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Gravitational wave

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Title: Gravitational wave  
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Gravitational wave

In physics, gravitational waves are ripples in the curvature of spacetime that propagate as a wave, travelling outward from the source. Predicted in 1916[1][2] by Albert Einstein to exist on the basis of his theory of general relativity,[3][4] gravitational waves theoretically transport energy as gravitational radiation. Sources of detectable gravitational waves could possibly include binary star systems composed of white dwarfs, neutron stars, or black holes. The existence of gravitational waves is a possible consequence of the Lorentz invariance of general relativity since it brings the concept of a limiting speed of propagation of the physical interactions with it. Gravitational waves cannot exist in the Newtonian theory of gravitation, in which physical interactions propagate at infinite speed.

Although gravitational radiation has not been directly detected, there is indirect evidence for its existence. For example, the 1993 Nobel Prize in Physics was awarded for measurements of the Hulse–Taylor binary system that suggests gravitational waves are more than mathematical anomalies. Various gravitational wave detectors exist and on 17 March 2014, astronomers at the Harvard–Smithsonian Center for Astrophysics claimed that they had detected and produced "the first direct image of gravitational waves across the primordial sky" within the cosmic microwave background, providing strong evidence for inflation and the Big Bang.[5][6][7][8] Peer review will be needed before there can be any scientific consensus about these new findings.[9][10] On 19 June 2014, lowered confidence in confirming the cosmic inflation findings was reported;[11][12][13] and on 19 September 2014, a further reduction in confidence was reported.[14][15]


  • Introduction 1
  • Effects of a passing gravitational wave 2
  • Sources of gravitational waves 3
    • Power radiated by orbiting bodies 3.1
    • Orbital decay from gravitational radiation 3.2
    • Orbital lifetime limits from gravitational radiation 3.3
    • Wave amplitudes from the Earth–Sun system 3.4
    • Radiation from other sources 3.5
  • Astrophysics and gravitational waves 4
    • Energy, momentum, and angular momentum carried by gravitational waves 4.1
  • Detecting gravitational waves 5
    • Difficulties in detection 5.1
    • Ground-based interferometers 5.2
    • Using pulsar timing arrays 5.3
    • Einstein@Home 5.4
    • Primordial gravitational waves 5.5
  • Mathematics 6
    • Linear approximation 6.1
    • Relation to the source 6.2
  • See also 7
  • References 8
  • Further reading 9
  • Bibliography 10
  • External links 11


History of the Universe - gravitational waves are hypothesized to arise from cosmic inflation, a faster-than-light expansion just after the Big Bang (17 March 2014).[5][7][8]


As a gravitational wave passes a distant observer, that observer will find spacetime distorted by the effects of strain. Distances between free objects increase and decrease rhythmically as the wave passes, at a frequency corresponding to that of the wave. This occurs despite such free objects never being subjected to an unbalanced force. The magnitude of this effect decreases inversely with distance from the source. Inspiralling binary neutron stars are predicted to be a powerful source of gravitational waves as they coalesce, due to the very large acceleration of their masses as they orbit close to one another. However, due to the astronomical distances to these sources the effects when measured on Earth are predicted to be very small, having strains of less than 1 part in 1020. Scientists are attempting to demonstrate the existence of these waves with ever more sensitive detectors. The current most sensitive measurement is about one part in 5×1022 (as of 2012) provided by the LIGO and VIRGO observatories.[16] The lack of detection in these observatories provides an upper limit on the frequency of such powerful sources.[17][18] A space based observatory, the Laser Interferometer Space Antenna, is currently under development by ESA.

Linearly polarised gravitational wave

Gravitational waves should penetrate regions of space that electromagnetic waves cannot. It is hypothesized that they will be able to provide observers on Earth with information about black holes and other exotic objects in the distant Universe. Such systems cannot be observed with more traditional means such as optical telescopes and radio telescopes. In particular, gravitational waves could be of interest to cosmologists as they offer a possible way of observing the very early universe. This is not possible with conventional astronomy, since before recombination the universe was opaque to electromagnetic radiation.[19] Precise measurements of gravitational waves will also allow scientists to test the general theory of relativity more thoroughly.

In principle, gravitational waves could exist at any frequency. However, very low frequency waves would be impossible to detect and there is no credible source for detectable waves of very high frequency. Stephen W. Hawking and Werner Israel list different frequency bands for gravitational waves that could be plausibly detected, ranging from 10−7 Hz up to 1011 Hz.[20]

Effects of a passing gravitational wave


The effects of a passing gravitational wave can be visualized by imagining a perfectly flat region of spacetime with a group of motionless test particles lying in a plane (the surface of your screen). As a gravitational wave passes through the particles along a line perpendicular to the plane of the particles (i.e. following your line of vision into the screen), the particles will follow the distortion in spacetime, oscillating in a "cruciform" manner, as shown in the animations. The area enclosed by the test particles does not change and there is no motion along the direction of propagation.

