Another way in which the speed of gravitational waves could
differ from
*c*
is if gravitation were propagated by a massive field (a massive
graviton), in which case
would be given by, in a local inertial frame,

where
and
*E*
are the graviton rest mass and energy, respectively. For a
recent review of the idea of a massive graviton along with a
model theory, see [132].

The most obvious way to test this is to compare the arrival
times of a gravitational wave and an electromagnetic wave from
the same event, e.g. a supernova. For a source at a distance
*D*, the resulting value of the difference
is

where
is the ``time difference'', where
and
are the differences in arrival time and emission time,
respectively, of the two signals, and
*Z*
is the redshift of the source. In many cases,
is unknown, so that the best one can do is employ an upper bound
on
based on observation or modelling. The result will then be a
bound on
.

For a massive graviton, if the frequency of the gravitational
waves is such that
, where
*h*
is Planck's constant, then
, where
is the graviton Compton wavelength, and the bound on
can be converted to a bound on
, given by

The foregoing discussion assumes that the source emits
*both*
gravitational and electromagnetic radiation in detectable
amounts, and that the relative time of emission can be
established to sufficient accuracy, or can be shown to be
sufficiently small.

However, there is a situation in which a bound on the graviton
mass can be set using gravitational radiation alone [151]. That is the case of the inspiralling compact binary. Because
the frequency of the gravitational radiation sweeps from low
frequency at the initial moment of observation to higher
frequency at the final moment, the speed of the gravitons emitted
will vary, from lower speeds initially to higher speeds (closer
to
*c*) at the end. This will cause a distortion of the observed
phasing of the waves and result in a shorter than expected
overall time
of passage of a given number of cycles. Furthermore, through the
technique of matched filtering, the parameters of the compact
binary can be measured accurately, (assuming that GR is a good
approximation to the orbital evolution, even in the presence of a
massive graviton), and thereby the emission time
can be determined accurately. Roughly speaking, the ``phase
interval''
in Eq. (76) can be measured to an accuracy
, where
is the signal-to-noise ratio.

Thus one can estimate the bounds on achievable for various compact inspiral systems, and for various detectors. For stellar-mass inspiral (neutron stars or black holes) observed by the LIGO/VIRGO class of ground-based interferometers, 200 Mpc, 100 Hz, and . The result is km. For supermassive binary black holes ( to ) observed by the proposed laser interferometer space antenna (LISA), 3 Gpc, Hz, and . The result is km.

A full noise analysis using proposed noise curves for the advanced LIGO and for LISA weakens these crude bounds by factors between two and 10. These potential bounds can be compared with the solid bound , [122] derived from solar system dynamics, which limit the presence of a Yukawa modification of Newtonian gravity of the form

and with the model-dependent bound from consideration of galactic and cluster dynamics [132].

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 |