like to emphasize^{35} is that the postNewtonian 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 postNewtonian 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 postNewtonian expression of the circularorbit energy (194), developed to the 3PN order, which is of the form
The first term, proportional to , is the Newtonian term, and then we have many postNewtonian corrections, the coefficients of which are known up to 3PN order [139, 140, 95, 97, 37, 38, 103]: For the discussion it is helpful to keep the Hadamard regularization ambiguity parameter present in the 3PN coefficient . Recall from Section 8.2 that this parameter was introduced in Refs. [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. [94] 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 [94] that might be precisely equal to , with
However, as reviewed in Sections 8.2 and 8.3, the computations performed using dimensional regularization, within the ADMHamiltonian formalism [96] and harmoniccoordinate approaches [30], and the independent computation of Refs. [133, 132], have settled the value of this parameter in general relativity to be We note that this result is quite different from , Equation (206). This already suggests that different resummation techniques, namely Padé approximants [92, 93, 94] and effectiveonebody methods [60, 61, 94], which are designed to “accelerate” the convergence of the postNewtonian 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,
The minimum of that function or Schwarzschild ICO occurs at , and we have . We know that the Schwarzschild ICO is also an innermost 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 (“lightring” singularity). This orbit can also be viewed as the last unstable circular orbit. We can check that the postNewtonian coefficients corresponding to Equation (208) are given by They increase with by roughly a factor 3 at each order. This is simply the consequence of the fact that the radius of convergence of the postNewtonian series is given by the Schwarzschild lightring singularity at the value . We may therefore recover the lightring orbit by investigating the limitLet us now discuss a few orderofmagnitude estimates. At the location of the ICO we have found (see Figure 1 in Section 9.5) that the frequencyrelated 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 higherorder 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 postNewtonian 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 postNewtonian order , one sees that the postNewtonian 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 testmass limit the postNewtonian series converges slowly, so the postNewtonian 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 postNewtonian 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 postNewtonian 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 postNewtonian approximation depends crucially on how rapidly the postNewtonian 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 lightring orbit. For continuing the discussion we shall say that the relativistic interaction between two bodies of comparable masses is “Schwarzschildlike” if the postNewtonian coefficients increase when . If this is the case this signals the existence of something like a lightring singularity which could be interpreted as the deformation, when the mass ratio is “turned on”, of the Schwarzschild lightring orbit. By analogy with Equation (210) we can estimate the location of this “pseudolightring” orbit by
Here is the highest known postNewtonian order. If the twobody problem is “Schwarzschildlike” then the righthand side of Equation (211) is small (say around ), the postNewtonian coefficients typically increase with , and most likely it should be difficult to get a reliable estimate by postNewtonian methods of the location of the ICO. So we ask: Is the gravitational interaction between two comparable masses Schwarzschildlike?

In Table 1 we present the values of the coefficients in the testmass limit (see Equation (209) for their analytic expression), and in the equalmass 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 testmass case . This indicates that the gravitational interaction in the case of looks like that in a onebody problem. The associated pseudolightring singularity is estimated using Equation (211) as
The pseudolightring orbit seems to be a very small deformation of the Schwarzschild lightring orbit given by Equation (210). In this Schwarzschildlike situation, we should not expect the postNewtonian series to be very accurate.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 postNewtonian coefficients in general relativity do not increase very much with . This is an interesting finding because it indicates that the actual twobody interaction in general relativity is not Schwarzschildlike. There does not seem to exist something like a lightring orbit which would be a deformation of the Schwarzschild one. Applying Equation (211) we obtain as an estimate of the “light ring”,
It is clear that if we believe the correctness of this estimate we must conclude that there is in fact no notion of a lightring orbit in the real twobody problem. Or, one might say (pictorially speaking) that the lightring 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 3PNbased knowledge, and until fuller information is available) to assume the existence of a lightring 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 spacetime. 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 socalled postNewtonian resummation techniques, i.e. Padé approximants [92, 93, 94], which aim at “boosting” the convergence of the postNewtonian series in the precoalescence stage, and the effectiveonebody (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 twobody interaction is a “deformation”  with being the deformation parameter  of the Schwarzschild spacetime. 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 [92]: 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 postNewtonian twobody dynamics (at the 2PN or 3PN orders) on the geodesic motion on some effective metric which happens to be a deformation of the Schwarzschild spacetime. In the EOB method the effective metric looks like Schwarzschild by definition, and we might of course expect the twobody interaction to own the main Schwarzschildlike features.
Our comment is that the validity of these postNewtonian resummation techniques does not seem to be compatible with the value , which suggests that the twobody interaction in general relativity is not Schwarzschildlike. This doubt is confirmed by the finding of Ref. [94] (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 lightring 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 [94]). Another confirmation comes from the lightring singularity which is determined from the Padé approximants at the 2PN order (see Equation (3.22) in [92]) as
This value is rather close to Equation (212) but strongly disagrees with Equation (213). Our explanation is that the Padé series converges toward a theory having ; such a theory is different from general relativity.Finally we come to the good news that, if really the postNewtonian coefficients when stay of the order of one (or minus one) as it seems to, this means that the standard postNewtonian approach, based on the standard Taylor approximants, is probably very accurate. The postNewtonian series is likely to “converge well”, with a “convergence radius” of the order of one^{36}. Hence the orderofmagnitude 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.
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