The oscillations depicted here in the animation are exaggerated for the purpose of discussion—in reality a gravitational wave has a very small amplitude (as formulated in linearized gravity). However they enable us to visualize the kind of oscillations associated with gravitational waves as produced for example by a pair of masses in a circular orbit. In this case the amplitude of the gravitational wave is a constant, but its plane of polarization changes or rotates at twice the orbital rate and so the time-varying gravitational wave size (or 'periodic spacetime strain') exhibits a variation as shown in the animation.[21] If the orbit is elliptical then the gravitational wave's amplitude also varies with time according Einstein's quadrupole formula.[22]

Like other waves, there are a few useful characteristics describing a gravitational wave:

  • Amplitude: Usually denoted h, this is the size of the wave — the fraction of stretching or squeezing in the animation. The amplitude shown here is roughly h=0.5 (or 50%). Gravitational waves passing through the Earth are many billions times weaker than this — h \approx 10^{-20}. Note that this is not the quantity that would be analogous to what is usually called the amplitude of an electromagnetic wave, which would be \frac{\mathrm{d}h}{\mathrm{d}t}.
  • Wavelength: Usually denoted \lambda, this is the distance along the wave between points of maximum stretch or squeeze.
  • Speed: This is the speed at which a point on the wave (for example, a point of maximum stretch or squeeze) travels. For gravitational waves with small amplitudes, this is equal to the speed of light, c.

The speed, wavelength, and frequency of a gravitational wave are related by the equation c = λ f, just like the equation for a light wave. For example, the animations shown here oscillate roughly once every two seconds. This would correspond to a frequency of 0.5 Hz, and a wavelength of about 600,000 km, or 47 times the diameter of the Earth.

In the example just discussed, we actually assume something special about the wave. We have assumed that the wave is linearly polarized, with a "plus" polarization, written h_{\,+}. Polarization of a gravitational wave is just like polarization of a light wave except that the polarizations of a gravitational wave are at 45 degrees, as opposed to 90 degrees. In particular, if we had a "cross"-polarized gravitational wave, h_{\,\times}, the effect on the test particles would be basically the same, but rotated by 45 degrees, as shown in the second animation. Just as with light polarization, the polarizations of gravitational waves may also be expressed in terms of circularly polarized waves. Gravitational waves are polarized because of the nature of their sources. The polarization of a wave depends on the angle from the source, as we will see in the next section.

Sources of gravitational waves

In general terms, gravitational waves are radiated by objects whose motion involves acceleration, provided that the motion is not perfectly spherically symmetric (like an expanding or contracting sphere) or cylindrically symmetric (like a spinning disk or sphere). A simple example of this principle is provided by the spinning dumbbell. If the dumbbell spins like wheels on an axle, it will not radiate gravitational waves; if it tumbles end over end like two planets orbiting each other, it will radiate gravitational waves. The heavier the dumbbell, and the faster it tumbles, the greater is the gravitational radiation it will give off. If we imagine an extreme case in which the two weights of the dumbbell are massive stars like neutron stars or black holes, orbiting each other quickly, then significant amounts of gravitational radiation would be given off.

Some more detailed examples:

  • Two objects orbiting each other in a quasi-Keplerian planar orbit (basically, as a planet would orbit the Sun) will radiate.

More technically, the third time derivative of the quadrupole moment (or the l-th time derivative of the l-th multipole moment) of an isolated system's stress–energy tensor must be nonzero in order for it to emit gravitational radiation. This is analogous to the changing dipole moment of charge or current necessary for electromagnetic radiation.

Power radiated by orbiting bodies



Suppose that the two masses are m_1 and m_2, and they are separated by a distance r. The power given off (radiated) by this system is:

P = \frac{\mathrm{d}E}{\mathrm{d}t} = - \frac{32}{5}\, \frac{G^4}{c^5}\, \frac{(m_1m_2)^2 (m_1+m_2)}{r^5} ,[23]

where G is the gravitational constant, c is the speed of light in vacuum and where the negative sign means that power is being given off by the system, rather than received. For a system like the Sun and Earth, r is about 1.5×1011 m and m_1 and m_2 are about 2×1030 and 6×1024 kg respectively. In this case, the power is about 200 watts. This is truly tiny compared to the total electromagnetic radiation given off by the Sun (roughly 3.86×1026 watts).



Orbital decay from gravitational radiation


\frac{\mathrm{d}r}{\mathrm{d}t} = - \frac{64}{5}\, \frac{G^3}{c^5}\, \frac{(m_1m_2)(m_1+m_2)}{r^3}\ ,

where the variables are the same as in the previous equation.

