Though the successive post-Newtonian approximations are really a consequence of general relativity, the final equations of motion must be interpreted in a Newtonian-like fashion. That is, once a convenient general-relativistic (Cartesian) coordinate system is chosen, we should express the results in terms of the coordinate positions, velocities, and accelerations of the bodies, and view the trajectories of the particles as taking place in the absolute Euclidean space of Newton. But because the equations of motion are actually relativistic, they must
We denote by the harmonic-coordinate distance between the two particles, with and , by the corresponding unit direction, and by and the coordinate velocity and acceleration of the particle 1 (and idem for 2). Sometimes we pose for the relative velocity. The usual Euclidean scalar product of vectors is denoted with parentheses, e.g., and . The equations of the body 2 are obtained by exchanging all the particle labels (remembering that and change sign in this operation):20: the dimensionless constant (e.g., a rational fraction), linked with an incompleteness of the Hadamard regularization as discussed in Section 8.2; and two arbitrary length scales and associated with the logarithms present at the 3PN order. It has been proved in Ref.  that the two constants and are merely linked with the choice of coordinates - we can refer to and as “gauge constants”. In our approach [21, 22], the harmonic coordinate system is not uniquely fixed by the coordinate condition . In fact there are infinitely many harmonic coordinate systems that are local. For general smooth sources, as in the general formalism of Part A, we expect the existence and uniqueness of a global harmonic coordinate system. But here we have some point-particles, with delta-function singularities, and in this case we don’t have the notion of a global coordinate system. We can always change the harmonic coordinates by means of the gauge vector , satisfying except at the location of the two particles (we assume that the transformation is at the 3PN level, so we can consider simply a flat-space Laplace equation). More precisely, we can show that the logarithms appearing in Eq. (131), together with the constants and therein, can be removed by the coordinate transformation associated with the 3PN gauge vector (with and ):
When retaining the “even” relativistic corrections at the 1PN, 2PN and 3PN orders, and neglecting the “odd” radiation reaction term at the 2.5PN order, we find that the equations of motion admit a conserved energy (and a Lagrangian, as we shall see), and that energy can be straightforwardly obtained by guess-work starting from Eq. (131), with the resultmotion, but concerning the radiation they are in fact Newtonian, because they contain merely the “Newtonian” radiation reaction force at the 2.5PN order.
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