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# Gravitational wave

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Simulation showing gravitational waves produced during the final moments before the collision of two black holes. In the video, the waves could be seen to propagate outwards as the black holes spin past each other.

Gravitational waves are ripples in the curvature of spacetime that propagate as waves at the speed of light, generated in certain gravitational interactions that propagate outward from their source. The possibility of gravitational waves was discussed in 1893 by Oliver Heaviside using the analogy between the inverse-square law in gravitation and electricity.[1] In 1905 Henri Poincaré first proposed gravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light as being required by the Lorentz transformations.[2] Predicted in 1916[3][4] by Albert Einstein on the basis of his theory of general relativity,[5][6] gravitational waves transport energy as gravitational radiation, a form of radiant energy similar to electromagnetic radiation.[7] Gravitational waves cannot exist in the Newton’s law of universal gravitation, since it is predicated on the assumption that physical interactions propagate at infinite speed.

Gravitational-wave astronomy is an emerging branch of observational astronomy which aims to use gravitational waves to collect observational data about objects such as neutron stars and black holes, events such as supernovae, and processes including those of the early universe shortly after the Big Bang.

Various gravitational-wave observatories (detectors) are under construction or in operation, such as Advanced LIGO which began observations in September 2015.[8]

Potential sources of detectable gravitational waves include binary star systems composed of white dwarfs, neutron stars, and black holes. On February 11, 2016, the LIGO Scientific Collaboration and Virgo Collaboration teams announced that they had made the first observation of gravitational waves, originating from a pair of merging black holes using the Advanced LIGO detectors.[9][10][11] On June 15, 2016, a second detection of gravitational waves from coalescing black holes was announced.[12][13][14]

## Introduction

In Einstein’s theory of general relativity, gravity is treated as a phenomenon resulting from the curvature of spacetime. This curvature is caused by the presence of mass. Generally, the more mass that is contained within a given volume of space, the greater the curvature of spacetime will be at the boundary of its volume.[15] As objects with mass move around in spacetime, the curvature changes to reflect the changed locations of those objects. In certain circumstances, accelerating objects generate changes in this curvature, which propagate outwards at the speed of light in a wave-like manner. These propagating phenomena are known as gravitational waves.

As a gravitational wave passes an observer, that observer will find spacetime distorted by the effects of strain. Distances between 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 proportional to the inverse distance from the source.[citation needed] Inspiraling 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 have demonstrated the existence of these waves with ever more sensitive detectors. The most sensitive detector accomplished the task possessing a sensitivity measurement of about one part in 5×1022 (as of 2012) provided by the LIGO and VIRGO observatories.[16] A space based observatory, the Laser Interferometer Space Antenna, is currently under development by ESA.

Linearly polarised gravitational wave

Gravitational waves can penetrate regions of space that electromagnetic waves cannot. They are able to allow the observation of the merger of black holes and possibly other exotic objects in the distant Universe. Such systems cannot be observed with more traditional means such as optical telescopes or radio telescopes, and so gravitational-wave astronomy gives new insights into the working of the Universe. 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.[17] Precise measurements of gravitational waves will also allow scientists to test more thoroughly the general theory of relativity.

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 Hawking and Werner Israel list different frequency bands for gravitational waves that could plausibly be detected, ranging from 10−7 Hz up to 1011 Hz.[18]

## History

Primordial gravitational waves are hypothesized to arise from cosmic inflation, a faster-than-light expansion just after the Big Bang (2014).[19][20][21]

In 1905, Henri Poincaré first suggested that in analogy to an accelerating electrical charge producing electromagnetic waves, accelarated masses in a relativistic field theory of gravity should produce gravitational waves.[22][23] When Einstein published his theory of general relativity in 1915, he was skeptical of Poincaré notion since the theory implied there were no “gravitational dipoles”. Nonetheless, he still pursued the idea and based on various approximations came to the conclusion there must in fact be three types of gravitational wave (dubbed longitudinal-longitudinal, transverse-longitudinal, and transverse-transverse by Hermann Weyl).[23]

However, the nature of Einstein’s approximations led many (including Einstein himself) to doubt the result. In 1922, Arthur Eddington showed that two Einstein’s types of wave were artifacts of coordinate system used, and could be made to propagate at any speed by choosing appropriate coordinates, leading him to jest that they “propagate at the speed of thought”. This also cast doubt on the physicality of the third (transverse-transverse) type (which Eddington showed always propagate at the speed of light regardless of coordinate system).[23] In 1936, Einstein and Nathan Rosen submitted a paper to Physical Review in which they claimed gravitational waves could not exist in the full theory of general relativity because any such solution of the field equations would have a singularity. The journal sent manuscript to be reviewed by Howard P. Robertson, who (anonymously) report that the singularities in question were simply the harmless coordinate singularities of the employed cylindrical coordinates. Einstein, who was unfamiliar with concept of peer review, angrily withdrew the manuscript, never to publish in Physical Review again. Nonetheless, his assistant Leopold Infeld, who had been in contact with Robertson, convinced Einstein that the criticism was correct and the paper was rewritten with the opposite conclusion (and published elsewhere).[23]

In 1956, Felix Pirani remedy the confusion caused by use of various coordinate systems by rephrasing the gravitational waves in terms of the manifestly observable Riemann curvature tensor. At the time this work was mostly ignored because the community was focussed on a different question: whether gravitational waves could transmit energy. This matter was settled by a thought experiment proposed by Richard Feynman during the first “GR” conference at Chapel Hill in 1957. In short his argument (now know as the “sticky bead argument“) notes that if one takes a stick with beads then the effect of a passing gravitational wave would be to move the beads along the stick, friction would then produce some heat implying that the passing wave had done work. Shortly after Hermann Bondi (a former gravitational wave skeptic) published a detailed version of the “sticky bead argument”.[23]

After the Chapel Hill conference, Joseph Weber started designing and building the first gravitational wave detectors now known as Weber bars. In 1969, Weber claimed to have detected the first gravitational waves and by 1970 he was “detecting” signal regularly. However,the frequency of detection soon raised doubts on their validity as the implied rate of energy loss of the Milky Way would drain our galaxy of energy on a timescale much shorter than its inferred age. This doubts were strengthened, when by the mid 1970s repeat experiments from other groups building their own Weber bars across the globe failed to find any signals, and by the late 1970s general consensus was that Weber’s results were spurious.[23]

In the same period, the first indirect evidence for the existence of gravitation waves. In 1974, Russell Alan Hulse and Joseph Hooton Taylor, Jr. had discover the first binary pulsar (a discovery that earned them the 1993 Nobel Prize in Physics). In 1979, results were published detailing measurement of the gradual decay of the orbital period of the Hulse-Taylor pulsar, which fitted precisely with the loss of energy and angular momentum in gravitational radiation predicted by general relativity.[23]

## Effects of passing

The effect of a polarized gravitational wave on a ring of particles.

Gravitational waves are constantly passing Earth; however, even the strongest have a minuscule effect and their sources are generally at a great distance. For example, the waves given off by the cataclysmic final merger of GW150914 reached Earth after travelling over a billion lightyears, as a ripple in spacetime that changed the length of a 4-km LIGO arm by a ten thousandth of the width of a proton, proportionally equivalent to changing the distance to the nearest star outside the Solar System by one hair’s width.[38] This tiny effect from even extreme gravitational waves makes them undetectable on Earth by any means other than the most sophisticated detectors.

The effects of a passing gravitational wave, in an extremely exaggerated form, can be visualized by imagining a perfectly flat region of spacetime with a group of motionless test particles lying in a plane (e.g., the surface of a computer screen). As a gravitational wave passes through the particles along a line perpendicular to the plane of the particles (i.e. following the observer’s 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.[citation needed]

The oscillations depicted 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 help illustrate 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 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.[39] If the orbit of the masses is elliptical then the gravitational wave’s amplitude also varies with time according to Einstein’s quadrupole formula.[4]

As with other waves, there are a number of characteristics used to describe 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 sextillion times weaker than this — h ≈ 10−20.
• Frequency: Usually denoted f, this is the frequency with which the wave oscillates (1 divided by the amount of time between two successive maximum stretches or squeezes)
• Wavelength: Usually denoted λ, 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 wave speed 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 above example, it is 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, in a “cross”-polarized gravitational wave, h×, 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.

## Sources

The gravitational wave spectrum with sources and detectors. Credit: NASA Goddard Space Flight Center[40]

In general terms, gravitational waves are radiated by objects whose motion involves acceleration and its change, provided that the motion is not perfectly spherically symmetric (like an expanding or contracting sphere) or rotationally symmetric (like a spinning disk or sphere). A simple example of this principle is a spinning dumbbell. If the dumbbell spins around its axis of symmetry, it will not radiate gravitational waves; if it tumbles end over end, as in the case of 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. In an extreme case, such as when 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, as a planet would orbit the Sun, will radiate.
• A spinning non-axisymmetric planetoid — say with a large bump or dimple on the equator — will radiate.
• A supernova will radiate except in the unlikely event that the explosion is perfectly symmetric.
• An isolated non-spinning solid object moving at a constant velocity will not radiate. This can be regarded as a consequence of the principle of conservation of linear momentum.
• A spinning disk will not radiate. This can be regarded as a consequence of the principle of conservation of angular momentum. However, it will show gravitomagnetic effects.
• A spherically pulsating spherical star (non-zero monopole moment or mass, but zero quadrupole moment) will not radiate, in agreement with Birkhoff’s theorem.

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 non-zero in order for it to emit gravitational radiation. This is analogous to the changing dipole moment of charge or current that is necessary for the emission of electromagnetic radiation.

### Binaries

Two stars of dissimilar mass are in circular orbits. Each revolves about their common center of mass (denoted by the small red cross) in a circle with the larger mass having the smaller orbit.

Two stars of similar mass are in circular orbits about their center of mass

Two stars of similar mass are in highly elliptical orbits about their center of mass

Gravitational waves carry energy away from their sources and, in the case of orbiting bodies, this is associated with an inspiral or decrease in orbit.[41][42] Imagine for example a simple system of two masses — such as the Earth–Sun system — moving slowly compared to the speed of light in circular orbits. Assume that these two masses orbit each other in a circular orbit in the xy plane. To a good approximation, the masses follow simple Keplerian orbits. However, such an orbit represents a changing quadrupole moment. That is, the system will give off gravitational waves.

In theory, the loss of energy through gravitational radiation could eventually drop the Earth into the Sun. However, the total energy of the Earth orbiting the Sun (kinetic energy + gravitational potential energy) is about 1.14×1036 joules of which only 200 Watts (joules per second) is lost through gravitational radiation, leading to a decay in the orbit by about 1×10−15 meters per day or roughly the diameter of a proton. At this rate, it would take the Earth approximately 1×1013 times more than the current age of the Universe to spiral onto the Sun. This estimate overlooks the decrease in r over time, but the majority of the time the bodies are far apart and only radiating slowly, so the difference is unimportant in this example.