The orbit decays at a rate proportional to the inverse third power of the radius. When the radius has shrunk to half its initial value, it is shrinking eight times faster than before. By Kepler's Third Law, the new rotation rate at this point will be faster by \sqrt{8}=2.828, or nearly three times the previous orbital frequency. As the radius decreases, the power lost to gravitational radiation increases even more. As can be seen from the previous equation, power radiated varies as the inverse fifth power of the radius, or 32 times more in this case.

If we use the previous values for the Sun and the Earth, we find that the Earth's orbit shrinks by 1.1×10−20 meter per second. This is 3.5×10−13 m per year, which is about 1/300 the diameter of a hydrogen atom. The effect of gravitational radiation on the size of the Earth's orbit is negligible over the age of the universe. This is not true for closer orbits.

A more practical example is the orbit of a Sun-like star around a heavy black hole. Our Milky Way has a, potential, 4 million M black hole at its center in Sagittarius A. Such supermassive black holes are being found in the center of almost all galaxies. For this example take a 2 million M black hole with a solar-mass star orbiting it at a radius of 1.89×1010 m (63 light-seconds). The mass of the black hole will be 4×1036 kg and its gravitational radius will be 6×109 m. The orbital period will be 1,000 seconds, or a little under 17 minutes. The solar-mass star will draw closer to the black hole by 7.4 meters per second or 7.4 km per orbit. A collision will not be long in coming.

Assume that a pair of 1 M neutron stars are in circular orbits at a distance of 1.89×108 m (189,000 km). This is a little less than 1/7 the diameter of the Sun or 0.63 light-seconds. Their orbital period would be 1,000 seconds. Substituting the new mass and radius in the above formula gives a rate of orbit decrease of 3.7×10−6 m/s or 3.7 mm per orbit. This is 116 meters per year and is not negligible over cosmic time scales.

Suppose instead that these two neutron stars were orbiting at a distance of 1.89×106 m (1890 km). Their period would be 1 second and their orbital velocity would be about 1/50 of the speed of light. Their orbit would now shrink by 3.7 meters per orbit. A collision is imminent. A runaway loss of energy from the orbit results in an ever more rapid decrease in the distance between the stars. They will eventually merge to form a black hole and cease to radiate gravitational waves. This is referred to as the inspiral.

The above equation can not be applied directly for calculating the lifetime of the orbit, because the rate of change in radius depends on the radius itself, and is thus non-constant with time. The lifetime can be computed by integration of this equation (see next section).

Orbital lifetime limits from gravitational radiation

Orbital lifetime is one of the most important properties of gravitational radiation sources. It determines the average number of binary stars in the universe that are close enough to be detected. Short lifetime binaries are strong sources of gravitational radiation but are few in number. Long lifetime binaries are more plentiful but they are weak sources of gravitational waves. LIGO is most sensitive in the frequency band where two neutron stars are about to merge. This time frame is only a few seconds. It takes luck for the detector to see this blink in time out of a million year orbital lifetime. It is predicted that such a merger will only be seen once per decade or so.

The lifetime of an orbit is given by:[23]

t= \frac{5}{256}\, \frac{c^5}{G^3}\, \frac{r^4}{(m_1m_2)(m_1+m_2)}\ ,

where r is the initial distance between the orbiting bodies. This equation can be derived by integrating the previous equation for the rate of radius decrease. It predicts the time for the radius of the orbit to shrink to zero. As the orbital speed becomes a significant fraction of the speed of light, this equation becomes inaccurate. It is useful for inspirals until the last few milliseconds before the merger of the objects.

Substituting the values for the mass of the Sun and Earth as well as the orbital radius gives a very large lifetime of 3.44×1030 seconds or 1.09×1023 years (that is approximately 1013 times larger than the age of the universe). The actual figure would be slightly less than that. The Earth will break apart from tidal forces if it orbits closer than a few radii from the Sun. This would form a ring around the Sun and instantly stop the emission of gravitational waves.

If we use a 2 million M black hole with a solar mass star orbiting it at 1.89×1010 meters, we get a lifetime of 6.50×108 seconds or 20.7 years.

Assume that a pair of solar mass neutron stars with a diameter of 10 kilometers are in circular orbits at a distance of 1.89×108 m (189,000 km). Their lifetime is 1.30×1013 seconds or about 414,000 years. Their orbital period will be 1,000 seconds and it could be observed by LISA if they were not too far away. A far greater number of white dwarf binaries exist with orbital periods in this range. White dwarf binaries have masses on the order of our Sun and diameters on the order of our Earth. They cannot get much closer together than 10,000 km before they will merge and cease to radiate gravitational waves. This results in the creation of either a neutron star or a black hole. Until then, their gravitational radiation will be comparable to that of a neutron star binary. LISA is the only gravitational wave experiment that is likely to succeed in detecting such types of binaries.

