This document summarizes the development of gravitational wave detection and possible sources. It discusses James Weber's early experiments using aluminum bar detectors in the 1960s, which reported the first detections of gravitational waves. It also describes proposals to use laser interferometers and resonant bar detectors, which could achieve greater sensitivity. Finally, it outlines potential sources of gravitational waves including bursts from stellar collapses and mergers, and continuous waves from rapidly spinning neutron stars.

1. DEVELOPMENT OF THE MEANS OF DETECTING GRAVITATIONAL RADIATION
AND POSSIBLE SOURCES FOR OBSERVATION
Michael L. Montagne
ASTR 6132: Cosmology

2. DEVELOPMENT OF THE MEANS OF DETECTING GRAVITATIONAL RADIATION
AND POSSIBLE SOURCES FOR OBSERVATION
This paper will review the theoretical and experimental
developments that have been made in the field of gravitational
wave astronomy since Webber's first proposal in 1960. Webber's
theoretical contributions and the observational results he
obtained in 1969 will be discussed. This paper will also
consider technical improvements on the original detectors in the
form of multimode resonant antennas. Various ideas have been
advanced for using laser interferometers to create more
sensitive and versatile antennae. These ideas will be studied.
Proposals that have been made for specific astronomical and
astrophysical observations that can be made when practical
detectors are developed will also be assessed.

3. THEORY OF GRAVITATIONAL RADIATION
Gravity waves are, according to General Relativity, ripples
in the fabric of spacetime propagating at the speed of light.
The waves are characterized by two dimensionless gravitational
wave amplitudes h+ and hx. These two amplitudes are scalar
fields which determine all properties of the wave. For a wave
moving through the solar system in Minkowskii space, the
gravitational wave field would be
hjk
TT
= h+e+
jk+hxex
jk .
This is a symmetric spacial tensor which is transverse to the
wave's direction of motion and trace free (denoted by "TT"), e+
and ex
are polarization tensors. This expression is analogous to
the Lorentz-gauge vector potential in electrodynamics. This is
also the metric perturbation associated with gravity waves and
is related to the Riemann tensor by
Rj0k0=-1/2h''TT
jk.
When propagating through interstellar space, which is very

4. nearly flat, the polarization constants will be constant and
amplitude will decrease linearly with distance from the source
h+=(1/r)A+(t-r/c), hx=(1/r)Ax(t-r/c).
It should be noted that waves from extremely distant sources
will experience the cosmological doppler shift.(1)
RESONANCE BAR ANTENNAS
In 1961, James Webber of the University of Maryland
proposed an antenna to detect Gravitational waves. The antenna
would consist of a bar of some elastic material such as
aluminum. This bar could be deformed by the dynamic derivatives
of the gravitational potentials, thus having its normal modes
excited. This would produce a measurement of some components of
the Riemann curvature tensor averaged over the volume of the
bar.(2) Webber proceeded to build such a device and reported
successful observation of gravitational radiation.
Webber established two antennae at the University of
Maryland at College Park and two at the Argonne National
Laboratory in Chicago. One of the detectors at College Park was
cryogenically cooled to improve its sensitivity. With a
baseline of 1000 kilometers he looked for coincidental responses
separated by .41 seconds. Piezoelectric crystals bonded to the
surface of the cylinders acted as transducers, coupling the
normal mode oscillations to an electromagnetic degree of

5. freedom. Over a period of 81 days Webber observed 17
significant coincidental responses between two of the detectors,
five between three detectors and three coincidences between all
four detectors. Probability analysis shows that the expected
period between such coincidences, if occurring randomly in the
natural oscillation of the antennae, range from 144 days to 48
years. Coincidences were observed at all times of day so it was
not possible to determine the direction of any particular source
of radiation.(3)
For a 96cm long detector, the current from the transducers
would start from zero and reach its peak 11 seconds later.
Detectors that were not cryogenically cooled would lack this
delay. Thus any coincidence of signal would be observed first
in the uncooled detectors and in the cooled one 11 seconds
later. This result was attained in about one quarter of
Webber's events. Webber concluded that the statistical
relationships between the coincidences observed overwhelmingly
indicated a common origin. He felt he had taken adequate steps
to rule out electromagnetic and seismic sources of excitation
and thus gravitational radiation was strongly indicated.(3)
Following Webber, theoretical work was done on detectors
similar to his but with multiple modes. Such a multimode
resonant antenna would consist of a string of resonators of
decreasing mass. The first one would be the actual antenna. It
would have a mass M1 which would be the mass of the quadrupole
mode which interacts with the gravity wave. The greater the
mass of this object the greater will be the gravity wave

