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

- stay manifestly invariant - at least in harmonic coordinates - when we perform a global post-Newtonian-expanded Lorentz transformation,
- possess the correct “perturbative” limit, given by the geodesics of the (post-Newtonian-expanded) Schwarzschild metric, when one of the masses tends to zero, and
- be conservative, i.e. to admit a Lagrangian or Hamiltonian formulation, when the gravitational radiation reaction is turned off.

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):

The 3PN equations of motion depend on three arbitrary constantsWhen 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 result

To the terms given above, we must add the terms corresponding to the relabelling . Actually, this energy is not conserved because of the radiation reaction. Thus its time derivative, as computed by means of the 3PN equations of motion themselves (i.e. order-reducing all the accelerations), is purely equal to the 2.5PN effect, The resulting “balance equation” can be better expressed by transfering to the left-hand side certain 2.5PN terms so that the right-hand side takes the familiar form of a total energy flux. Posing we find agreement with the standard Einstein quadrupole formula (4, 5): where the Newtonian trace-free quadrupole moment is . As we can see, the 3PN equations of motion (131) are highly relativistic when describing the motion, but concerning the radiation they are in fact Newtonian, because they contain merely the “Newtonian” radiation reaction force at the 2.5PN order.http://www.livingreviews.org/lrr-2002-3 |
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