If the orbit of a neutron star binary has decayed to 1.89×106m (1890 km), its remaining lifetime is 130,000 seconds or about 36 hours. The orbital frequency will vary from 1 revolution per second at the start and 918 revolutions per second when the orbit has shrunk to 20 km at merger. The gravitational radiation emitted will be at twice the orbital frequency. Just before merger, the inspiral can be observed by LIGO if the binary is close enough. LIGO has only a few minutes to observe this merger out of a total orbital lifetime that may have been billions of years. The chance of success with LIGO as initially constructed is quite low despite the large number of such mergers occurring in the universe, because the sensitivity of the instrument does not 'reach' out to enough systems to see events frequently. No mergers have been seen in the few years that initial LIGO has been in operation, and it is thought that a merger should be seen about once per several tens of years of observing time with initial LIGO.[24] The upgraded Advanced LIGO detector, with a ten times greater sensitivity, 'reaches' out 10 times further—encompassing a volume 1000 times greater, and seeing 1000 times as many candidate sources. Thus, the expectation is that detections will be made at the rate of tens per year.

Wave amplitudes from the Earth–Sun system

We can also think in terms of the amplitude of the wave from a system in circular orbits. Let \theta be the angle between the perpendicular to the plane of the orbit and the line of sight of the observer. Suppose that an observer is outside the system at a distance R from its center of mass. If R is much greater than a wavelength, the two polarizations of the wave will be

h_{+} = -\frac{1}{R}\, \frac{G^2}{c^4}\, \frac{2 m_1 m_2}{r} (1+\cos^2\theta) \cos\left[2\omega(t - R)\right],
h_{\times} = -\frac{1}{R}\, \frac{G^2}{c^4}\, \frac{4 m_1 m_2}{r}\, (\cos{\theta})\sin \left[2\omega(t-R)\right].



For example, if the observer is in the x-y plane then \theta=\pi/2, and \cos (\theta) = 0, so the h_\times polarization is always zero. We also see that the frequency of the wave given off is twice the rotation frequency. If we put in numbers for the Earth-Sun system, we find:

h_{+} =-\frac{1}{R}\, \frac{G^2}{c^4}\, \frac{4m_1 m_2}{r} = -\frac{1}{R}\, 1.7\times 10^{-10}\, \mathrm{m}.

In this case, the minimum distance to find waves is R ≈ 1 light-year, so typical amplitudes will be h ≈ 10−26. That is, a ring of particles would stretch or squeeze by just one part in 1026. This is well under the detectability limit of all conceivable detectors.

Radiation from other sources

Although the waves from the Earth-Sun system are minuscule, astronomers can point to other sources for which the radiation should be substantial. One important example is the Hulse-Taylor binary — a pair of stars, one of which is a pulsar.[25] The characteristics of their orbit can be deduced from the Doppler shifting of radio signals given off by the pulsar. Each of the stars are about 1.4 M and the size of their orbit is about 1/75 of the Earth-Sun orbit. This means the distance between the two stars is just a few times larger than the diameter of our own Sun. The combination of greater masses and smaller separation means that the energy given off by the Hulse-Taylor binary will be far greater than the energy given off by the Earth-Sun system — roughly 1022 times as much.

The information about the orbit can be used to predict just how much energy (and angular momentum) should be given off in the form of gravitational waves. As the energy is carried off, the stars should draw closer to each other. This effect is called an inspiral, and it can be observed in the pulsar's signals. The measurements on the Hulse-Taylor system have been carried out over more than 30 years. It has been shown that the gravitational radiation predicted by general relativity allows these observations to be matched within 0.2 percent. In 1993, Russell Hulse and Joe Taylor were awarded the Nobel Prize in Physics for this work, which was the first indirect evidence for gravitational waves. Unfortunately, the orbital lifetime of this binary system before merger is about 1.84 billion years. This is a substantial fraction of the age of the universe.


The only difficulty is that most systems like the Hulse-Taylor binary are so far away. The amplitude of waves given off by the Hulse-Taylor binary as seen on Earth would be roughly h ≈ 10−26. There are some sources, however, that astrophysicists expect to find with much larger amplitudes of h ≈ 10−20. At least eight other binary pulsars have been discovered.[27]

Astrophysics and gravitational waves

List of unsolved problems in physics
Can gravitational waves be detected experimentally?




Energy, momentum, and angular momentum carried by gravitational waves

Waves familiar from other areas of physics such as water waves, sound waves, and electromagnetic waves are able to carry energy, momentum, and angular momentum. By carrying these away from a source, waves are able to rob that source of its energy as well as its linear and angular momentum. Gravitational waves perform the same function. Thus, for example, a binary system loses angular momentum as the two orbiting objects spiral towards each other—the angular momentum is radiated away by gravitational waves.