6. absorption cross section. Something in the neighborhood of 1000
kg would be in order.
Each of the resonators would be attached to the next one in
sequence by a spring. The spring constant Kj is chosen so as to
obtain the desired uncoupled resonant frequency w=(Kj/Mj)1/2
. The
mass of each resonator is less than the one before it. The last
(and smallest) resonator in the series becomes the transducer.
In such a detector the optimum ratio of the masses of
successive resonators is
uopt(N)=(B /1+ 2
)(Oth+B ).
Where B is the energy coupling coefficient, Oth represents a
systemic thermal noise source and N is the total number of
resonators. The optimum number of resonators will be
N=1+[ln(mr/M1B)/ln(uopt)
in which mr is the coupling strength of the transducer. This
equation will generally yield a number from 2 to 4 for most
cases of practical interest. If the proper energy coupling
coefficient is chosen the thermal noise of the mechanical
oscillators can be rendered negligible. This will allow an
increase in the sensitivity of the detector over Webber's single
mode devices.(4)
LASER INTERFEROMETRIC DETECTORS
A more promising technique may exist in the use of laser
interferometry. The simplest form of interferometric detector
is a Michelson interferometer consisting of three test masses
arranged in a right isosceles triangle. The laser enters at the

7. apex of the triangle and is split into two beams that travel
along the legs of the triangle. The test masses are suspended
as pendulums but act as free masses for horizontal motions with
frequencies significantly greater than 1 Hz. An incident
gravitational wave will shorten one arm slightly while slightly
lengthening the other. If the lengthened arm is taken to be on
the x axis, then dl/l=(1/2)hTT
xx=(1/2)h+; for the other arm
dl/l=(1/2)hTT
yy=(1/2)h+. This difference in the path length of the
two halves of the laser beam will obviously produce a relative
phase shift and thus an intensity change of the recombined
light. The sensitivity can be improved by increasing the phase
shift with repeated reflections of the beams in the arms of the
detector. For b number of reflections the relative phase shift
will be dO=2(b=1)*(dl/ ). (1)
Theoretical work has shown that this technique of recycling
can be accomplished in several different ways. The standard
method is one of broadband recycling but it has been shown that
this can be tuned to a narrow band. All narrow band techniques
cause the laser light and a gravitationally induced sideband to
be resonant in the optical system. Resonant recycling, which
was the original narrow band method has now been shown to have
useful broadband applications including various sensitivity-
bandwidth combinations. (5)
Standard broadband recycling consists merely in placing a
mirror between the laser source and the beam splitter. If
properly placed this will coherently reflect back into the
interferometer any light that was traveling back toward the

8. laser source. The power of the beam is increased by the number
of times the light is recycled through the cavity while the
shot-noise limited sensitivity increases by the square root of
the same factor. Due to losses in the system from absorption
and scattering by the mirrors the maximum power gain P is
P=(1-R2
eff)-1
.
The term in parentheses is the total loss in one round trip
through the system.
A tradeoff between signal and losses must be made in
determining the storage time. The optimum choice is a time just
short of producing the maximum phase shift. This broadband
recycling is most useful when searching for unexpected events or
short bursts. For observing continuous monochromatic sources of
radiation such as that from pulsars or accreting neutron stars a
narrow band detector is more desirable.(5)
Such a narrow band detector can be produced by using
resonant recycling. In resonant recycling the reflected light
is coupled to the phase shift of the gravity wave so that the
signal builds up coherently. In such a system the gain in
sensitivity is roughly the square of the gain in standard
recycling. This is because it is the signal rather than the
intensity that is recycled. This effect can be achieved in two
ways depending on the type of interferometer used.
For a delay line interferometer the light receives the
maximum phase shift possible from the gravity wave by having a
storage time equal to one half the period of the wave. The
light is then reflected directly in to the other arm of the

