Summary

8.6 Summary

To deal with strongly self-gravitating objects such as neutron stars, we have used the surface integral approach with the strong field point particle limit. The surface integral approach is achieved by using the local conservation of the energy momentum, which led us to the general form of the equations of motion that are expressed entirely in terms of surface integrals. The use of the strong field point particle limit and the surface integral approach makes our 3 PN equations of motion applicable to inspiraling compact binaries which consist of strongly self-gravitating regular stars (modulo the scalings imposed on the initial hypersurface). Our 3 PN equations of motion depend only on the masses of the stars and are independent of their internal structure such as their density profiles or radii. Thus our result supports the strong equivalence principle up to 3 PN order.

At 3 PN order, it does not seem possible to derive the field in a closed form. This is because not all the super-potentials required are available, and thus we could not evaluate all the Poisson-type $N ∕B$ integrals. Some of the integrands allow us to derive super-potentials in a series form in the neighborhood of the stars. For others, we have adopted an idea that Blanchet and Faye have used in [25 , 26 , 27 ]. The idea is that while abandoning the complete derivation of the 3 PN gravitational field valid throughout $N ∕B$ , one exchanges the order of integrals¹⁵ . We first evaluate the surface integrals in the evolution equation for the energy of a star and the general form of equations of motion, and then we evaluate the remaining volume integrals. Using these methods, we first derived the 3 PN mass-energy relation and the momentum-velocity relation. The 3 PN mass-energy relation admits a natural interpretation. We then evaluated the surface integrals in the general form of equations of motion, and obtained the equations of motion up to 3 PN order of accuracy.

At 3 PN order, our equations of motion contain logarithms of the body zone radii $RA$ . We showed that we could remove the logarithmic terms by a suitable redefinition of the representative points of the stars. Thus we could transform our 3 PN equations of motion into unambiguous equations which do not contain any arbitrarily introduced free parameters.

Our so-obtained 3 PN equations of motion agree physically (modulo a definition of the representative points of the stars) with the result derived by Blanchet and Faye [27 ] with $λ = − 1987 ∕3080$ , which is consistent with Equation (1 ) and $ωstatic = 0$ reported by Damour, Jaranowski, and Schäfer [54 ]. This result indirectly supports the validity of the dimensional regularization in the ADM canonical approach in the ADMTT gauge.

Blanchet and Faye [25 , 27 ] introduced four arbitrary parameters. In the Hadamard partie finie regularization, one has to introduce a sphere around each singular point (representing a point mass) whose radius is a free parameter. In their framework, regularizations are employed in the evaluation of both the gravitational field having two singular points and the two equations of motion. Since, in their formalism, there is a priori no reason to expect that the spheres introduced for the evaluation of the field and the equations of motion coincide, there arise four arbitrary parameters. This is in contrast to our formalism where each body zone introduced in the evaluation of the field is inevitably the same as the body zone with which we defined the energy and the three-momentum of each star for which we derived our equations of motion.

Actually, the redefinition of the representative points in our formalism corresponds to the gauge transformation in [27 ], and only two of the four parameters remain in [27 ]. Then they have used one of the remaining two free parameters to ensure the energy conservation, and there remains only one arbitrary parameter $λ$ which they could not fix in their formalism.

On the other hand, our 3 PN equations of motion have no ambiguous parameter, admit conservation of an orbital energy of the binary system (when we neglect the 2.5 PN radiation reaction effect), and respect Lorentz invariance in the post-Newtonian perturbative sense. We emphasize that we do not need to a posteriori adjust some parameters to make our 3 PN equations of motion to satisfy the above three physical features.

We here note that Blanchet et al. [22 ], who computed the 3 PN equations of motion in the harmonic gauge using the dimensional regularization, have recently obtained the same value for $λ$ .

The gauge condition in a harmonic gauge is related to the equations of motion. One may ask if the 3 PN equations of motion that have been derived so far guarantee the harmonic gauge condition through the corresponding post-Newtonian accuracy. This has not been tested yet. Let us call the $n$ PN accurate metric components to be the components that are needed to compute the $n$ PN equations of motion. Then the harmonic condition for the $n$ PN field requires that matter obeys the $n − 1$ PN equations of motion. Thus, we need the 4 PN field to check if our resulting 3 PN equations of motion are a necessary condition to fullfil the harmonic gauge condition. This is beyond our current knowledge.