Gravitational Waves

Erik Elfgren, Tomas Samuelsson


A review on the properties of gravitational waves is presented along with comparisons with electromagnetic theory. Gravitational waves behave very much like light waves, except for that they are second rank tensors instead of being vectors, like light-waves. The different sources of gravitational radiation are discussed, and their origins and strengths. Finally, the development on the experimental front is treated. This is done mainly on the two experiments LISA and ALLEGRO representing the interferometer and the resonance detector types, respectively.



Ever since the introduction of Einstien's general theory of relativity, gravtational waves has been of interest. They arise quite naturally from the theory and were concieved by Einstein himself early after the general relativity. They are, if not an inevitable, but at least a very plausible part of the theory. If they were not to exist, we would have to reconsider Einstein's equations and try to explain why this would be so. This could be an exciting subject for further studies in the area.

Gravitational waves are a natural part of general relativity and astrophysics as they mainly stem from huge gravitational objects rapidly changing form, like supernovae explosions, binary stars revolving about each other, masses falling into black holes and so on.

In 1993, Taylor and Hulse were awarded with the Nobel prize for the discovery of indirect evidence of the existence of gravitational waves. A neutron star looses an amount of rotational energy which currently can only be concieved of as being caused by gravitational radiation. Their measurements of radio waves from a binary pulsar designated PSR1913+16 show that the pulsar's 8-hour orbit around the neutron star is gradually contracting; the faster the pulsar revolves around the neutron star, the smaller its orbit gets. As the rate of decrease agrees to within 0.5 percent with predictions derived from the general theory of relativity, the finding is excellent circumstantial evidence for orbital decay being a result of energy lost by gravitational radiation. Even though the gravitational radiation itself was not detected, Taylor and Hulse shared the 1993 Nobel Prize in Physics for this work.

In this brief review on the subject, firstly some general properties of gravitational waves will be presented. This is done as to acquaint the reader with the concept, and to show the many similarities with electromagnetism, thus facilitating the comprehension of the phenomenon. The second section treats the sources for gravitational radiation without any deeper theoretical argumentation. The principles can be seen easily enough from the end of the first section. Two types of sources for gravitational waves are discussed, periodical and catastrophical. The first being more permanent, the second more intensive. Gravitational waves are currently a hot issue and several attemps of detection is being deviced and put into action. There are mainly two types of gravitational wave's detectors, one technique from the 60's, resonance monitoring, and a second one, more modern, interferometric method. Primarly the next generation of detectors are discussed, where more sophisticated methods for detection are used.



The theory of gravitational waves was originally developed by Albert Einstein 1916 as a part of his extensive study on gravitation. The theory of gravitational radiation resembles a lot the theory of electromagnetic theory. Throughout this section comparisons will be made to show the similarities. When vizualizing what really happens it is often helpfull to think about the gravitational wave as an electromagnetic wave and mass as the charge affected by the field. The reader is supposed to be famaliar with tensor analysis and somewhat with general, or a least special, relativity. If not, Basic Relativity by Mould gives a good introduction to these subjects. We will denote






Normally, indices tex2html_wrap_inline403 and tex2html_wrap_inline405 will be used as dummy indices, while tex2html_wrap_inline407 and tex2html_wrap_inline409 will be used for vectors and tensors.


If we neglect the curvature of general relativitygif the metric of space-time be written on the form:


where tex2html_wrap_inline411 is a small perturbation in space-time caused by gravitational waves and tex2html_wrap_inline413 is the normal Minkowski metric tensor:


Starting from the Einstein field equations


where tex2html_wrap_inline415 is the contraction of the Ricci tensor tex2html_wrap_inline417 , G is the universal constant of gravitation, and c is the speed of light in vacuum. The 16 terms of tex2html_wrap_inline423 is reduced to ten with the symmetry condition tex2html_wrap_inline425 we have. In the linearized theory of gravity the Ricci tensor can be written in terms of the metric as


which gives the Einstein's equations:


We make a transformation:


and the Einstein's equations takes simpler form:


We can impose the guage condition:


without loss of generality. This can be compared with the guage condition in electromagnetic theory tex2html_wrap_inline427 . The guage condition in gravitational theory eliminates all but the first term in equation (gif) leaving us with the equation:


which can be compared with the equivalent in electromagnetism tex2html_wrap_inline429 . With the four guage-conditions, the ten field equations are reduced to only six.

