where
is the flat-spacetime wave operator,
is a ``gravitational tensor potential'' related to the deviation
of the spacetime metric from its Minkowski form by the formula
,
g
is the determinant of
, and a particular coordinate system has been specified by the
de Donder or harmonic gauge condition
(summation on repeated indices is assumed). This form of
Einstein's equations bears a striking similarity to Maxwell's
equations for the vector potential
in Lorentz gauge:
,
. There is a key difference, however: The source on the right
hand side of Eq. (45
) is given by the ``effective'' stress-energy pseudotensor
where
is the non-linear ``field'' contribution given by terms
quadratic (and higher) in
and its derivatives (see [94], Eqs. (20.20, 20.21) for formulae). In general relativity,
the gravitational field itself generates gravity, a reflection of
the nonlinearity of Einstein's equations, and in contrast to the
linearity of Maxwell's equations.
Equation (45) 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
where the integration is over the past flat-spacetime null
cone
of the field point
. The motion of the source is then determined either by the
equation
(which follows from the harmonic gauge condition), or from the
usual covariant equation of motion
, where the subscript
denotes a covariant divergence. This formal solution can then be
iterated in a slow motion (v
<1) weak-field (
) approximation. One begins by substituting
into the source
in Eq. (47
), and solving for the first iterate
, and then repeating the procedure sufficiently many times to
achieve a solution of the desired accuracy. For example, to
obtain the 1PN equations of motion,
two
iterations are needed (i.e.
must be calculated); likewise, to obtain the leading
gravitational waveform for a binary system, two iterations are
needed.
At the same time, just as in electromagnetism, the formal
integral (47) 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
), the field can be expanded in inverse powers of
in a multipole expansion, evaluated at the ``retarded time''
t
-
R
. The leading term in 1/
R
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
t, 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 [19,
20,
21,
45,
22,
15] 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. The
``Direct Integration of the Relaxed Einstein Equations'' (DIRE)
approach of Will, Wiseman and Pati [152
,
105] retains Eq. (47
) 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.
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
strong equivalence principle 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 [39
].
Methods such as these have resolved most of the issues that led to criticism of the foundations of gravitational radiation theory during the 1970s.
![]() |
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 |