New derivation of a third post-Newtonian equation of motion for relativistic compact binaries without ambiguity

Renewed attention has been paid to a high order post-Newtonian equation of motion governing inspiralling compact binaries in the context of the efforts for direct detection of gravitational waves [1,2]. It is well-known that detectability of the gravitational waves emitted by the binaries and quality of measurements of astrophysical information (e.g. masses) depend on accuracy of theoretical knowledge of the waveforms [1], and hence partly of dynamics of the binaries. The 3 PN approximation has been a subject of much discussion because of its ambiguity reported originally in Jaranowski and Schäfer [3]. In fact, the 3 PN ADM Hamiltonian in the ADM-type gauge obtained in [3] has two undetermined coefficients (ωkinetic and ωstatic) and the 3 PN equation of motion in the harmonic gauge derived by Blanchet and Faye [4] has one coefficient λ undetermined within their framework. Both groups have used Dirac delta distributions, which cause divergences in general relativity, to express the point particles and inevitably they have resorted to mathematical regularizations. Damour et al. [5] pointed out that the undetermined coefficients may arise due to unsatisfactory features of the regularizations they have used in [3,4]. Indeed, using the dimensional regularization, the work [5] have succeeded in determining both of the coefficients, namely, ωstatic = 0, which means λ = −1987/3080 via relationship established in [6]. (ωkinetic is related with Lorentz invariance and was fixed in [5,7]. Blanchet and Faye have developed a Lorentz invariant Hadamard Patie Finie regularization [8,9] and do not have any ambiguity other than λ.) In gravitational wave data analysis, the reduction of predictability of the equation of motion due to the undetermined coefficient can become a problem. In fact, the 3.5 PN phase evolution equation and luminosity [10] unfortunately have four undetermined coefficients, one of which is λ. Theoretically, a use of Dirac delta distributions and inevitable regularization should be verified in some manner. The perfect (physical) agreements among the results obtained by various authors with various methods [11–13] give a direct theoretical confirmation of the 2.5 PN result first derived by Damour and Deruelle [14]. It is important to achieve 3 PN iteration without introducing singular sources to derive unambiguous result and support the previous 3 PN works which have used Dirac delta distributions. Based on our previous papers [12,15], we derive a 3 PN equation of motion for two spherical compact stars in harmonic gauge without introducing singular sources. Instead, we apply the strong field point particle limit [16] to deal with strong internal gravity of the stars. Our derivation is satisfactory in a sense that the equation admits conserved energy, is Lorentz invariant, and is unambiguous. In this paper, we shall show both of the 3 PN equation of motion and an associated 3 PN energy of the orbital motion in the center of mass frame and in the case of circular orbit. Below, we shall explain briefly our yet another derivation of a 3 PN equation of motion. Since this method is different from others, we mention some details specific to our method at the 3 PN order. After deriving an invariant