More generally, the rate of orbital decay can be approximated by[43]

d r d t = − 64 5 G 3 c 5 ( m 1 m 2 ) ( m 1 + m 2 ) r 3   , {\displaystyle {\frac {\mathrm {d} r}{\mathrm {d} t}}=-{\frac {64}{5}}\,{\frac {G^{3}}{c^{5}}}\,{\frac {(m_{1}m_{2})(m_{1}+m_{2})}{r^{3}}}\ ,}

where r is the separation between the bodies, t time, G Newton’s constant, c the speed of light, and m1 and m2 the masses of the bodies. This leads to an expected time to merger of [43]

t = 5 256 c 5 G 3 r 4 ( m 1 m 2 ) ( m 1 + m 2 ) . {\displaystyle t={\frac {5}{256}}\,{\frac {c^{5}}{G^{3}}}\,{\frac {r^{4}}{(m_{1}m_{2})(m_{1}+m_{2})}}.}

For example, a pair of solar mass neutron stars in a circular orbit at a separation of 1.89×108 m (189,000 km) has an orbital period of 1,000 seconds, and an expected lifetime of 1.30×1013 seconds or about 414,000 years. Such a system could be observed by LISA if it 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 in the order of the Sun, and diameters in the order of the Earth. They cannot get much closer together than 10,000 km before they will merge and explode in a supernova which would also end the emission of gravitational waves. Until then, their gravitational radiation would be comparable to that of a neutron star binary.

When the orbit of a neutron star binary has decayed to 1.89×106 m (1890 km), its remaining lifetime is about 130,000 seconds or 36 hours. The orbital frequency will vary from 1 orbit per second at the start, to 918 orbits per second when the orbit has shrunk to 20 km at merger. The majority of gravitational radiation emitted will be at twice the orbital frequency. Just before merger, the inspiral would be observed by LIGO if such a binary were 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. Advanced LIGO detector should be able to detect these events up to 200 megaparsec away. Within this range of the order 40 events are expected per year.[44]

### Black holes

Black hole binaries emit gravitational waves during their in-spiral, merger, and ring-down phases. The largest amplitude of emission occurs during the merger phase, which can be modeled with the techniques of numerical relativity.[31][32][33] The first direct detection of gravitational waves, GW150914, came from the merger of two black holes.

### Supernova

Main article: supernova

A supernova is an astronomical event that occurs during the last stellar evolutionary stages of a massive star’s life, whose dramatic and catastrophic destruction is marked by one final titanic explosion. This explosion can happen in one of many ways, but in all of them a significant proportion of the matter in the star is blown away into the surrounding space at extremely high velocities (up to 10% of the speed of light). Unless there is perfect spherical symmetry in these explosions (i.e., unless matter is spewed out evenly in all directions), there will be gravitational radiation from the explosion. This is because gravitational waves are generated by a changing quadrupole moment, which can happen only when there is asymmetrical movement of masses. Since the exact mechanism by which supernovae take place is not fully understood, it is not easy to model the gravitational radiation emitted by them.

### Rotating neutron stars

As noted above, a mass distribution will emit gravitational radiation only when there is spherically asymmetric motion among the masses. A spinning neutron star will generally emit no gravitational radiation because neutron stars are highly dense objects with a strong gravitational field that keeps them almost perfectly spherical. In some cases, however, there might be slight deformities on the surface called “mountains”, which are bumps extending no more than 10 centimeters (4 inches) above the surface,[45] that make the spinning spherically asymmetric. This gives the star a quadrupole moment that changes with time, and it will emit gravitational waves until the deformities are smoothed out.

### Inflation

Main article: inflation (cosmology)

Many models of the Universe postulate that there was an inflationary epoch in the early history of the Universe when space expanded by a large factor in a very short amount of time. If this expansion was not symmetric in all directions, it may have emitted gravitational radiation detectable today as a gravitational wave background. This background signal is too weak for any currently operational gravitational wave detector to observe, and it is thought it may be decades before such an observation can be made.

## Properties and behaviour

### Energy, momentum, and angular momentum

Water waves, sound waves, and electromagnetic waves are able to carry energy, momentum, and angular momentum and by doing so they carry those away from the source. 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.[46] 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.[47] A kicked black hole can also carry a star cluster with it, forming a hyper-compact stellar system.[48] 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 thought to contain a recoiling supermassive black hole.[49]

### Redshifting and blueshifting

Like electromagnetic waves, gravitational waves should exhibit shifting of wavelength due to the relative velocities of the source and observer, but also due to distortions of space-time, such as cosmic expansion.[citation needed] This is the case even though gravity itself is a cause of distortions of space-time.[citation needed] Redshifting of gravitational waves is different from redshifting due to gravity.

### Quantum gravity, wave-particle aspects, and graviton

At present, and unlike all other known forces in the universe, no “force carrying” particle has been identified as mediating gravitational interactions.

In the framework of quantum field theory, the graviton is the name given to a hypothetical elementary particle speculated to be the force carrier that mediates gravity. However the graviton is not yet proven to exist and no reconciliation yet exists between general relativity which describes gravity, and the Standard Model which describes all other fundamental forces. (For scientific models which attempt to reconcile these, see quantum gravity).

