76 Tests of General Relativity
In a Newtonian approximation the strength of the waves from a nonspherical body
of mass푀, oscillating size퐿(푡), and quadrupole moment푄(푡)≈푀퐿^2 at a distance푟
from Earth is
ℎ(푡)≈퐺
푐^4 푟d^2 푄(푡)
d푡^2= 퐺
푐^4 푟
2 Mv(푡)^2 =^4 퐺
푐^4 푟퐸(푡), (4.16)
where퐺is the Newtonian constant,푣is the internal velocity, and퐸=^12 푀푣^2 is the
nonspherical part of the internal kinetic energy. The factor푐^4 is introduced only to
makeℎ(푡)dimensionless.
Sources of Gravitational Waves. From this formula one can work out that a
nonspherical symmetric supernova collapse at the center of our Galaxy will give
rise to waves of amplitudeℎ≈ 10 −^19 causing a subnuclear stretch and squeeze of
an object 1km in length by 10−^16 m. A spherically symmetric supernova collapse
causes no waves. In a catastrophic event such as the collision of two neutron stars
or two stellar-mass black holes in which퐸∕푐^2 is of the order of one solar mass,
Equation (4.16) givesℎ≈ 10 −^20 at the 16Mpc distance of the Virgo cluster of galaxies,
andℎ≈ 10 −^21 at a distance of approximately 200Mpc.
The signals one can expect to observe in the amplitude rangeℎ≈ 10 −^21 –10−^20 with
the present generation of detectors are bursts due to the coalescence of neutron-star
binaries during their final minutes and seconds (in the high frequency band 1–10^4 Hz),
and periodic waves from slowly merging galactic binaries and extragalactic mas-
sive black hole binaries (low-frequency band 10−^4 –10−^2 Hz), which are stable over
hundreds to millions of years. The timing of millisecond binary pulsars such as the
PSR 1913+16 belong to the very low-frequency band of 10−^9 –10−^7 Hz. In this band,
processes in the very early Universe may also act as sources.
Merger waves from superheavy black holes with 10^6 푀⊙mass may be so strong that
both their direction and their amplitude can be determined by monitoring the waves
while the detector rotates around the Sun. This may permit researchers to identify
the source with the parallax method and to determine the distance to it with high
precision. Combined with redshift measurements of the source, one could determine
not only퐻 0 but even the deceleration parameter푞 0 of the Universe. Thus the detection
of gravitational waves from black holes would go beyond testing general relativity to
determining fundamental cosmological parameters of the Universe.
The dynamics of a hole–hole binary can be divided into three epochs: inspiral,
merger and ringdown. The inspiral epoch ends when the holes reach their last sta-
ble orbit and begin plunging toward each other. Then the merger epoch commences,
during which the binary is a single nonspherical black hole undergoing highly nonlin-
ear space-time oscillations and vibrations of large amplitude. In the ringdown epoch,
the oscillations decay due to gravitational wave emission, leaving finally a spherical,
spinning black hole.
Gravitational Wave Detection. Detection with huge metal bars as resonant anten-
nas was started byJoseph Weberin 1969. These detectors couple to one axis of the