The waves can also carry off linear momentum, a possibility that has some interesting implications for astrophysics.[31] After two supermassive black holes coalesce, emission of linear momentum can produce a "kick" with amplitude as large as 4000 km/s. This is fast enough to eject the coalesced black hole completely from its host galaxy. Even if the kick is too small to eject the black hole completely, it can remove it temporarily from the nucleus of the galaxy, after which it will oscillate about the center, eventually coming to rest.[32] A kicked black hole can also carry a star cluster with it, forming a hyper-compact stellar system.[33] Or it may carry gas, allowing the recoiling black hole to appear temporarily as a "naked quasar". The quasar SDSS J092712.65+294344.0 is believed to contain a recoiling supermassive black hole.[34]

Detecting gravitational waves

Difficulties in detection

Evidence of gravitational waves in the infant universe may have been uncovered by the BICEP2 radio telescope. The microscopic examination of the focal plane of the BICEP2 detector is shown here.[5][6][7][8][10]

Gravitational waves are not easily detectable. This knowledge gap is primarily due to the massive presence of noise in the low frequencies where antennas currently operate. Gravitational waves are expected to have frequencies 10^{-16}\, \mathrm{Hz}.[35]

Ground-based interferometers




A schematic diagram of a laser interferometer

A more sensitive class of detector uses laser interferometry to measure gravitational-wave induced motion between separated 'free' masses.[39] This allows the masses to be separated by large distances (increasing the signal size); a further advantage is that it is sensitive to a wide range of frequencies (not just those near a resonance as is the case for Weber bars). Ground-based interferometers are now operational. Currently, the most sensitive is LIGO — the Laser Interferometer Gravitational Wave Observatory. LIGO has three detectors: one in Livingston, Louisiana; the other two (in the same vacuum tubes) at the Hanford site in Richland, Washington. Each consists of two light storage arms that are 2 to 4 kilometers in length. These are at 90 degree angles to each other, with the light passing through 1m diameter vacuum tubes running the entire 4 kilometers. A passing gravitational wave will slightly stretch one arm as it shortens the other. This is precisely the motion to which an interferometer is most sensitive.

Even with such long arms, the strongest gravitational waves will only change the distance between the ends of the arms by at most roughly 10−18 meters. LIGO should be able to detect gravitational waves as small as h \sim 5\times 10^{-20}. Upgrades to LIGO and other detectors such as Virgo, GEO 600, and TAMA 300 should increase the sensitivity still further; the next generation of instruments (Advanced LIGO and Advanced Virgo) will be more than ten times more sensitive. Another highly sensitive interferometer (LCGT) is currently in the design phase. A key point is that a tenfold increase in sensitivity (radius of 'reach') increases the volume of space accessible to the instrument by one thousand times. This increases the rate at which detectable signals should be seen from one per tens of years of observation, to tens per year.[24]



There are currently two detectors focusing on detection at the higher end of the gravitational wave spectrum (10−7 to 105 Hz): one at University of Birmingham, England, and the other at INFN Genoa, Italy. A third is under development at Chongqing University, China. The Birmingham detector measures changes in the polarization state of a microwave beam circulating in a closed loop about one meter across. Two have been fabricated and they are currently expected to be sensitive to periodic spacetime strains of h\sim{2 \times 10^{-13}/\sqrt{\mathrm{Hz}}} , given as an amplitude spectral density. The INFN Genoa detector is a resonant antenna consisting of two coupled spherical superconducting harmonic oscillators a few centimeters in diameter. The oscillators are designed to have (when uncoupled) almost equal resonant frequencies. The system is currently expected to have a sensitivity to periodic spacetime strains of h\sim{2 \times 10^{-17}/\sqrt{\mathrm{Hz}}} , with an expectation to reach a sensitivity of h\sim{2 \times 10^{-20}/\sqrt{\mathrm{Hz}}} . The Chongqing University detector is planned to detect relic high-frequency gravitational waves with the predicted typical parameters ?g ~1010 Hz (10 GHz) and h ~10−30-10−31.

Using pulsar timing arrays



In some sense, the easiest signals to detect should be constant sources. Supernovae and neutron star or black hole mergers should have larger amplitudes and be more interesting, but the waves generated will be more complicated. The waves given off by a spinning, aspherical neutron star would be "monochromatic"—like a pure tone in acoustics. It would not change very much in amplitude or frequency.

The Einstein@Home project is a distributed computing project similar to SETI@home intended to detect this type of simple gravitational wave. By taking data from LIGO and GEO, and sending it out in little pieces to thousands of volunteers for parallel analysis on their home computers, Einstein@Home can sift through the data far more quickly than would be possible otherwise.[40]

Primordial gravitational waves

Primordial gravitational waves are gravitational waves observed in the cosmic microwave background. They were allegedly detected by the BICEP2 instrument, an announcement being made on 17 March 2014.