If such a particle exists, it is expected to be massless (because the gravitational force appears to have unlimited range) and must be a spin-2 boson. It can be shown that any massless spin-2 field would give rise to a force indistinguishable from gravitation, because a massless spin-2 field must couple to (interact with) the stress–energy tensor in the same way that the gravitational field does; therefore if a massless spin-2 particle were ever discovered, it would be likely to be the graviton without further distinction from other massless spin-2 particles.[50] Such a discovery would unite quantum theory with gravity.[51]

### Absorption, re-emission, refraction (lensing), superposition, and other wave effects

Due to the weakness of the coupling of gravity to matter, gravitational waves experience very little absorption or scattering, even as they travel over astronomical distances. In particular, gravitational waves are expected to be unaffected by the opacity of the very early universe before space became “transparent”; observations based upon light, radio waves, and other electromagnetic radiation further back into time is limited or unavailable. Therefore, gravitational waves are expected to have the potential to open a new means of observation to the very early universe.

### Determining direction of travel

The difficulty in directly detecting gravitational waves, means it is also difficult for a single detector to identify by itself the direction of a source. Therefore, multiple detectors are used, both to distinguish signals from other “noise” by confirming the signal is not of earthly origin, and also to determine direction by means of triangulation. This technique uses the fact that the waves travel at the speed of light and will reach different detectors at different times depending on their source direction. Although the differences in arrival time may be just a few milliseconds, this is sufficient to identify the direction of the origin of the wave with considerable precision.

In the case of GW150914, only two detectors were operating at the time of the event, therefore, the direction is not so precisely defined and it could lie anywhere within an arc-shaped region of space rather than being identified as a single point.

## Astrophysics implications

Two-dimensional representation of gravitational waves generated by two neutron stars orbiting each other.

During the past century, astronomy has been revolutionized by the use of new methods for observing the universe. Astronomical observations were originally made using visible light. Galileo Galilei pioneered the use of telescopes to enhance these observations. However, visible light is only a small portion of the electromagnetic spectrum, and not all objects in the distant universe shine strongly in this particular band. More useful information may be found, for example, in radio wavelengths. Using radio telescopes, astronomers have found pulsars, quasars, and other extreme objects that push the limits of our understanding of physics. Observations in the microwave band have opened our eyes to the faint imprints of the Big Bang, a discovery Stephen Hawking called the “greatest discovery of the century, if not all time”. Similar advances in observations using gamma rays, x-rays, ultraviolet light, and infrared light have also brought new insights to astronomy. As each of these regions of the spectrum has opened, new discoveries have been made that could not have been made otherwise. Astronomers hope that the same holds true of gravitational waves.[52]

Gravitational waves have two important and unique properties. First, there is no need for any type of matter to be present nearby in order for the waves to be generated by a binary system of uncharged black holes, which would emit no electromagnetic radiation. Second, gravitational waves can pass through any intervening matter without being scattered significantly. Whereas light from distant stars may be blocked out by interstellar dust, for example, gravitational waves will pass through essentially unimpeded. These two features allow gravitational waves to carry information about astronomical phenomena never before observed by humans.

The sources of gravitational waves described above are in the low-frequency end of the gravitational-wave spectrum (10−7 to 105 Hz). An astrophysical source at the high-frequency end of the gravitational-wave spectrum (above 105 Hz and probably 1010 Hz) generates[clarification needed] relic gravitational waves that are theorized to be faint imprints of the Big Bang like the cosmic microwave background (see gravitational wave background).[53] At these high frequencies it is potentially possible that the sources may be “man made”[18] that is, gravitational waves generated and detected in the laboratory.[54][55]

## Detection

Now disproved evidence allegedly showing gravitational waves in the infant universe was found by the BICEP2 radio telescope. The microscopic examination of the focal plane of the BICEP2 detector is shown here.[19][20] In 2015, however, the BICEP2 findings were confirmed to be the result of cosmic dust.[56]

### Indirect detection

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.[57] The characteristics of their orbit can be deduced from the Doppler shifting of radio signals given off by the pulsar. Each of the stars is about 1.4 M and the size of their orbits is about 1/75 of the Earth–Sun orbit, 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 how much energy (and angular momentum) would be radiated 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 change in the orbit period, as predicted from the assumed gravitational radiation and general relativity, and the observations 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. The lifetime of this binary system, from the present to merger is estimated to be a few hundred million years.[58]

Inspirals are very important sources of gravitational waves. Any time two compact objects (white dwarfs, neutron stars, or black holes) are in close orbits, they send out intense gravitational waves. As they spiral closer to each other, these waves become more intense. At some point they should become so intense that direct detection by their effect on objects on Earth or in space is possible. This direct detection is the goal of several large scale experiments.[59]

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 at Earth would be roughly h ≈ 10−26. There are some sources, however, that astrophysicists expect to find that produce much greater amplitudes of h ≈ 10−20. At least eight other binary pulsars have been discovered.[60]

### Difficulties

Gravitational waves are not easily detectable. When they reach the Earth, they have a small amplitude with strain approximates 10−21, meaning that an extremely sensitive detector is needed, and that other sources of noise can overwhelm the signal.[61] Gravitational waves are expected to have frequencies 10−16 Hz < f < 104 Hz.[62]

### Ground-based detectors

A schematic diagram of a laser interferometer

Though the Hulse–Taylor observations were very important, they give only indirect evidence for gravitational waves. A more conclusive observation would be a direct measurement of the effect of a passing gravitational wave, which could also provide more information about the system that generated it. Any such direct detection is complicated by the extraordinarily small effect the waves would produce on a detector. The amplitude of a spherical wave will fall off as the inverse of the distance from the source (the 1/R term in the formulas for h above). Thus, even waves from extreme systems like merging binary black holes die out to very small amplitudes by the time they reach the Earth. Astrophysicists expect that some gravitational waves passing the Earth may be as large as h ≈ 10−20, but generally no bigger.[63]

