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9.6 Accuracy of the post-Newtonian approximation

In this section we want to assess the validity of the post-Newtonian approximation, and, more precisely, to address, and to some extent to answer, the following questions: How accurate is the post-Newtonian expansion for describing the dynamics of binary black hole systems? Is the ICO of binary black holes, defined by the minimum of the energy function E(w), accurately determined at the highest currently known post-Newtonian order? The latter question is pertinent because the ICO represents a point in the late stage of evolution of the binary which is very relativistic (orbital velocities of the order of 50% of the speed of light). How well does the 3PN approximation as compared with the prediction provided by numerical relativity (see Section 9.5)? What is the validity of the various post-Newtonian resummation techniques [92Jump To The Next Citation Point93Jump To The Next Citation Point60Jump To The Next Citation Point61Jump To The Next Citation Point] which aim at “boosting” the convergence of the standard post-Newtonian approximation?

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., [187188189]. 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. [92Jump To The Next Citation Point60Jump To The Next Citation Point61Jump To The Next Citation Point]), is able to located the ICO of two black holes, in the case of comparable masses (m1 -~ m2), 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 (194View Equation), developed to the 3PN order, which is of the form

m c2x { } E(x) = - ----- 1 + a1(n) x + a2(n)x2 + a3(n)x3 + O(x4) . (203) 2
The first term, proportional to x, is the Newtonian term, and then we have many post-Newtonian corrections, the coefficients of which are known up to 3PN order [139Jump To The Next Citation Point140Jump To The Next Citation Point959737Jump To The Next Citation Point38Jump To The Next Citation Point103]:
3 n a1(n) = - --- --, 4 12 2 a (n) = - 27-+ 19n - n--, (204) 2 8 8 24 [ ] 675- 209323- 205- 2 110- 155-2 -35-- 3 a3(n) = - 64 + 4032 - 96 p - 9 c n - 96 n - 5184 n .
For the discussion it is helpful to keep the Hadamard regularization ambiguity parameter c present in the 3PN coefficient a3(n). Recall from Section 8.2 that this parameter was introduced in Refs. [3738Jump To The Next Citation Point] and is equivalent to the parameter wstatic of Refs. [139140]. We already gave in Equation (133View Equation) the relation linking them,
-3- 1987- c = - 11 wstatic- 3080 . (205)

Before its actual computation in general relativity, it has been argued in Ref. [94Jump To The Next Citation Point] that the numerical value of wstatic could be -~ -9, 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 [94Jump To The Next Citation Point] that wstatic might be precisely equal to * wstatic, with

* 47- 41- 2 w static = - 3 + 64p = -9.34 .... (206)
However, as reviewed in Sections 8.2 and 8.3, the computations performed using dimensional regularization, within the ADM-Hamiltonian formalism [96] and harmonic-coordinate approaches [30], and the independent computation of Refs. [133132], have settled the value of this parameter in general relativity to be
1987- wstatic = 0 <====> c = - 3080 . (207)
We note that this result is quite different from * wstatic, Equation (206View Equation). This already suggests that different resummation techniques, namely Padé approximants [92Jump To The Next Citation Point93Jump To The Next Citation Point94Jump To The Next Citation Point] and effective-one-body methods [60Jump To The Next Citation Point61Jump To The Next Citation Point94Jump To The Next Citation Point], 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 n --> 0, the expression (203View Equation, 204View Equation) reduces to the 3PN approximation of the energy for a test particle in the Schwarzschild background,

[ ] ESch(x) = m c2 V~ 1--2x--- 1 . (208) 1- 3x
The minimum of that function or Schwarzschild ICO occurs at Sch xICO = 1/6, and we have Sch 2( V~ ---- ) E ICO = mc 8/9- 1. 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 (208View Equation) is the singularity at the value xSch = 1/3 lightring 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 Sch an =_ an(0) corresponding to Equation (208View Equation) are given by
n Sch 3-(2n---1)!!(2n---1) a n = - 2n(n + 1)! . (209)
They increase with n by roughly a factor 3 at each order. This is simply the consequence of the fact that the radius of convergence of the post-Newtonian series is given by the Schwarzschild light-ring singularity at the value 1/3. We may therefore recover the light-ring orbit by investigating the limit
aSch lim -n-1-= 1-= xSlcighhtring. (210) n-->+ oo aScnh 3

Let us now discuss a few order-of-magnitude estimates. At the location of the ICO we have found (see Figure 1View Image in Section 9.5) that the frequency-related parameter x defined by Equation (192View Equation) is approximately of the order of x ~ (0.1)2/3 ~ 20% 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 v/c ~ x1/2 ~ 50%. By the same argument we infer that the 2PN approximation should do much better, with fractional errors of the order of 2 x ~ 5%, while 3PN will be even better, with the accuracy 3 x ~ 1%.

