9.1 The 3PN accelerations 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. [25]), which consists of reducing
the 3PN metric of an extended regular source, worked out in Eqs. (109, 110, 111, 112, 113,
114, 115), 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. [23] for a
proof).
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
constants:
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. [22
] 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
):
Therefore, the “ambiguity” in the choice of the constants
and
is completely innocuous on the
physical point of view, because the physical results must be gauge invariant. Indeed we shall verify that
and
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 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.