#### Resonant antennae

Main article: Weber bar

A simple device theorised to detect the expected wave motion is called a Weber bar — a large, solid bar of metal isolated from outside vibrations. This type of instrument was the first type of gravitational wave detector. Strains in space due to an incident gravitational wave excite the bar’s resonant frequency and could thus be amplified to detectable levels. Conceivably, a nearby supernova might be strong enough to be seen without resonant amplification. With this instrument, Joseph Weber claimed to have detected daily signals of gravitational waves. His results, however, were contested in 1974 by physicists Richard Garwin and David Douglass. Modern forms of the Weber bar are still operated, cryogenically cooled, with superconducting quantum interference devices to detect vibration. Weber bars are not sensitive enough to detect anything but extremely powerful gravitational waves.[64]

MiniGRAIL is a spherical gravitational wave antenna using this principle. It is based at Leiden University, consisting of an exactingly machined 1150 kg sphere cryogenically cooled to 20 mK.[65] The spherical configuration allows for equal sensitivity in all directions, and is somewhat experimentally simpler than larger linear devices requiring high vacuum. Events are detected by measuring deformation of the detector sphere. MiniGRAIL is highly sensitive in the 2–4 kHz range, suitable for detecting gravitational waves from rotating neutron star instabilities or small black hole mergers.[66]

There are currently two detectors focused on the higher end of the gravitational wave spectrum (10−7 to 105 Hz): one at University of Birmingham, England,[67] 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. Both detectors are expected to be sensitive to periodic spacetime strains of h ∼ 2 × 10 − 13 / H z {\displaystyle 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 ∼ 2 × 10 − 17 / H z {\displaystyle h\sim {2\times 10^{-17}/{\sqrt {\mathrm {Hz} }}}} , with an expectation to reach a sensitivity of h ∼ 2 × 10 − 20 / H z {\displaystyle 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 ~1011 Hz (100 GHz) and h ~10−30 to 10−32.[68]

#### Interferometers

Simplified operation of a gravitational wave observatory

Figure 1: A beamsplitter (green line) splits coherent light (from the white box) into two beams which reflect off the mirrors (cyan oblongs); only one outgoing and reflected beam in each arm is shown, and separated for clarity. The reflected beams recombine and an interference pattern is detected (purple circle).
Figure 2: A gravitational wave passing over the left arm (yellow) changes its length and thus the interference pattern.

A more sensitive class of detector uses laser interferometry to measure gravitational-wave induced motion between separated ‘free’ masses.[69] 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, one at the Hanford site in Richland, Washington and a third (formerly installed as a second detector at Hanford) that is planned to be moved to India. Each observatory has two light storage arms that are 4 kilometers in length. These are at 90 degree angles to each other, with the light passing through 1 m 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 ∼ 5 × 10 − 22 {\displaystyle h\sim 5\times 10^{-22}} . 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, KAGRA, is under construction in the Kamiokande mine in Japan. 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 might be seen from one per tens of years of observation, to tens per year.[70]

Interferometric detectors are limited at high frequencies by shot noise, which occurs because the lasers produce photons randomly; one analogy is to rainfall—the rate of rainfall, like the laser intensity, is measurable, but the raindrops, like photons, fall at random times, causing fluctuations around the average value. This leads to noise at the output of the detector, much like radio static. In addition, for sufficiently high laser power, the random momentum transferred to the test masses by the laser photons shakes the mirrors, masking signals of low frequencies. Thermal noise (e.g., Brownian motion) is another limit to sensitivity. In addition to these ‘stationary’ (constant) noise sources, all ground-based detectors are also limited at low frequencies by seismic noise and other forms of environmental vibration, and other ‘non-stationary’ noise sources; creaks in mechanical structures, lightning or other large electrical disturbances, etc. may also create noise masking an event or may even imitate an event. All these must be taken into account and excluded by analysis before detection may be considered a true gravitational wave event.

#### Einstein@Home

Main article: Einstein@Home

The simplest gravitational waves are those with constant frequency. The waves given off by a spinning, non-axisymmetric neutron star would be approximately monochromatic: a pure tone in acoustics. Unlike signals from supernovae of binary black holes, these signals evolve little in amplitude or frequency over the period it would be observed by ground-based detectors. However, there would be some change in the measured signal, because of Doppler shifting caused by the motion of the Earth. Despite the signals being simple, detection is extremely computationally expensive, because of the long stretches of data that must be analysed.

The Einstein@Home project is a distributed computing project similar to SETI@home intended to detect this type of 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.[71]

### Space-based interferometers

Space-based interferometers, such as LISA and DECIGO, are also being developed. LISA’s design calls for three test masses forming an equilateral triangle, with lasers from each spacecraft to each other spacecraft forming two independent interferometers. LISA is planned to occupy a solar orbit trailing the Earth, with each arm of the triangle being five million kilometers. This puts the detector in an excellent vacuum far from Earth-based sources of noise, though it will still be susceptible to heat, shot noise, and artifacts caused by cosmic rays and solar wind.

### Using pulsar timing arrays

Pulsars are rapidly rotating stars. A pulsar emits beams of radio waves that, like lighthouse beams, sweep through the sky as the pulsar rotates. The signal from a pulsar can be detected by radio telescopes as a series of regularly spaced pulses, essentially like the ticks of a clock. Gravitational waves affect the time it takes the pulses to travel from the pulsar to a telescope on Earth. A pulsar timing array uses millisecond pulsars to seek out perturbations due to gravitational waves in measurements of pulse arrival times at a telescope, in other words, to look for deviations in the clock ticks. In particular, pulsar timing arrays can search for a distinct pattern of correlation and anti-correlation between the signals over an array of different pulsars (resulting in the name “pulsar timing array”). Although pulsar pulses travel through space for hundreds or thousands of years to reach us, pulsar timing arrays are sensitive to perturbations in their travel time of much less than a millionth of a second.

Globally there are three active pulsar timing array projects. The North American Nanohertz Gravitational Wave Observatory uses data collected by the Arecibo Radio Telescope and Green Bank Telescope. The Parkes Pulsar Timing Array at the Parkes radio-telescope has been collecting data since March 2005. The European Pulsar Timing Array uses data from the four largest telescopes in Europe: the Lovell Telescope, the Westerbork Synthesis Radio Telescope, the Effelsberg Telescope and the Nancay Radio Telescope. (Upon completion the Sardinia Radio Telescope will be added to the EPTA also.) These three projects have begun collaborating under the title of the International Pulsar Timing Array project.

### Primordial

Primordial gravitational waves are gravitational waves observed in the cosmic microwave background. They were allegedly detected by the BICEP2 instrument, an announcement made on 17 March 2014, which was withdrawn on 30 January 2015 (“the signal can be entirely attributed to dust in the Milky Way”[56]).

### LIGO observations

LIGO measurement of the gravitational waves at the Hanford (left) and Livingston (right) detectors, compared to the theoretical predicted values.

On 11 February 2016, the LIGO collaboration announced the detection of gravitational waves, from a signal detected at 09:50:45 GMT on 14 September 2015[72] of two black holes with masses of 29 and 36 solar masses merging about 1.3 billion light years away. During the final fraction of a second of the merger, it released more than 50 times the power of all the stars in the observable universe combined.[73] The signal increased in frequency from 35 to 250 Hz over 10 cycles (5 orbits) as it rose in strength for a period of 0.2 second.[10] The mass of the new merged black hole was 62 solar masses. Energy equivalent to three solar masses was emitted as gravitational waves.[74] The signal was seen by both LIGO detectors in Livingston and Hanford, with a time difference of 7 milliseconds due to the angle between the two detectors and the source. The signal came from the Southern Celestial Hemisphere, in the rough direction of (but much further away than) the Magellanic Clouds.[9] The confidence level of this being an observation of gravitational waves was 99.99994%.[74]

On 15 June 2016, the LIGO group announced the detection of a second set of gravitational waves, which was observed at 03:38:53 GMT on 26 December 2015. The signal was seen by the Hanford LIGO detector 1.1 milliseconds after the Livingston detector. The signal rose from 35 to 450 Hz over the course of 55 cycles (27 orbits) during the period of observation of about a second. Analysis of the signal indicates that this event represented the merger of two black holes about 1.4 billion light years distant, with masses of about 14.2 and 7.5 solar masses, yielding a combined black hole of approximately of 20.8 solar masses, with one solar mass radiated away. The estimated spin parameter (ratio of angular momentum to theoretical limit) of the final black hole is 0.74, slightly higher than for the first detection (0.67); it was also found that at least one of the premerger black holes had a spin of greater than 0.2. This measurement provided additional support for general relativity.[12][13]

## Mathematics

### Framework

Gravitational waves are presently understood to be described by Albert Einstein‘s theory of general relativity. In the simplest cases, and certain less-dynamic situations, the energy implications of gravitational waves can be deduced from other conservation laws such as those governing conservation of energy or conservation of momentum.

Beyond these simple cases, Einstein’s equations show how the curvature of spacetime can be expressed mathematically using the metric tensor — denoted g μ ν {\displaystyle 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.

### Basic mathematics

#### Power radiated by orbiting bodies

Suppose that the two masses are m 1 {\displaystyle m_{1}} and m 2 {\displaystyle m_{2}} , and they are separated by a distance r {\displaystyle r} . The power given off (radiated) by this system is:

P = d E d t = − 32 5 G 4 c 5 ( m 1 m 2 ) 2 ( m 1 + m 2 ) r 5 {\displaystyle P={\frac {\mathrm {d} E}{\mathrm {d} t}}=-{\frac {32}{5}}\,{\frac {G^{4}}{c^{5}}}\,{\frac {(m_{1}m_{2})^{2}(m_{1}+m_{2})}{r^{5}}}} ,[43]

where G is the gravitational constant, c is the speed of light in vacuum and where the negative sign means that power is leaving the system, rather than entering. For a system like the Sun and Earth, r {\displaystyle r} is about 1.5×1011 m and m 1 {\displaystyle m_{1}} and m 2 {\displaystyle m_{2}} are about 2×1030 and 6×1024 kg respectively. In this case, the power leaving the Earth, Sun system is about 200 watts. This is truly tiny compared to the total electromagnetic radiation given off by the Sun (roughly 3.86×1026 watts, or almost 400 million, million, million, million watts).

#### 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 θ 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 + = − 1 R G 2 c 4 2 m 1 m 2 r ( 1 + cos 2 ⁡ θ ) cos ⁡ [ 2 ω ( t − R / c ) ] , {\displaystyle h_{+}=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {2m_{1}m_{2}}{r}}(1+\cos ^{2}\theta )\cos \left[2\omega (t-R/c)\right],}
h × = − 1 R G 2 c 4 4 m 1 m 2 r ( cos ⁡ θ ) sin ⁡ [ 2 ω ( t − R / c ) ] . {\displaystyle h_{\times }=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {4m_{1}m_{2}}{r}}\,(\cos {\theta })\sin \left[2\omega (t-R/c)\right].}

Here, we use the constant angular velocity of a circular orbit in Newtonian physics:

ω = G ( m 1 + m 2 ) / r 3 . {\displaystyle \omega ={\sqrt {G(m_{1}+m_{2})/r^{3}}}.}

For example, if the observer is in the xy plane then θ = π / 2 {\displaystyle \theta =\pi /2} , and cos ⁡ ( θ ) = 0 {\displaystyle \cos(\theta )=0} , so the h × {\displaystyle 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 + = − 1 R G 2 c 4 4 m 1 m 2 r = − 1 R 1.7 × 10 − 10 m . {\displaystyle 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 ≈ 1/4π light-year, so typical amplitudes will be h ≈ 10−25. That is, a ring of particles would stretch or squeeze by just one part in 1025. This is well under the detectability limit of all conceivable detectors.