Einstein's equations form the fundamental law of general relativity. The curvature of spacetime can be expressed mathematically using the metric tensor — denoted g_{\mu \nu} \,. The metric holds information regarding how distances are measured in the space under consideration. Because the propagation of gravitational waves through space and time change distances, we will need to use this to find the solution to the wave equation.

Spacetime curvature is also expressed with respect to a covariant derivative, \nabla \,, in the form of the Einstein tensor, G_{\mu \nu}. This curvature is related to the stress–energy tensor, T_{\mu\nu}, by the key equation


where G_N is Newton's gravitational constant, and c is the speed of light. We assume geometrized units, so G_N = 1 = c.

With some simple assumptions, Einstein's equations can be rewritten to show explicitly that they are wave equations. To begin with, we adopt some coordinate system, like (t,r,\theta,\phi) \,. We define the "flat-space metric" \eta_{\mu\nu} \, to be the quantity that — in this coordinate system — has the components we would expect for the flat space metric. For example, in these spherical coordinates, we have

\eta_{\mu \nu} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2\theta \end{bmatrix}.


Now, we can also think of the physical metric g_{\mu \nu} as a matrix, and find its determinant, \det g. Finally, we define a quantity

\bar{h}^{\alpha \beta} \equiv \eta^{\alpha \beta} - \sqrt{|\det g|} g^{\alpha \beta} \, .

This is the crucial field, which will represent the radiation. It is possible (at least in an asymptotically flat spacetime) to choose the coordinates in such a way that this quantity satisfies the "de Donder" gauge conditions (conditions on the coordinates):

\nabla_\beta\, \bar{h}^{\alpha \beta} = 0,

where \nabla represents the flat-space derivative operator. These equations say that the divergence of the field is zero. The linear Einstein equations can now be written[41] as

\Box \bar{h}^{\alpha \beta} = -16\pi \tau^{\alpha \beta} \, ,

where \Box = -\partial_t^2 + \Delta \, represents the flat-space d'Alembertian operator, and \tau^{\alpha \beta} \, represents the stress–energy tensor plus quadratic terms involving \bar{h}^{\alpha \beta} \,. This is just a wave equation for the field with a source, despite the fact that the source involves terms quadratic in the field itself. That is, it can be shown that solutions to this equation are waves traveling with velocity 1 in these coordinates.

Linear approximation

The equations above are valid everywhere — near a black hole, for instance. However, because of the complicated source term, the solution is generally too difficult to find analytically. We can often assume that space is nearly flat, so the metric is nearly equal to the \eta^{\alpha \beta} \, tensor. In this case, we can neglect terms quadratic in \bar{h}^{\alpha \beta} \,, which means that the \tau^{\alpha \beta} \, field reduces to the usual stress–energy tensor T^{\alpha \beta} \,. That is, Einstein's equations become

\Box \bar{h}^{\alpha \beta} = -16\pi T^{\alpha \beta} \, .


\Box \bar{h}^{\alpha \beta} = 0 \, .

Now, this is the usual homogeneous wave equation — one for each component of \bar{h}^{\alpha \beta} \,. Solutions to this equation are well known. For a wave moving away from a point source, the radiated part (meaning the part that dies off as 1/r \, far from the source) can always be written in the form A(t-r,\theta,\phi)/r \,, where A \, is just some function. It can be shown[42] that — to a linear approximation — it is always possible to make the field traceless. Now, if we further assume that the source is positioned at r=0, the general solution to the wave equation in spherical coordinates is

\begin{array}{lcl} \bar{h}^{\alpha \beta} & = & \frac{1}{r}\, \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & A_{+}(t-r,\theta,\phi) & A_{\times}(t-r,\theta,\phi) \\ 0 & 0 & A_{\times}(t-r,\theta,\phi) & -A_{+}(t-r,\theta,\phi) \end{bmatrix} \\ \\ & \equiv & \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & h_{+}(t-r,r,\theta,\phi) & h_{\times}(t-r,r,\theta,\phi) \\ 0 & 0 & h_{\times}(t-r,r,\theta,\phi) & -h_{+}(t-r,r,\theta,\phi) \end{bmatrix} \end{array} \,

where we now see the origin of the two polarizations.

Relation to the source



we can write the solution in terms of the tensorial Green's function for the d'Alembertian operator:[41]

\bar{h}^{\alpha \beta} (t,\vec{x}) = -16\pi \int\, G^{\alpha \beta}_{\gamma \delta} (t,\vec{x};t',\vec{x}')\, \tau^{\gamma \delta}(t',\vec{x}')\, \mathrm{d}t'\, \mathrm{d}^3x' .


