General relativity  
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In physics, a gravitational wave is a fluctuation in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted by Einstein's theory of general relativity, the waves transport energy known as gravitational radiation. Sources of gravitational waves include binary star systems composed of white dwarfs, neutron stars, or black holes.
Although gravitational radiation has not yet been directly detected, it has been indirectly shown to exist. This was the basis for the 1993 Nobel Prize in Physics, awarded for measurements of the HulseTaylor binary system. Various gravitational wave detectors exist.
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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 massive objects. Roughly speaking, the more massive the object is, the greater the curvature it produces and hence the more intense the gravity. As massive objects move around in spacetime, the curvature changes to reflect the changed locations of those objects. In certain circumstances, objects that are accelerated generate a disturbance in spacetime which spreads, as the metaphor goes, "like ripples on the surface of a pond", although perhaps a better analogy would be electromagnetic waves. This disturbance is known as gravitational radiation. Gravitational radiation is thought to travel through the Universe at the speed of light, diminishing in strength but never stopping or slowing down.
As waves of gravitational radiation pass a distant observer, that observer will find spacetime distorted by the effects of strain. Distances between free objects will increase and decrease rhythmically as the wave passes. The magnitude of this effect will decrease the further the observer is from the source. Binary neutron stars are predicted to be a strong source of such waves owing to the acceleration of their enormous masses as they orbit each other and yet even those waves are expected to be very weak by the time they reach the Earth, resulting in strains of less than 1 part in 10^{20}. Scientists are attempting to prove the existence of these waves with ever more sensitive detectors; the current best upper limit thus found (as of September 2009), provided by the LIGO detector, is a wave amplitude of 2.3 x 10^{−26}.^{[1]}
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 mysterious objects in the distant Universe. Such systems cannot be observed with more traditional means such as optical telescopes and radio telescopes. 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 10^{11} Hz.^{[2]}
Imagine a perfectly flat region of spacetime with a group of motionless test particles lying in a plane (the surface of your screen). Then a weak gravitational wave arrives, passing 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 oscillate 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 of course immensely 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 timevarying gravitational wave size (or 'periodic spacetime strain') exhibits a variation as shown in the animation.^{[3]} If the orbit is elliptical then the gravitational wave’s amplitude also varies with time according to an equation called the “quadrupole”.^{[4]}
Like other waves, there are a few useful characteristics describing a gravitational wave:
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 . 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, , 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 actually depends on the angle from the source, as we will see in the next section.
In general terms, gravitational waves are radiated by objects whose motion involves acceleration, provided that the motion is not perfectly spherically symmetric (like a spinning, expanding or contracting sphere) or cylindrically symmetric (like a spinning disk). A simple example of this principle is provided by the spinning dumbbell. The dumbbell can spin like two planets rotating around a shared pole or it can spin by tumbling end over end, like two planets revolving around each other. It will not radiate if it spins in the former manner but it will radiate if it tumbles endoverend. 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:
More technically, the third time derivative of the quadrupole moment (or the lth time derivative of the lth multipole moment) of an isolated system's stressenergy 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.
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. Imagine for example a simple system of two masses — such as the EarthSun system — moving slowly compared to the speed of light. 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.
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:
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 × 10^{11} m and m_{1} and m_{2} are about 2 × 10^{30} and 6 × 10^{24} 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 × 10^{26} Watts).
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 plus gravitational potential energy) is about 1.14 × 10^{36} joules of which only 200 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 × 10^{13} times more than the current age of the Universe to spiral onto the Sun and the Earth is predicted to be swallowed by the Sun in the red giant stage of its life in a few billion years time.
A more dramatic example of radiated gravitational energy is represented by two solar mass neutron stars orbiting each other at a distance of 189 × 10^{6} m (0.63 lightseconds). Plugging their masses into the above equation shows that the gravitational radiation from them would be 1.38 × 10^{28} Watts, which is about 100 times more than the Sun's electromagnetic radiation.