9. detector where it then sees the same sign of the gravity wave it
saw before. This is because the gravity wave has, of course,
changed sign after half a period. The increase in signal is
restricted to a narrow bandwidth because the other frequencies
become out of step with the gravitational wave. (6)
If an interferometer with optical cavities (Fabrey-Perot
cavities) is used, the detector can be regarded as a system of
coupled cavities which have two normal modes. The laser
resonates with one mode while the gravity wave pumps energy into
the other. As the light moves through a cavity, the gravity
wave acts to change the effective length of the cavity. This
creates two sidebands on the carrier light emanating from the
cavity. When the light enters the second cavity and is
reflected, one of the sidebands will experience a phase shift
180o
different from the carrier signal. If the sideband and
carrier were both shifted 180o
in the original cavity then they
will be resonant and the sideband will have the phase necessary
for its amplitude to be increased by the gravitational wave. (5)
As noted before, these narrow band recyclers can be made
broadband. If the lowest possible resonant frequency is vg0, so
that wgots=2, then the bandwidth of a tuned detector will be
delta vg= 2v0A2
vg/pivg0
proper adjustment of the detector will give a bandwidth roughly
equal to the carrier frequency. In this case the sensitivity
gain S will be equal to that obtained with a standard recycler
and thus the recycler will be broadband.
In order to cause a standard broadband recycler to work in

10. a narrow band the center and interferometer cavities are
coupled, creating a two mode system. This is done by adjusting
the cavities so that one of the sidebands rather than the laser
light which is on resonance with the isolated cavities. This
produces a phase shift of one half the maximum possible and
reduces the losses for the laser light. This allows a greater
buildup of intensity in the center cavity and an improvement in
the shot-noise-limited sensitivity. In such a case the gain in
sensitivity is exactly the same as that achieved in resonant
recyclers. (5, 6)
TYPES AND SOURCES OF GRAVITATIONAL RADIATION
The possible sources of gravitational radiation can be
divided into three main classes: burst, continuous wave, and the
stochastic background. A burst is generally defined as any
event of sufficiently short duration that the doppler shift
produced by the Earth's rotation can be neglected. This is
roughly 30 minutes for radiation with a frequency around 1 kHz.
A continuous wave is any coherent wave train longer than this.
Other, incoherent waves contribute to the stochastic background.
(7)
The only events likely to produce sufficient radiation to
be detected by an instrument such as Webber's are the collapse
of a body with stellar mass or the capture of one collapsed
object by another (8). Most of the gravitational energy from
such an event would be released in a time M/MO sec. This time
period is determined by the dynamics of the collapse or capture.

11. Most of the Gravitational energy would be released as matter
neared the Schwarzchild radius. This would be as infalling
matter neared the Schwarzchild radius of a collapsing object or
as a captured object neared the Schwarzchild radius of the
capturing body. Most of the energy would be released in a
period t of the order of the dynamical time at this stage,
specifically, t=GMc-3
=10-5
MM0
-1
sec.
For a burst of finite energy the time integral of the
Riemann tensor components over the duration of the burst will be
zero. This is shown in the formula for the energy flux in the
wave at the observer;
F(t)=c7
(4piG-1
{[St
R1010(u)du]2
+ [St
R1020(u)du]2}erg/cm2
sec.
This clearly requires the sign of the Riemann tensor components
to reverse during the burst. During the indicated period only a
double or triple pulse of gravitational radiation is likely to
be detected because the Riemann tensor components will only
reverse sign a few times. It is likely that a collapse will
show three reversals while a capture would only show one during
the critical period (8). This gives us a characteristic by
which we can judge the nature of the source of a particular
burst.
The gravitational wave form, like its electromagnetic
counterpart, carries information about the direction to its
source but also detailed information about the motions of the
mass that generated it. This is due to the fact that such waves
are produced by coherent motions of large masses rather than the
emissions of individual charged particles which are incoherently

12. superimposed on one another. With more sensitive detectors we
may eventually be able to make detailed observations of various
astrophysical phenomena. It should be possible to study things
such as; the dynamics of the cores of supernovae, the evolution
of very young rapidly rotating stars, surface quakes in neutron
stars, the dynamics of black hole formation and the pulsations
of newly formed holes, collisions between high density objects
such as neutron stars and black holes and the internal
structures of common envelope binary stars. (7)
In the event of the collapse of a neutron star with the
mass of the sun and all other mass expelled then a burst of
gravitational radiation will be produced with
h<2*10-21
(15Mpc/r) at a frequency f=1Kh
In such an event the total radiated energy is likely to be only
E<0.05MO, i.e. less than half the binding energy. For the
collapse of roughly 10 MO into a black hole we could quite
reasonably expect a 10% efficiency giving
h=1.5*10-20
(15Mpc/r) for f=1kHz.
This would result in total energy release of E=1MO.(7)
Theoretically a collapse can be perfectly spherical and
produce no gravitational radiation whatsoever. In the actual
event however, a range of initial conditions will produce
varying results. We can take these numbers as being an upper
limit.
The frequency of occurrence of such collapses is still a
matter of some debate. Certainly the rate of supernova events
can be taken as a minimum. Theoretical calculations have shown