Now the previous estimate makes sense only if the numerical values of the post-Newtonian coefficients in Equations (204View Equation) stay roughly of the order of one. If this is not the case, and if the coefficients increase dangerously with the post-Newtonian order n, 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. [77Jump To The Next Citation Point183Jump To The Next Citation Point92Jump To The Next Citation Point]) that in the test-mass limit n --> 0 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 n = 0 the radius of convergence of the series is 1/3 (not so far from xSch = 1/6 ICO), and that accordingly the post-Newtonian coefficients increase by a factor ~ 3 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 (n = 1 4)? It is clear that the accuracy of the post-Newtonian approximation depends crucially on how rapidly the post-Newtonian coefficients increase with n. 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 an(14) increase when n --> +o o. 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 n is “turned on”, of the Schwarzschild light-ring orbit. By analogy with Equation (210View Equation) we can estimate the location of this “pseudo-light-ring” orbit by

a (n) -n-1----~ xlightring(n) with n = 3. (211) an(n)
Here n = 3 is the highest known post-Newtonian order. If the two-body problem is “Schwarzschild-like” then the right-hand side of Equation (211View Equation) is small (say around 1/3), the post-Newtonian coefficients typically increase with n, and most likely it should be difficult to get a reliable estimate by post-Newtonian methods of the location of the ICO. So we ask: Is the gravitational interaction between two comparable masses Schwarzschild-like?

Newtonian a1(n) a2(n) a3(n)

n = 0 1 -0.75 - 3.37 - 10.55
n = 1 4, w*static - ~ -9.34 1 -0.77 - 2.78 - 8.75
1 n = 4, wstatic = 0 (GR) 1 -0.77 - 2.78 - 0.97

Table 1: Numerical values of the sequence of coefficients of the post-Newtonian series composing the energy function E(x) as given by Equations (203View Equation, 204View Equation).

In Table 1 we present the values of the coefficients an(n) in the test-mass limit n = 0 (see Equation (209View Equation) for their analytic expression), and in the equal-mass case n = 14 when the ambiguity parameter takes the “uncorrect” value w*static defined by Equation (206View Equation), and the correct one wstatic = 0 predicted by general relativity. When n = 0 we clearly see the expected increase of the coefficients by roughly a factor 3 at each step. Now, when 1 n = 4 and * wstatic = wstatic we notice that the coefficients increase approximately in the same manner as in the test-mass case n = 0. This indicates that the gravitational interaction in the case of w*static looks like that in a one-body problem. The associated pseudo-light-ring singularity is estimated using Equation (211View Equation) as

(1 ) xlightring --,w*static ~ 0.32. (212) 4
The pseudo-light-ring orbit seems to be a very small deformation of the Schwarzschild light-ring orbit given by Equation (210View Equation). In this Schwarzschild-like situation, we should not expect the post-Newtonian series to be very accurate.

Now in the case n = 14 but when the ambiguity parameter takes the correct value wstatic = 0, we see that the 3PN coefficient a3( 1) 4 is of the order of - 1 instead of being ~ -10. This suggests, unless 3PN happens to be quite accidental, that the post-Newtonian coefficients in general relativity do not increase very much with n. 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 (211View Equation) we obtain as an estimate of the “light ring”,

( ) 1 xlightring 4-,GR ~ 2.88. (213)
It is clear that if we believe the correctness of this estimate we must conclude that there is in fact 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 (211View Equation) using the 2PN approximation n = 2 instead of the 3PN one n = 3, we get the value ~ 0.28 instead of Equation (213View Equation). 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 a3(14) = - 0.97 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 [92Jump To The Next Citation Point9394Jump To The Next Citation Point], which aim at “boosting” the convergence of the post-Newtonian series in the pre-coalescence stage, and the effective-one-body (EOB) method [606194Jump To The Next Citation Point], 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 n < 1 4 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 (203View Equation, 204View Equation)) would admit a singularity at some reasonable value of x (e.g., < 0.5). In the Schwarzschild case, for which Equation (210View Equation) holds, the Padé series converges rapidly [92Jump To The Next Citation Point]: The Padé constructed only from the 2PN approximation of the energy - keeping only aSch 1 and aSch 2 - already coincide with the exact result given by Equation (208View Equation). 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 n-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 wstatic = 0, which suggests that the two-body interaction in general relativity is not Schwarzschild-like. This doubt is confirmed by the finding of Ref. [94Jump To The Next Citation Point] (already alluded to above) that in the case of the wrong ambiguity parameter w* -~ - 9.34 static 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 w*static. By contrast, in the case of general relativity, wstatic = 0, the Padé and EOB methods give quite different results (cf. the Figure 2 in [94]). Another confirmation comes from the light-ring singularity which is determined from the Padé approximants at the 2PN order (see Equation (3.22) in [92]) as

(1 ) xlightring -,Pade´ ~ 0.44. (214) 4
This value is rather close to Equation (212View Equation) but strongly disagrees with Equation (213View Equation). Our explanation is that the Padé series converges toward a theory having * wstatic - ~ wstatic; such a theory is different from general relativity.

Finally we come to the good news that, if really the post-Newtonian coefficients when n = 14 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.

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