G^{\alpha \beta}_{\gamma \delta} (t,\vec{x};t',\vec{x}') = \frac{1}{4\pi} \delta_{\gamma}^\alpha\, \delta_{\delta}^\beta\, \frac{\delta(t\pm|\vec{x}-\vec{x}'|-t')} {|\vec{x}-\vec{x}'|} ,


\bar{h}^{\alpha \beta} (t,\vec{x}) = -4 \int\, \frac{\tau^{\alpha \beta}(t-|\vec{x}-\vec{x}'|,\vec{x}')}{|\vec{x}-\vec{x}'|}\, \mathrm{d}^3x' .


\bar{h}^{\alpha \beta} (t,\vec{x}) \approx -\frac{4}{r}\, \int\, \tau^{\alpha \beta}(t-r,\vec{x}')\, \mathrm{d}^3x' ,

where r=|\vec{x}| .


\int\, \tau^{i j}(t-r,\vec{x}')\, \mathrm{d}^3x' = \int\, x'^i x'^j \nabla_k \nabla_l \tau^{k l} (t-r,\vec{x}')\, \mathrm{d}^3x' ,

Inserting this into the above equation, we arrive at

\bar{h}^{i j} (t,\vec{x}) \approx -\frac{4}{r}\, \int\, x'^i x'^j \nabla_k \nabla_l \tau^{k l} (t-r,\vec{x}')\, \mathrm{d}^3x' ,

Finally, because we have chosen to work in coordinates for which \nabla_\beta\, \bar{h}^{\alpha \beta} = 0, we know that \nabla_\beta\, \tau^{\alpha \beta} = 0. With a few simple manipulations, we can use this to prove that

\nabla_0 \nabla_0 \tau^{00} = \nabla_j \nabla_k \tau^{jk} .

With this relation, the expression for the radiated field is

\bar{h}^{i j} (t,\vec{x}) \approx -\frac{4}{r}\, \frac{\mathrm{d}^2}{\mathrm{d}t^2}\, \int\, x'^i x'^j \tau^{0 0} (t-r,\vec{x}')\, \mathrm{d}^3x' .

In the linear case, \tau^{00} = \rho, the density of mass-energy.

To a very good approximation, the density of a simple binary can be described by a pair of delta-functions, which eliminates the integral. Explicitly, if the masses of the two objects are M_1 and M_2, and the positions are \vec{x}_1 and \vec{x}_2, then

\rho(t-r,\vec{x}') = M_1 \delta^3(\vec{x}'-\vec{x}_1(t-r)) + M_2 \delta^3(\vec{x}'-\vec{x}_2(t-r)) .

We can use this expression to do the integral above:

\bar{h}^{i j} (t,\vec{x}) \approx -\frac{4}{r}\, \frac{\mathrm{d}^2}{\mathrm{d}t^2}\, \left\{ M_1 x_1^i(t-r) x_1^j(t-r) + M_2 x_2^i(t-r) x_2^j(t-r) \right\} .


\bar{h}^{i j} (t,\vec{x}) \approx -\frac{4}{r}\, \frac{M_1 M_2}{R}\, n^i(t-r) n^j(t-r) ,

where \vec{n} = \vec{x}_1 / |\vec{x}_1|. Plugging in the known values of \vec{x}_1(t-r), we obtain the expressions given above for the radiation from a simple binary.

