6.5 Other strong-gravity tests6 Gravitational Wave Tests of 6.3 Gravitational radiation back-reaction

6.4 Speed of gravitational waves

According to GR, in the limit in which the wavelength of gravitational waves is small compared to the radius of curvature of the background spacetime, the waves propagate along null geodesics of the background spacetime,i.e. they have the same speed c as light (in this section, we do not set c =1). In other theories, the speed could differ from c because of coupling of gravitation to ``background'' gravitational fields. For example, in the Rosen bimetric theory with a flat background metric tex2html_wrap_inline3749 , gravitational waves follow null geodesics of tex2html_wrap_inline3749 , while light follows null geodesics of tex2html_wrap_inline5665 (TEGP 10.1 [147Jump To The Next Citation Point In The Article]).

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 tex2html_wrap_inline5669 would be given by, in a local inertial frame,


where tex2html_wrap_inline5671 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 [132Jump To The Next Citation Point In The Article].

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 tex2html_wrap_inline5677 is


where tex2html_wrap_inline5679 is the ``time difference'', where tex2html_wrap_inline5681 and tex2html_wrap_inline5683 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, tex2html_wrap_inline5683 is unknown, so that the best one can do is employ an upper bound on tex2html_wrap_inline5683 based on observation or modelling. The result will then be a bound on tex2html_wrap_inline5677 .

For a massive graviton, if the frequency of the gravitational waves is such that tex2html_wrap_inline5693, where h is Planck's constant, then tex2html_wrap_inline5697, where tex2html_wrap_inline5699 is the graviton Compton wavelength, and the bound on tex2html_wrap_inline5677 can be converted to a bound on tex2html_wrap_inline5703, 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 tex2html_wrap_inline5681 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 tex2html_wrap_inline5683 can be determined accurately. Roughly speaking, the ``phase interval'' tex2html_wrap_inline5711 in Eq. (76Popup Equation) can be measured to an accuracy tex2html_wrap_inline5713, where tex2html_wrap_inline4407 is the signal-to-noise ratio.

Thus one can estimate the bounds on tex2html_wrap_inline5703 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, tex2html_wrap_inline5719 200 Mpc, tex2html_wrap_inline5721 100 Hz, and tex2html_wrap_inline5723 . The result is tex2html_wrap_inline5725  km. For supermassive binary black holes (tex2html_wrap_inline4187 to tex2html_wrap_inline5729) observed by the proposed laser interferometer space antenna (LISA), tex2html_wrap_inline5719 3 Gpc, tex2html_wrap_inline5733  Hz, and tex2html_wrap_inline5735 . The result is tex2html_wrap_inline5737  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 tex2html_wrap_inline5739, [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 tex2html_wrap_inline5741 from consideration of galactic and cluster dynamics [132].

6.5 Other strong-gravity tests6 Gravitational Wave Tests of 6.3 Gravitational radiation back-reaction

image The Confrontation between General Relativity and Experiment
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de