where
is the total mass,
,
, and
. The notation
indicates that the term is
relative to the Newtonian term
. Explicit formulae for terms through various orders have been
calculated by various authors: non-radiative terms through 2PN
order [40,
38,
72,
39,
27], radiation reaction terms at 2.5PN and 3.5PN order [78,
79,
17], and non-radiative 3PN terms [80,
81,
46,
26,
25]. Here we quote only the first PN corrections and the leading
radiation-reaction terms at 2.5PN order:
where
. These terms are sufficient to analyse the orbit and evolution
of the binary pulsar (Sec.
5.1). For example, the 1PN terms are responsible for the periastron
advance of an eccentric orbit, given by
, where
a
and
e
are the semi-major axis and eccentricity, respectively, of the
orbit, and
is the orbital frequency, given to the needed order by Kepler's
third law
.
Another product is a formula for the gravitational field far from the system, written schematically in the form
where R is the distance from the source, and the variables are to be evaluated at retarded time t - R . The leading term is the so-called quadrupole formula
where
is the quadrupole moment of the source, and overdots denote time
derivatives. For a binary system this leads to
For binary systems, explicit formulae for all the terms
through 2.5PN order have been derived by various authors [135,
155,
106,
23
,
24
,
152
,
16,
18]. Given the gravitational waveform, one can compute the rate at
which energy is carried off by the radiation (schematically
, the gravitational analog of the Poynting flux). The
lowest-order quadrupole formula leads to the gravitational wave
energy flux
Formulae for fluxes of angular and linear momentum can also be
derived. The 2.5PN radiation-reaction terms in the equation of
motion (48) result in a damping of the orbital energy that precisely
balances the energy flux (54
) determined from the waveform. Averaged over one orbit, this
results in a rate of increase of the binary's orbital frequency
given by
where
is the so-called ``chirp'' mass, given by
, and
. Notice that by making precise measurements of the phase
of either the orbit or the gravitational waves (for which
for the dominant component) as a function of the frequency, one
in effect measures the ``chirp'' mass of the system.
These formalisms have also been generalized to include the leading effects of spin-orbit and spin-spin coupling between the bodies [83, 82].
Another approach to gravitational radiation is applicable to
the special limit in which one mass is much smaller than the
other. This is the method of black-hole perturbation theory. One
begins with an exact background spacetime of a black hole, either
the non-rotating Schwarzschild or the rotating Kerr solution, and
perturbs it according to
. The particle moves on a geodesic of the background spacetime,
and a suitably defined source stress-energy tensor for the
particle acts as a source for the gravitational perturbation and
wave field
. This method provides numerical results that are exact in
v, as well as analytical results expressed as series in powers of
v, both for non-rotating and for rotating black holes. For
non-rotating holes, the analytical expansions have been carried
to 5.5PN order, or
beyond the quadrupole approximation. All results of black hole
perturbation agree precisely with the
limit of the PN results, up to the highest PN order where they
can be compared (for a detailed review see [93]).
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The Confrontation between General Relativity and
Experiment
Clifford M. Will http://www.livingreviews.org/lrr-2001-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |