Equation (62) is exact, and depends only on the assumption that spacetime can be covered by harmonic
coordinates. It is called “relaxed” because it can be solved formally as a functional of source variables
without specifying the motion of the source, in the form
At the same time, just as in electromagnetism, the formal integral (64) must be handled differently,
depending on whether the field point is in the far zone or the near zone. For field points in the far zone or
radiation zone,
(
is the gravitational wavelength divided by
), the field
can be expanded in inverse powers of
in a multipole expansion, evaluated at the
“retarded time”
. The leading term in
is the gravitational waveform. For field
points in the near zone or induction zone,
, the field is expanded in powers of
about the local time
, yielding instantaneous potentials that go into the equations of
motion.
However, because the source contains
itself, it is not confined to a compact region, but
extends over all spacetime. As a result, there is a danger that the integrals involved in the various
expansions will diverge or be ill-defined. This consequence of the non-linearity of Einstein’s equations has
bedeviled the subject of gravitational radiation for decades. Numerous approaches have been developed
to try to handle this difficulty. The “post-Minkowskian” method of Blanchet, Damour, and
Iyer [35, 36, 37, 76, 38, 33] solves Einstein’s equations by two different techniques, one in the near
zone and one in the far zone, and uses the method of singular asymptotic matching to join the
solutions in an overlap region. The method provides a natural “regularization” technique to
control potentially divergent integrals (see [34
] for a thorough review). The “Direct Integration
of the Relaxed Einstein Equations” (DIRE) approach of Will, Wiseman, and Pati [291
, 208]
retains Equation (64
) as the global solution, but splits the integration into one over the near
zone and another over the far zone, and uses different integration variables to carry out the
explicit integrals over the two zones. In the DIRE method, all integrals are finite and convergent.
Itoh and Futamase have used an extension of the Einstein-Infeld-Hoffman matching approach
combined with a specific method for taking a point-particle limit [134], while Damour, Jaranowski,
and Schäfer have pioneered an ADM Hamiltonian approach that focuses on the equations of
motion [139, 140, 77, 79, 78].
These methods assume from the outset that gravity is sufficiently weak that and harmonic
coordinates exists everywhere, including inside the bodies. Thus, in order to apply the results to cases where
the bodies may be neutron stars or black holes, one relies upon the SEP to argue that, if tidal
forces are ignored, and equations are expressed in terms of masses and spins, one can simply
extrapolate the results unchanged to the situation where the bodies are ultrarelativistic. While no
general proof of this exists, it has been shown to be valid in specific circumstances, such as at
2PN order in the equations of motion, and for black holes moving in a Newtonian background
field [70].
Methods such as these have resolved most of the issues that led to criticism of the foundations of gravitational radiation theory during the 1970s.
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