See also


  2. ^ Einstein, A (1918). "Über Gravitationswellen". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 154-167. 
  4. ^ The Detection of Gravitational Waves using LIGO, B. Barish
  6. ^ a b ""First Direct Evidence of Cosmic Inflation"". Harvard-Smithsonian Center for Astrophysics. 17 March 2014. Retrieved 17 March 2014. 
  8. ^ a b c  
  9. ^ Cosmic inflation: 'Spectacular' discovery hailedAstronomers discover echoes from expansion after Big BangGravitational Waves: The Big Bang's Smoking GunGravitational Waves from Big Bang Detected
  10. ^ a b  
  11. ^  
  15. ^  
  17. ^ LIGO Scientific Collaboration; Virgo Collaboration (2012). "All-sky search for gravitational-wave bursts in the second joint LIGO-Virgo run". Physical Review D 85: 122007.  
  18. ^ LIGO Scientific Collaboration; Virgo Collaboration (2013). "Search for gravitational waves from binary black hole inspiral, merger, and ringdown in LIGO-Virgo data from 2009-2010". Physical Review D 87: 022002.  
  19. ^ Krauss, LM; Dodelson, S; Meyer, S (2010). "Primordial Gravitational Waves and Cosmology". Science 328 (5981): 989–992.  
  20. ^ a b Hawking, S. W. and Israel, W., General Relativity: An Einstein Centenary Survey, Cambridge University Press, Cambridge, 1979, 98.
  21. ^ Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields. Fourth Revised English Edition, Pergamon Press., 1975, 356–357.
  22. ^ Einstein, A., "Über Gravitationswellen." Sitzungsberichte, Preussische Akademie der Wissenschaften, 154, (1918).
  23. ^ a b c Gravitational Radiation
  24. ^ a b LIGO Scientific Collaboration; Virgo Collaboration (2010). "Predictions for the rates of compact binary coalescences observable by ground-based gravitational-wave detectors". Classical and Quantum Gravity 27: 17300.  
  25. ^ Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis
  26. ^ Crashing Black Holes
  27. ^ Binary and Millisecond Pulsars
  30. ^ Li, Fangyu, Baker, R. M L, Jr., and Woods, R. C., "Piezoelectric-Crystal-Resonator High-Frequency Gravitational Wave Generation and Synchro-Resonance Detection", in the proceedings of Space Technology and Applications International Forum (STAIF-2006), edited by M.S. El-Genk, American Institute of Physics Conference Proceedings, Melville NY 813: 2006.
  31. ^  
  32. ^ Gualandris, A.;  
  33. ^  
  34. ^ Komossa, S.; Zhou, H.; Lu, H. (May 2008). "A Recoiling Supermassive Black Hole in the Quasar SDSS J092712.65+294344.0?". The Astrophysical Journal 678 (2): L81–L84.  
  35. ^ Thorne, Kip S. (1995). "Gravitational Waves". Cornell University Library. 
  36. ^ For a review of early experiments using Weber bars, see Levine, J. (April 2004). "Early Gravity-Wave Detection Experiments, 1960–1975". Physics in Perspective (Birkhäuser Basel) 6 (1): 42–75.  
  37. ^ Gravitational Radiation Antenna In Leiden
  38. ^ de Waard, Arlette; Luciano Gottardi; Giorgio Frossati (July 2000). "Spherical Gravitational Wave Detectors: cooling and quality factor of a small CuAl6% sphere" (PDF). Marcel Grossmann meeting on General Relativity. Rome, Italy: World Scientific Publishing Co. Pte. Ltd. (published December 2002). pp. 1899–1901.  
  40. ^ Einstein@Home
  41. ^ a b Thorne, Kip (April 1980). "Multipole expansions of gravitational radiation". Reviews of Modern Physics 52 (2): 299.  
  42. ^ C. W. Misner, K. S. Thorne, and J. A. Wheeler (1973). Gravitation. W. H. Freeman and Co. 

Further reading

  • Chakrabarty, Indrajit, "Gravitational Waves: An Introduction". arXiv:physics/9908041 v1, Aug 21, 1999.
  • Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields (Pergamon Press),(1987).
  • Will, Clifford M., The Confrontation between General Relativity and Experiment. Living Rev. Relativity 9 (2006) 3.
  • Peter Saulson, "Fundamentals of Interferometric Gravitational Wave Detectors", World Scientific, 1994.
  • J. Bicak, W.N. Rudienko, "Gravitacionnyje wolny w OTO i probliema ich obnarużenija", Izdatielstwo Moskovskovo Universitieta, 1987.
  • A. Kułak, "Electromagnetic Detectors of Gravitational Radiation", PhD Thesis, Cracow 1980 (In Polish).
  • P. Tatrocki, "On intuitive description of graviton detector", .
  • P. Tatrocki, "Can the LIGO, VIRGO, GEO600, AIGO, TAMA, LISA detectors really detect?", .


  • Collins, Harry, Gravity's Shadow: the search for gravitational waves, University of Chicago Press, 2004.

External links

  • Gravitational waves at Encyclopædia Britannica
  • Gravitational Waves on In Our Time at the BBC. (listen now)
  • The LISA Brownbag – Selection of the most significant e-prints related to LISA science
  • To know everything about astroparticle physics, including gravitational waves
  • Caltech's Physics 237-2002 Gravitational Waves by Kip Thorne Video plus notes: Graduate level but does not assume knowledge of General Relativity, Tensor Analysis, or Differential Geometry; Part 1: Theory (10 lectures), Part 2: Detection (9 lectures)
  • January 14, 2008 Episode 71: Gravitational Waves
  • Laser Interferometer Gravitational Wave Observatory. LIGO Laboratory, operated by the California Institute of Technology and the Massachusetts Institute of Technology
  • The LIGO Scientific Collaboration
  • Einstein's Messengers – The LIGO Movie by NSF
  • Home page for Einstein@Home project, a distributed computing project processing raw data from LIGO Laboratory, searching for gravitational waves
  • The National Center for Supercomputing Applications – a numerical relativity group
  • Caltech Relativity Tutorial – A basic introduction to gravitational waves, and astrophysical systems giving off gravitational waves
  • Resource Letter GrW-1: Gravitational waves – a list of books, journals and web resources compiled by Joan Centrella for research into gravitational waves
  • Mathematical and Physical Perspectives on Gravitational Radiation – written by B F Schutz of the Max Planck Institute explaining the significance and background of some key concepts in gravitational radiation
  • Binary BH Merger – estimating the radiated power and merger time of a BH binary using dimensional analysis