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

G μ ν = 8 π G N c 4 T μ ν , {\displaystyle G_{\mu \nu }={\frac {8\pi G_{N}}{c^{4}}}T_{\mu \nu },}

where G N {\displaystyle G_{N}} is Newton’s gravitational constant, and c {\displaystyle c} is the speed of light. We assume geometrized units, so G N = 1 = c {\displaystyle 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 , θ , ϕ ) {\displaystyle (t,r,\theta ,\phi )} . We define the flat-space metric η μ ν {\displaystyle \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

η μ ν = [ − 1 0 0 0 0 1 0 0 0 0 r 2 0 0 0 0 r 2 sin 2 ⁡ θ ] . {\displaystyle \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}}.}

This flat-space metric has no physical significance; it is a purely mathematical device necessary for the analysis. Tensor indices are raised and lowered using this “flat-space metric”.

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

h ¯ α β ≡ η α β − g α β | det g | . {\displaystyle {\bar {h}}^{\alpha \beta }\equiv \eta ^{\alpha \beta }-g^{\alpha \beta }{\sqrt {|\det g|}}.}

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 condition (conditions on the coordinates):

∇ β h ¯ α β = 0 , {\displaystyle \nabla _{\beta }\,{\bar {h}}^{\alpha \beta }=0,}

where ∇ {\displaystyle \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[75] as

◻ h ¯ α β = − 16 π τ α β {\displaystyle \Box {\bar {h}}^{\alpha \beta }=-16\pi \tau ^{\alpha \beta }\,} ,

where ◻ = − ∂ t 2 + Δ {\displaystyle \Box =-\partial _{t}^{2}+\Delta \,} represents the flat-space d’Alembertian operator, and τ α β {\displaystyle \tau ^{\alpha \beta }\,} represents the stress–energy tensor plus quadratic terms involving h ¯ α β {\displaystyle {\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 η α β {\displaystyle \eta ^{\alpha \beta }\,} tensor. In this case, we can neglect terms quadratic in h ¯ α β {\displaystyle {\bar {h}}^{\alpha \beta }\,} , which means that the τ α β {\displaystyle \tau ^{\alpha \beta }\,} field reduces to the usual stress–energy tensor T α β {\displaystyle T^{\alpha \beta }\,} . That is, Einstein’s equations become

◻ h ¯ α β = − 16 π T α β {\displaystyle \Box {\bar {h}}^{\alpha \beta }=-16\pi T^{\alpha \beta }\,} .

If we are interested in the field far from a source, however, we can treat the source as a point source; everywhere else, the stress–energy tensor would be zero, so

◻ h ¯ α β = 0 {\displaystyle \Box {\bar {h}}^{\alpha \beta }=0\,} .

Now, this is the usual homogeneous wave equation — one for each component of h ¯ α β {\displaystyle {\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 {\displaystyle 1/r\,} far from the source) can always be written in the form A ( t − r , θ , ϕ ) / r {\displaystyle A(t-r,\theta ,\phi )/r\,} , where A {\displaystyle A\,} is just some function. It can be shown[76] 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 {\displaystyle r=0} , the general solution to the wave equation in spherical coordinates is

h ¯ α β = 1 r [ 0 0 0 0 0 0 0 0 0 0 A + ( t − r , θ , ϕ ) A × ( t − r , θ , ϕ ) 0 0 A × ( t − r , θ , ϕ ) − A + ( t − r , θ , ϕ ) ] ≡ [ 0 0 0 0 0 0 0 0 0 0 h + ( t − r , r , θ , ϕ ) h × ( t − r , r , θ , ϕ ) 0 0 h × ( t − r , r , θ , ϕ ) − h + ( t − r , r , θ , ϕ ) ] , {\displaystyle {\begin{aligned}{\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{aligned}}}

where we now see the origin of the two polarizations.

#### Relation to the source

If we know the details of a source — for instance, the parameters of the orbit of a binary — we can relate the source’s motion to the gravitational radiation observed far away. With the relation

◻ h ¯ α β = − 16 π τ α β {\displaystyle \Box {\bar {h}}^{\alpha \beta }=-16\pi \tau ^{\alpha \beta }\,} ,

we can write the solution in terms of the tensorial Green’s function for the d’Alembertian operator:[75]

h ¯ α β ( t , x → ) = − 16 π ∫ G γ δ α β ( t , x → ; t ′ , x → ′ ) τ γ δ ( t ′ , x → ′ ) d t ′ d 3 x ′ {\displaystyle {\bar {h}}^{\alpha \beta }(t,{\vec {x}})=-16\pi \int \,G_{\gamma \delta }^{\alpha \beta }(t,{\vec {x}};t’,{\vec {x}}’)\,\tau ^{\gamma \delta }(t’,{\vec {x}}’)\,\mathrm {d} t’\,\mathrm {d} ^{3}x’} .

