Going further

8.7 Going further

The 4 PN templates may be required to detect gravitational wave directly with high signal to noise ratio and extract astrophysical information from the wave. We have to derive the 4 PN equations of motion to derive the 4 PN templates if we use the energy balance.

At this moment, we should say that it is difficult to derive the 4 PN equations of motion. The technical obstacles regarding the derivation of the 4 PN equations of motion are the following. First, to derive the 4 PN equations of motion, we have to derive the 4 PN gravitational field at least in the neighborhood of a star. This requires the 3 PN gravitational field valid throughout the near zone. As seen in this article, however, it seems impossible to derive the 3 PN accurate gravitational field in harmonic coordinates in a closed form completely. We have not yet found the super-potentials necessary for us to evaluate the 3 PN Poisson integrals (or, retarded integrals).

Second, the amounts of calculations required would be too large to derive the 4 PN equations of motion successfully. For example, the Newtonian field consists of only two terms. The Landau–Lifshitz pseudotensor at the Newtonian order consists of basically one term. The gravitational field at 1 PN order consists of about $10$ terms, while the Landau–Lifshitz pseudotensor has about $5$ terms and thus we have to handle $50$ terms to derive the 1 PN equations of motion. Next, the 2 PN field has about $102$ terms. The Landau–Lifshitz pseudotensor in terms of $h μν$ consists of about $50$ terms and thus $∼ 103$ terms must be treated to derive the 2 PN equations of motion. The number of terms in the 3 PN field are of order $3 4 10 ∼ 10$ , while Landau–Lifshitz pseudotensor has about $100$ terms. Thus we may encounter $5 ∼ 10$ terms to derive the 3 PN equations of motion. Then at the 4 PN order, we may expect $104 ∼ 105$ terms for the field, $500$ terms for the integrand, and $∼ 106 ∼ 107$ terms for the equations of motion. Furthermore, for each term spatial and temporal derivative generate additional terms. The number of newly generated terms are about three or four for each term. Thus the number of terms in the intermediate expressions to be dealt with are about ten-fold the number quoted above at each order¹⁶ . Such an enormous number of terms forces us to use algebraic computing softwares to deal with them. In this work, we have extensively used the algebraic computing software Maple [118 ], Mathematica [166 ], and MathTensor [120 ] to deal with tensors. However, quantity changes quality. When some equations are produced through a sequence of black-boxes (that is, calculations done by computers) we could not check the equations term by term even if the calculations follow trivial procedures such as taking a Taylor expansion of equations around a star and extracting off some coefficients from the Taylor-expanded equations. Then, how could one confirm one’s result? Some possible tests include the following: Do the resulting equations of motion respect Lorentz invariance? Do the equations of motion admit the existence of a conserved energy? Does the gravitational field (if obtainable) satisfy the harmonic condition? Do the results obtained by more than two groups agree with each other? Then do all these consist of a set of necessary and sufficient criteria for confirmation?

One way to derive the 4 PN equations of motion is to use brute force (if the first difficulty – how to derive the required super-potentials at 3 PN order – was successfully dealt with). On the other hand, one may find a scaling appropriate to the late inspiralling phase and construct an approximation scheme based on such a scaling, as the post-Newtonian approximation is based on the Newtonian scaling. In the post-Newtonian approximation, the lowest order term is just the Newtonian term, and the first order term is the 1 PN correction. In the post-Minkowskian approximation, (the lowest order is just a straight line and) the first order correction is valid for any velocity but only for a weak field. Likewise, a new approximation scheme (if any) would give (the lowest order of the conservative dynamics and) the first order correction to the radiation reaction effect. Such an approximation may produce a smaller number of terms than the post-Newtonian approximation and thus give easier-to-treat equations of motion. Also the relatively lower order equations of motion obtained by such an approximation are expected to give templates with the same accuracy as the accuracy achieved by the relatively higher order post-Newtonian equations of motion. Thus, such an approximation scheme is an attractive alternative to the post-Newtonian approximation. The construction of such an approximation remains to be done in future work.