Gravitational radiation robs the orbiting bodies of energy. As the energy of the orbit reduces, the distance between the bodies decreases, and they rotate more rapidly. The rate of decrease of distance between the bodies versus time is given by:
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 , 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 Sunlike star around a huge black hole. Our Milky Way has a 4 million solar mass 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 solar mass black hole with a solar mass star orbiting it at a radius of 1.89 × 10^{10} m (63 lightseconds). The mass of the black hole will be 4 × 10^{36} kg and its gravitational radius will be 6 × 10^{9} 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 solar mass neutron stars with a diameter of 10 kilometers are in circular orbits at a distance of 1.89 × 10^{8}m (189,000 km). This is a little less than 1/7 the diameter of the Sun or 0.63 lightseconds. 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 × 10^{6} m {1890 kilometers}. 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 gravity waves. This is referred to as the inspiral.
The above equation is not too useful for calculating the lifetime of the orbit because the rate of change in radius is so nonlinear with time. Another equation can be derived from this one for that purpose.^{[citation needed]}
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 gravity 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: ^{[5]}, 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 lifetime of 3.44 × 10^{30} seconds or 1.09 × 10^{23} years. 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 gravity waves.
If we use a 2 million solar mass black hole with a solar mass star orbiting it at 1.89 × 10^{10} meters, we get a lifetime of 6.50 × 10^{8} seconds or 20.7 years. The solar mass star may be torn apart from tidal forces as it approaches the event horizon of the black hole.
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 × 10^{8} (189,000 km). Their lifetime is 1.30 × 10^{13} 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 which is likely to succeed in detecting such types of binaries.
If the orbit of a neutron star binary has decayed to 1.89 × 10^{6}m (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. Its chances of success are quite low despite the large number of such mergers occurring in the universe. No mergers have been seen in the few years that LIGO has been in operation. It is thought that a merger should be seen about once per decade of observing time.
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
Here, we use the constant angular velocity of a circular orbit in Newtonian physics. For example, if the observer is in the xy plane then θ = π / 2, and cosθ = 0, so the 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 EarthSun system, we find : .
In this case, the minimum distance to find waves is light year, so typical amplitudes will be . That is, a ring of particles would stretch or squeeze by just one part in 10^{−26}. This is well under the detectability limit of all conceivable detectors.
Although the waves from the EarthSun system are minuscule, astronomers can point to other sources for which the radiation should be substantial. One important example is the HulseTaylor binary — a pair of stars, one of which is a pulsar.^{[6]} The characteristics of their orbit can be deduced from the Doppler shifting of radio signals given off by the pulsar. Each of the stars has a mass about 1.4 times that of the Sun. Also, their orbit is about one seventyfifth the distance between the Earth and Sun — which means the distance between the two stars is just a few times larger than the diameter of our own Sun. This combination of greater masses and smaller separation means that the energy given off by the HulseTaylor binary will be far greater than the energy given off by the EarthSun system — roughly 10^{22} 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 HulseTaylor 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.
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.^{[7]}
The only difficulty is that most systems like the HulseTaylor binary are so far away. The amplitude of waves given off by the HulseTaylor 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.^{[8]}
Is our universe filled with gravitational radiation from the big bang? From astrophysical sources, such as inspiraling neutron stars? What can this tell us about quantum gravity and general relativity? 
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 which 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, xrays, 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.
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 lowfrequency end of the gravitationalwave spectrum (10^{−7} to 10^{5} Hz). An astrophysical source at the highfrequency end of the gravitationalwave spectrum (above 10^{5} Hz and probably 10^{10} Hz) generates relic gravitational waves that are theorized to be faint imprints of the Big Bang like the cosmic microwave background.^{[9]} At these high frequencies it is potentially possible that the sources may be “man made”^{[2]} that is, gravitational waves generated and detected in the laboratory.^{[10]}^{[11]}
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, linear or 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.^{[12]} Consider for instance a cluster of stars with a binary black hole system in the center. The holes orbit each other, but their center of mass doesn't move with respect to the cluster at first. However, as the binary inspirals, the radiated gravitational waves carry away linear momentum in some direction. In keeping with Newton's third law of motion, the binary will gain some linear momentum in the opposite direction. Thus, it may be shot out of the cluster.