In empty space-time, the energy-momentum tensor of matter is zero, tex2html_wrap_inline431 . The Einstein's equations then has the form:


which is a partial differential equation of second order with the solution


where tex2html_wrap_inline433 is a constant tensor dermining the orientation of the wave in space. Once again, the analogy with electromagnetism is obvious with tex2html_wrap_inline435 is replaced with tex2html_wrap_inline437 .

Combining equation (gif) with equation ( tex2html_wrap_inline439 ) the guage condition can now be written as


which means that in the direction of the wave, the field is zero, or in other words, the gravitational waves are transversal. Even with the guage condition there is still a freedom in the equations, since the coordinate transformation


with tex2html_wrap_inline441 leaves the coordinate system unchanged. This can be used to further reduce the number of equations, arriving at only two left. These conditions can be arranged in such a manner that:


which also can be written as


This means that the time-components are all zero and that tex2html_wrap_inline411 is trace-less. With this choice of coordinates, we have defined the Transversal-Traceless (TT) guage. If we let the wave propagate in the x-direction, tex2html_wrap_inline445 the TT-guage along with equation (gif) and symmetry gives


which can be decomposed to




Hence, the wave equation can be split up in rwo components


which are the two polarization states of gravitational radiation. Let us see what the two states really means. In tex2html_wrap_inline447 the y-direction decrease when the z-direction increase and vice versa, see figure gif. To visualize tex2html_wrap_inline449 we rotate the coordinate system 45 degrees where it takes the same form as tex2html_wrap_inline447 is the normal coordinate system. Thus, it represents the same kind of oscillation but rotated 45 degrees with respect to tex2html_wrap_inline447 .

Figure: Deformation of a test ring of matter produced by linearly polarized waves.

With a polarized state tex2html_wrap_inline455 , the metric of space-time would become


This gives the strain in a material as


This is a summary of what we have seen in the previous section, to better see the similarities.


We have also seen that gravitational waves only have two polarized states which are both transversal, that is, perpendicular to the direction of propagation.

Electric Quadrupole Radiation

In electromagnetic theory the electric dipole radiation is given by


where L is the luminosity, tex2html_wrap_inline477 is the electric dipole moment and tex2html_wrap_inline479 is the charge of particle i. The equivalent in the theory of gravity is obtained by replacing the charge, tex2html_wrap_inline479 with the mass, tex2html_wrap_inline485 :


However, tex2html_wrap_inline487 by conservation of momentum:


That is, there is no gravitational dipole radiation.

The second strongest type of radiation in electromagnetic theory is the magnetic dipole radiation which is given by the second derivative of the magnetic dipole moment:


which would give a gravitational "magnetic" dipole moment of


where tex2html_wrap_inline489 is the angular momentum. Hence, conservation of angular momentum gives that there is no dipole radiation for gravitation what so ever.

The first type of radiation that can occur is from the quadrupole moment, and the gravitational luminosity can be expressed as:


where tex2html_wrap_inline491 is the mass-energy quadrupole tensor of the source:



Gravitational waves, as predicted through general relativity has a number of properties simliar to those of electromagenitc radiation:

  1. The velocity of the gravitaional waves is the speed of light, v = c.
  2. Gravitational radiation has two different polarized states.
    Gravitational radiation has only transversal components.
  3. Gravitational waves are emitted through quadrupole radiation.

Sources of Gravitational Waves

There are two types of sources of considerable gravitational waves, periodical and catastrophic. Examples of periodical sources are spinning rods, spinning stars, spinning of asymmetric black holes and double stars. Catastrophic events can be the explosion of a supernova or a strong gravitational collapse. Generally, the energy emitted from the catastrophic events is much greater than that from the periodic sources. However, the very fact that they are periodical makes it relatively easy to monitor them and fine tune our detectors to reduce the noise of the system. Let us start with investigating a couple of periodical sources of gravitational waves.

Spinning Rod

The energy radiated from a system is principally given by equation (gif). To show the order of magnitude of gravitational waves, we will illustrate the power output from a steel rod of 20 meters length and 1 meter radius spinning as fast as permitted without breaking. The power ouput for a rotating solid cylinder is given by:


where I is the moment of inertia about the spin-axis and tex2html_wrap_inline497 is the angular velocity of the object. For the steel rod, the result is very insignificant:


This tiny energy is then to be distributed among all the particles that it passes which gives a small effect indeed. Of course this was to be expected as we haven't seen any direct evidence of gravitational waves, which we would have observed had the effect been greater. Hence, laboratory-created gravitational waves will not be detected yet for many a year. As we cannot deduce the existence of gravitational waves through a laboratory let us turn to other (natural) sources of gravitational waves.