Though it is possible to expand the Green’s function in tensor spherical harmonics, it is easier to simply use the form

G γ δ α β ( t , x → ; t ′ , x → ′ ) = 1 4 π δ γ α δ δ β δ ( t ± | x → − x → ′ | − t ′ ) | x → − x → ′ | , {\displaystyle G_{\gamma \delta }^{\alpha \beta }(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}}’|}},}

where the positive and negative signs correspond to ingoing and outgoing solutions, respectively. Generally, we are interested in the outgoing solutions, so

h ¯ α β ( t , x → ) = − 4 ∫ τ α β ( t − | x → − x → ′ | , x → ′ ) | x → − x → ′ | d 3 x ′ {\displaystyle {\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} ^{3}x’} .

If the source is confined to a small region very far away, to an excellent approximation we have:

h ¯ α β ( t , x → ) ≈ − 4 r ∫ τ α β ( t − r , x → ′ ) d 3 x ′ {\displaystyle {\bar {h}}^{\alpha \beta }(t,{\vec {x}})\approx -{\frac {4}{r}}\,\int \,\tau ^{\alpha \beta }(t-r,{\vec {x}}’)\,\mathrm {d} ^{3}x’} ,

where r = | x → | {\displaystyle r=|{\vec {x}}|} .

Now, because we will eventually only be interested in the spatial components of this equation (time components can be set to zero with a coordinate transformation), and we are integrating this quantity — presumably over a region of which there is no boundary — we can put this in a different form. Ignoring divergences with the help of Stokes’ theorem and an empty boundary, we can see that

∫ τ i j ( t − r , x → ′ ) d 3 x ′ = ∫ x ′ i x ′ j ∇ k ∇ l τ k l ( t − r , x → ′ ) d 3 x ′ . {\displaystyle \int \,\tau ^{ij}(t-r,{\vec {x}}’)\,\mathrm {d} ^{3}x’=\int \,x’^{i}x’^{j}\nabla _{k}\nabla _{l}\tau ^{kl}(t-r,{\vec {x}}’)\,\mathrm {d} ^{3}x’.}

Inserting this into the above equation, we arrive at

h ¯ i j ( t , x → ) ≈ − 4 r ∫ x ′ i x ′ j ∇ k ∇ l τ k l ( t − r , x → ′ ) d 3 x ′ . {\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,\int \,x’^{i}x’^{j}\nabla _{k}\nabla _{l}\tau ^{kl}(t-r,{\vec {x}}’)\,\mathrm {d} ^{3}x’.}

Finally, because we have chosen to work in coordinates for which ∇ β h ¯ α β = 0 {\displaystyle \nabla _{\beta }\,{\bar {h}}^{\alpha \beta }=0} , we know that ∇ β τ α β = 0 {\displaystyle \nabla _{\beta }\,\tau ^{\alpha \beta }=0} . With a few simple manipulations, we can use this to prove that

∇ 0 ∇ 0 τ 00 = ∇ j ∇ k τ j k {\displaystyle \nabla _{0}\nabla _{0}\tau ^{00}=\nabla _{j}\nabla _{k}\tau ^{jk}} .

With this relation, the expression for the radiated field is

h ¯ i j ( t , x → ) ≈ − 4 r d 2 d t 2 ∫ x ′ i x ′ j τ 00 ( t − r , x → ′ ) d 3 x ′ {\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}\,\int \,x’^{i}x’^{j}\tau ^{00}(t-r,{\vec {x}}’)\,\mathrm {d} ^{3}x’} .

In the linear case, τ 00 = ρ {\displaystyle \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 {\displaystyle M_{1}} and M 2 {\displaystyle M_{2}} , and the positions are x → 1 {\displaystyle {\vec {x}}_{1}} and x → 2 {\displaystyle {\vec {x}}_{2}} , then

ρ ( t − r , x → ′ ) = M 1 δ 3 ( x → ′ − x → 1 ( t − r ) ) + M 2 δ 3 ( x → ′ − x → 2 ( t − r ) ) {\displaystyle \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:

h ¯ i j ( t , x → ) ≈ − 4 r d 2 d t 2 { M 1 x 1 i ( t − r ) x 1 j ( t − r ) + M 2 x 2 i ( t − r ) x 2 j ( t − r ) } {\displaystyle {\bar {h}}^{ij}(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\}} .

Using mass-centered coordinates, and assuming a circular binary, this is

h ¯ i j ( t , x → ) ≈ − 4 r M 1 M 2 R n i ( t − r ) n j ( t − r ) {\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {M_{1}M_{2}}{R}}\,n^{i}(t-r)n^{j}(t-r)} ,

where n → = x → 1 / | x → 1 | {\displaystyle {\vec {n}}={\vec {x}}_{1}/|{\vec {x}}_{1}|} . Plugging in the known values of x → 1 ( t − r ) {\displaystyle {\vec {x}}_{1}(t-r)} , we obtain the expressions given above for the radiation from a simple binary.

## In fiction

An episode of the Russian science-fiction novel Space Apprentice by Arkady and Boris Strugatsky shows the experiment monitoring the propagation of gravitational waves at the expense of annihilating a chunk of asteroid 15 Eunomia the size of Everest.[77]

## References

1. ME Gerstenstein; VI Pustovoit (1962). “On the Detection of Low-Frequency Gravitational Waves”. ZhETF (in Russian). 16 (8): 605–607. Bibcode:1963JETP…16..433G.

## Bibliography

My name is Patrick Ireland, living in the Philippines with my wife and two daughters. I have been studying the web for over a decade. Now that I am 60 years old, I am starting to apply some of the knowledge that I have gained. "Learn from yesterday, live for today, hope for tomorrow. The important thing is to never stop questioning." -Einstein.

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