Though the HulseTaylor 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 which 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 amplitude by the time they reach the Earth. Astrophysicists expect that some gravitational waves passing the Earth may be as large as , but generally no bigger.^{[citation needed]}
A simple device 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.^{[13]}
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.^{[14]} 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.^{[15]}
A more sensitive detector uses laser interferometry to measure gravitationalwave induced motion between separated 'free' masses^{[16]}. 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). Groundbased 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 which 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 . 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.
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 at low frequencies. Thermal noise—e.g., Brownian motion  is another limit to sensitivity. In addition to these 'stationary' (constant) noise sources, all groundbased detectors are also limited at low frequencies by seismic noise and other forms of environmental vibration, and other 'nonstationary' 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 a detection may be considered a true gravitational wave event.
Spacebased 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 Earthbased sources of noise, though it will still be susceptible to shot noise, as well as artifacts caused by cosmic rays and solar wind.
There are currently two detectors focusing on detection at the higher end of the gravitational wave spectrum (10^{−7} to 10^{5} 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 , 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 , with an expectation to reach a sensitivity of . The Chongqing University detector is planned to detect relic highfrequency gravitational waves with the predicted typical parameters ?_{g} ~ 10^{10} Hz (10 GHz) and h ~ 10^{−30}10^{−31}.
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, bumpy 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.^{[17]}
Einstein's equations form the fundamental law of general relativity. The curvature of spacetime can be expressed mathematically using the metric tensor — denoted . 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, , in the form of the Einstein tensor — G_{μν}. This curvature is related to the stressenergy tensor — T_{μν} — by the key equation
where is Newton's gravitational constant, and c is the speed of light. We assume geometrized units, so .
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 . We define the "flatspace metric" to be the quantity which — in this coordinate system — has the components we would expect for the flat space metric. For example, in these spherical coordinates, we have
This flatspace metric has no physical significance; it is a purely mathematical device necessary for the analysis. Tensor indices are raised and lowered using this "flatspace metric".
Now, we can also think of the physical metric as a matrix, and find its determinant, . Finally, we define a quantity
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):
where represents the flatspace derivative operator. These equations say that the divergence of the field is zero. The full, nonlinear Einstein equations can now be written^{[18]} as
where represents the flatspace d'Alembertian operator, and represents the stressenergy tensor plus quadratic terms involving . 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.
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 tensor. In this case, we can neglect terms quadratic in , which means that the field reduces to the usual stressenergy tensor . That is, Einstein's equations become
If we are interested in the field far from a source, however, we can treat the source as a point source; everywhere else, the stressenergy tensor would be zero, so
Now, this is the usual homogeneous wave equation — one for each component of . 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 far from the source) can always be written in the form , where is just some function. It can be shown^{[19]} 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
where we now see the origin of the two polarizations.
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
we can write the solution in terms of the tensorial Green's function for the d'Alembertian operator:^{[18]}
Though it is possible to expand the Green's function in tensor spherical harmonics, it is easier to simply use the form
where the positive and negative signs correspond to ingoing and outgoing solutions, respectively. Generally, we are interested in the outgoing solutions, so
If the source is confined to a small region very far away, to an excellent approximation we have:
where .
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
Inserting this into the above equation, we arrive at
Finally, because we have chosen to work in coordinates for which , we know that . With a few simple manipulations, we can use this to prove that
With this relation, the expression for the radiated field is
In the linear case, τ^{00} = ρ, the density of massenergy.
To a very good approximation, the density of a simple binary can be described by a pair of deltafunctions, which eliminates the integral. Explicitly, if the masses of the two objects are M_{1} and M_{2}, and the positions are and , then
We can use this expression to do the integral above:
Using masscentered coordinates, and assuming a circular binary, this is
where . Plugging in the known values of , we obtain the expressions given above for the radiation from a simple binary.