Spinning Star

As previously stated, it is an asymmetry around the axis of revolution that gives rise to the quadrupole gravitational radiation. Hence, it is only emitted be non-spherical starsgif with an asymmetry around the axis of rotation. The asymmetry, e gives a power output of about


The asymmetry is tex2html_wrap_inline501 , where a and b are the principal axis in the equatorial plane. The highest power output will certainly occur for stars with a low radius because this will make the star rotate even faster, by conservation of angular momentum, thus increasing tex2html_wrap_inline497 and consequently the power output. This fact makes dense objects like neutron stars and black holes a primary target for detectors. However, our scope of measurement still lies several orders of magnitude above the calculated power output.

Double-Star System

The power output for two objectsgif of masses tex2html_wrap_inline509 and tex2html_wrap_inline511 rotating about each other with an angular velocity of tex2html_wrap_inline497 and a distance between mass-centers of r is


Once again, the angular velocity tex2html_wrap_inline497 depends on the radius of rotation, thus conserving angular momentum. Hence, the closer the two objects are to each other, the higher will the power output (obviously, they will evetually crash making the formula invalid).

Even if the power output is rather large, it is greatly reduced by the distance between us and all double-star systems of significance. For example, the binary tex2html_wrap_inline405 Per has a radiated energy of about


which gives


here on earth, which is still a very small value.

About fifty percent of all stars in the universe are belonging to a multiple system. It is estimated [1] that there should be more than tex2html_wrap_inline521 double stars in our galaxies with collapsed bodies, which are more interesting due to their higher angular velocity.

Body Falling into a Black Hole

Let us now move on to some catastrophic events, which on one hand gives more energy, but on the other hand are rather short-lived. For an object falling into a collapsed body of mass M, the gravitational energy emitted is


where m is the mass of the object falling into the collapsed body. This energy is peaked around the frequency


For example, a neutron star with mass tex2html_wrap_inline527 falling into a black hole of mass tex2html_wrap_inline529 would produce an energy of roughly tex2html_wrap_inline531 Joules with a peak-frequency of about 6 kHz. If this burst would occur in the relative proximity of earth ( tex2html_wrap_inline533 pc) within a millisecond, it would be possible to observe it with current detectors.

Gravitational Collapse

Objects that collapse symmetrically give no gravitational waves. However, the star is rotating while collapsing and bursts of gravitational waves may be emitted during the process. The total energy emitted will depend on the particulars of the collapse, but its spectrum is expected to be continous from zero to the critical frequency


where tex2html_wrap_inline535 is the density of the final stage collapse and G is the constant of gravitation. This gives the total emitted energy as


This energy could be considerable and could even be observed if it would occur a major collapse within our own galaxy. From the Virgo-cluster a few such events are expected per month with an flux in the order of tex2html_wrap_inline539 J/s which is within reach of near-future detectors.


There exists two main kinds of gravitational wave detectors: Resonance detectors and interferometer detectors. The first one is able to detect signals at their single resonance modes and the last one can detect signals over a whole bandwith.

Resonance detectors

In the beginning of the 60's, Joseph Weber build the first gravitational wave detector. It was a mechanically isolated cylinder of solid aluminum weighing 1.5 tons. Some piezoelectric strain transducers at its circumference measured the vibrations induced by passing gravitational waves. The bar resonated around the frequence of 1 kHz, which is expected of a gravitational wave from a supernova. The main problem with resonant-bar antennas is their insensitivity. The sensitivity was a distorsion of about tex2html_wrap_inline541 meters. However, that is too small to detect gravitational waves from any but the nearest and most violent events. The noise-level of the detector is measured in strain sensitivity, tex2html_wrap_inline543 and dimension is tex2html_wrap_inline545 . For an ordinary resonant detector a noise of tex2html_wrap_inline547 is a typical number for the resonance modes, but better technology will hopefully decrease the noise-level to tex2html_wrap_inline549 in the future.


The detectors are often build of aluminium. To reduce the thermal noise they are cooled down to the absolute temperature and to reduce the seismic noise they are isolated from vibrations.


For an example we have taken the resonance detector, ALLEGRO, in Louisiana. Schematic description in figure gif. It is a cylindric bar cooled down to 4.2K and has a tex2html_wrap_inline551 for f=920Hz and f=896Hz. The detector is not unique, there are several others of similar character.

