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9.1 The 3PN acceleration and energy

We present the acceleration of one of the particles, say the particle 1, at the 3PN order, as well as the 3PN energy of the binary, which is conserved in the absence of radiation reaction. To get this result we used essentially a “direct” post-Newtonian method (issued from Ref. [42]), which consists of reducing the 3PN metric of an extended regular source, worked out in Equations (115View Equation), to the case where the matter tensor is made of delta functions, and then curing the self-field divergences by means of the Hadamard regularization technique. The equations of motion are simply the geodesic equations associated with the regularized metric (see Ref. [39] for a proof). The Hadamard ambiguity parameter c is computed from dimensional regularization in Section 8.3. We also add the 3.5PN terms which are known from Refs. [136Jump To The Next Citation Point137Jump To The Next Citation Point138Jump To The Next Citation Point174Jump To The Next Citation Point148Jump To The Next Citation Point164Jump To The Next Citation Point].

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

  1. stay manifestly invariant - at least in harmonic coordinates - when we perform a global post-Newtonian-expanded Lorentz transformation,
  2. 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
  3. be conservative, i.e. to admit a Lagrangian or Hamiltonian formulation, when the gravitational radiation reaction is turned off.

We denote by r12 = |y1(t)- y2(t)| the harmonic-coordinate distance between the two particles, with y1 = (yi1) and y2 = (yi2), by ni12 = (yi1- yi2)/r12 the corresponding unit direction, and by vi = dyi/dt 1 1 and ai= dvi/dt 1 1 the coordinate velocity and acceleration of the particle 1 (and idem for 2). Sometimes we pose vi = vi - vi 12 1 2 for the relative velocity. The usual Euclidean scalar product of vectors is denoted with parentheses, e.g., (n12v1) = n12 .v1 and (v1v2) = v1 .v2. The equations of the body 2 are obtained by exchanging all the particle labels 1 <--> 2 (remembering that ni12 and vi12 change sign in this operation):

ai = ... (168) 1

The 2.5PN and 3.5PN terms are associated with gravitational radiation reaction. The 3PN harmonic-coordinates equations of motion depend on two arbitrary length scales r'1 and r'2 associated with the logarithms present at the 3PN order31. It has been proved in Ref. [38Jump To The Next Citation Point] that r'1 and r'2 are merely linked with the choice of coordinates - we can refer to r' 1 and r' 2 as “gauge constants”. In our approach [37Jump To The Next Citation Point38Jump To The Next Citation Point], the harmonic coordinate system is not uniquely fixed by the coordinate condition am @mh = 0. 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 do not have the notion of a global coordinate system. We can always change the harmonic coordinates by means of the gauge vector a a j = dx, satisfying a Dj = 0 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 Equation (168View Equation), together with the constants ' r1 and ' r2 therein, can be removed by the coordinate transformation associated with the 3PN gauge vector (with r1 = |x- y1(t)| and r2 = |x - y2(t)|):
[ ( ) ( )] a 22-G2m1m2--- a Gm1-- r12 Gm2-- r12 j = - 3 c6 @ r2 ln r'1 + r1 ln r'2 . (169)
Therefore, the “ambiguity” in the choice of the constants ' r1 and ' r2 is completely innocuous on the physical point of view, because the physical results must be gauge invariant. Indeed we shall verify that r'1 and r'2 cancel out in our final results.

When retaining the “even” relativistic corrections at the 1PN, 2PN and 3PN orders, and neglecting the “odd” radiation reaction terms at the 2.5PN and 3.5PN orders, 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 Equation (168View Equation), with the result

E = ... (170)

To the terms given above, we must add the terms corresponding to the relabelling 1 <--> 2. 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,
dE 4 G2m2 m [ ( Gm Gm ) ( Gm 52 Gm )] --- = -------13-2- (v1v12) -v212 + 2----1- 8----2 + (n12v1)(n12v12) 3v212- 6----1+ -------2 dt 5 c5r12 ( ) r12 r12 r12 3 r12 1 +1 <--> 2 + O -7 . (171) c
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
2 2 [ ] E = E + 4G--m-1m2--(n12v1) v212- 2G(m1----m2)-- + 1 <--> 2, (172) 5c5r212 r12
we find agreement with the standard Einstein quadrupole formula (4View Equation, 5View Equation):
3 3 ( ) dE- = - -G-d-Qij-d-Qij-+ O -1 , (173) dt 5c5 dt3 dt3 c7
where the Newtonian trace-free quadrupole moment is i j 1 ij 2 Qij = m1(y 1y1- 3d y 1) + 1 <--> 2. We refer to Iyer and Will [136Jump To The Next Citation Point137Jump To The Next Citation Point] for the discussion of the energy balance equation at the next 3.5PN order. As we can see, the 3.5PN equations of motion (168View Equation) are highly relativistic when describing the motion, but concerning the radiation they are in fact 1PN, because they contain merely the radiation reaction force at the 2.5PN + 3.5PN orders.
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