13. a possibility of as many as one collapse every one to five years
in our galaxy. The unseen collapses may produce little or no
electromagnetic radiation or may be obscured by dust clouds. If
the collapse rate is in the range of this upper limit then
broadband detectors should eventually be able to detect them.(7)
Bursts can also be produced by coalescing binaries. As the
two bodies spiral toward each other the frequency of radiation
and the total energy produced will grow quite rapidly. Consider
two neutron stars each having a mass of 1.4MO with orbits
decaying due to gravitational radiation. When the radius of the
system is 80 Km they will be radiating at a frequency of
approximately f=100Hz and have h=5*10-23
(100Mpc/r). Within 500
revolutions the frequency will double. The energy radiated in
this period will be approximately E=6*10-3
MO. A system containing
a 1.4MO neutron star and a 10MO black hole in a similar
situation would have h=3*10-22
(100Mpc/r) and would radiate E=3*10-
2
MO in just 130 revolutions.
Continuous gravitational radiation can be produced by the
spindown of a neutron star with gravitationally induced
instabilities. This can occur in stars with periods of less
than one millisecond. For any given eigenvalue m a sufficiently
large angular momentum will produce unstable modes for that
eigenvalue. For m=2 and m=3 the modes have time scales of 1 to
100 seconds. The timescales of the modes for m=4 and m=5 are
more like several hours to several weeks. These modes could
produce radiation of a frequency of 1 to 5 kHz and release total
energies on the order of 3*10-3
MO. Thus if a burst were observed

14. followed by a period of continuous waves of this frequency range
it would indicate the presence of a neutron star. (7)
Continuous waves can also be produced by slow rotating
neutron stars accreting material from a binary companion. This
will occur if the neutron star has a fairly weak magnetic field
and acquires enough angular momentum from the accreted matter to
become gravitationally unstable. These will produce radiation
with h=3*10-27
at frequencies of 200 to 800 Hz. This sort of
system is probably best detected with narrow band highly
sensitive laser interferometers. (9)
Many possible sources may contribute to a gravitational
stochastic background: binary stars, phase transitions in the
early universe, primordial nucleosynthesis, formation of black
holes at in the period of galaxy formation, and cosmic strings.
This radiation has been predicted to have a frequency of
approximately 1kHz. This background noise can best be observed
by using two detectors within 1500 km of one another. Their
responses would be correlated to determine if the observed noise
is higher than that predicted, based on the Gaussian noise
statistics of the individual detectors. Observation of excess
noise would be evidence of a stochastic background of
gravitational radiation.(10)
CONCLUSION
The prediction of Gravitational Radiation is a natural
consequence of the Theory of General Relativity. James Webber
first proposed a method of detecting such radiation in 1960 and

15. actually conducted experimental observations in 1969. He
reported successful results that strongly indicated the presence
of gravity waves. Since that time assorted work has been done
on the theory and practice of building gravity wave detectors.
In addition to improvements on Webber's resonance bar antenna,
ideas have been developed for laser interferometric detectors.
Specific theoretical predictions have been made concerning the
sources of gravitational burst, continuous waves and a
stochastic background. As practical detectors of various types
are built, these predictions will be subjected to observational
confirmation.
REFERENCES
1
K. S. Thorne, Rev. of Mod. Phys. 52, 285 (1980).
2
J. Webber, Phys. Rev. 117, 306 (1960).
3
J. Webber, Phys. Rev. Letters, 22, 1320 (1969).
4
Massimo Bassan, Phys. Rev. D 38, 2327 (1988).
5
Brian J. Meers, Phys. Rev. D 38, 2317 (1988).
6
Jean-Yves Vinet, Brian Meers, Catharine Nary Man, Alain Brillet,
Phys. Rev. D 38, 433 (1988).
7
B. F. Schutz, Class. Quantum Grav. 6, 1761 (1989).

16. 8
G. W. Gibbons, S. W. Hawking, Phys. Rev. D 4, 2191 (1971).
9
Robert V. Wagoner, Astrophys. Jrnl. 278, 345 (1984).
10
Peter F. Michelson, Mon. Not. R. Astr. Soc. 227, 933 (1987).