Interferometer detectors

In 1978, the first gravitational wave detector to be based on a laser interferometery was designed by Forward. It had legs which were two meters long. Five years later, in 1983, a prototype with 40-meter-long legs was built at CalTech. The sensitivity was roughly tex2html_wrap_inline559 . Today the senitivity is even greater, tex2html_wrap_inline541 being is a rather normal value of the sensitivity.

The first generations of large scale-interferometers is under construction; VIRGO in Italy and LIGO in USA. They are expected to start operate in 2002. Two other smaller ones are also being built; GEO300 in Germany and TAMA300 in Japan. Lowfrequency gravitational waves cannot be detected on earth because of the seismic noise. Therefore a space interferometer, LISA, is planned by ESA. One possible target with LISA is to explore the background of gravitational waves from the origin of the universe and probe deeper than we are able to do today. One reason why that is possible is that the graviton interacts very weakly and have a small cross subsection.


The construction of an interferometer detector is simply a Michelson interferometer with very long arms. It is made of optical resonant evacuated cavities with two mirrors, with test masses, at their ends. After going through very long optical path's inside the cavities, two beams of laser light, produced by the same source, are again combined out of phase so that no light gets into the detector. The variation of the optical path length, caused by the changed distance between the mirrors, affected by a gravitational wave, produces a partial phase shift of the beams and, thus, an alteration in the observed luminous intensity which is proportional to the amplitude of the wave.


The lenth of the arms of LIGO are 4 km and for VIRGO 3 km and for the two smaller ones 300 m. Interferometer detectors are wide-band detectors. They can measure from a few Hz to several kHz. On the Earth the band is lowered by the seismic noise. For VIRGO this gives a limit of 2 Hz and for LIGO 40 Hz. Up to several hundred Hz the thermal noise dominates and after that the laser shot noise will overshadow this effect. The minimum sensitivity for LIGO is about tex2html_wrap_inline563 for 200 Hz and the corresponding for VIRGO is tex2html_wrap_inline549 Hz. All these detectors will be in a network to increase the reliability and the sensitvity. To be able to detect a cosmic gravitational wave width, a singel detector with tex2html_wrap_inline567 is needed. Obviously, no single detector can detect a gravitational wave. The second or maybe the third generation of detectors would be able to detect gravitational wave, due to reduced thermal noise thanks to better technology such as better materials in the test masses and increase the laserpower. To reduce the noise, another way is to have a space interferometer, like LISA, which will have a sensitivity of tex2html_wrap_inline569 .

LISA (Laser Interferometer Space Antenna)

LISA will primarly detect gravitional waves from galactic and extragalactic binary systems and generating gravitational waves from massive black holes in the centre of galaxies.

The three flying space craft will act like a giant Michelson-interferometer, figure gif, measuring the distortion of space caused by passing gravitional waves.


Each spacecraft will contain two freely floating test masses. The test masses will define optical paths which are 5 million kilometers long, with a 60 degree angle between them. The precession in optical path will be 20 picometers. The spacecraft are planned to be launched in 2008.

The main problem for LISA is to make sure that the distance between the test masses is changed only by gravitional waves, and not by photons and other particles. Therefore the test masses are carefully isolated from any distortion from space. Each of the instument can act like a beamsplitter or a detector, and the other two will act like the test masses in figure gif. Since thay are so far away from each other, they can't just reflect the beam. Instead the light will be transmitted to a new beam.

Figure: Gravitaional waves from objects: Coalescence of black holes (left), black holes orbiting massive black holes (centre), galactic binaries (right).

LISA will observe gravitational waves from massive black holes; bursts which come from the terminal stages of binary coalescences and continues waves which come from binaries.


Pantaleo, M. and de Finis, F., Relativity, Quanta, and Cosmology Johnson Reprint Corporation, 1979.
Popular science information source about LISA:
Comparison of different detectors:
Good source for the future:

About this document ...

Gravitational Waves

This document was generated using the LaTeX2HTML translator Version 96.1 (Feb 5, 1996) Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.

The command line arguments were:
latex2html -split 0 -show_section_numbers GW.tex.

The translation was initiated by Erik Elfgren on Tue Mar 28 09:28:44 MET DST 2000

This is called a linearized theory of gravity in vacuum. The universe is supposed to be a flat, empty space-time, at least from a local viewpoint. This approximation only holds for domains with weak gravitational forces.

Star means a star in any state of evolution, from a young star to a neutron star and a black hole.

Objects in this context means a star in any state of evolution, from a young star to a neutron star and a black hole. However, the formula would also be valid for any type of massive objects.


Erik Elfgren
Tue Mar 28 09:28:44 MET DST 2000