Gravitational wave

Not to be confused with Gravity wave.

In physics, a gravitational wave is a fluctuation in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves transport energy as gravitational radiation. Sources of gravitational waves include binary star systems composed of white dwarfs, neutron stars, or black holes.

Introduction

In relativistic quantum theory, a system cannot be localized to a precision better than its Compton wavelength, expressed as λ = hc/E = c/f.

At f = 1 Hz, the Compton wavelength (the radius of nonlocality, instantaneous propagation) of a quantum is 1 light-second (300 thousand kilometres). According to the theory of relativity, superluminal propagation is propagation into the past. Therefore:

• A relativistic wave, whose frequency is above 1 Hz, begins its second period in the future; the third period, still further in the future, and so on. Such a wave is an electromagnetic wave; in its quantum—the photon—the particle-like magnetic phase dominates over the wave-like electric phase.
• A relativistic wave, whose frequency is below 1 Hz, begins its second period in the past; the third period, still further in the past, and so on. Such a wave is a gravitoelectromagnetic wave; in its quantum—the graviton—the wave-like electric phase dominates over the particle-like magnetic phase.

Conclusion:

1. Gravitoelectromagnetic waves, whose frequencies are below 1 Hz, are undetectable.
2. Gravitoelectromagnetic waves, whose frequencies are above 1 Hz, do not exist.

Effects of a passing gravitational wave

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

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

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 time-varying gravitational wave size (or 'periodic spacetime strain') exhibits a variation as shown in the animation. If the orbit is elliptical then the gravitational wave’s amplitude also varies with time according to an equation called the “quadrupole”.

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

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

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

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

Sources of gravitational waves

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

Some more detailed examples:

• Two objects orbiting each other in a quasi-Keplerian planar orbit (basically, as a planet would orbit the Sun) will radiate.
• 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 speed 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 nonzero in order for it to emit gravitational radiation. This is analogous to the changing dipole moment of charge or current necessary for electromagnetic radiation.

Power radiated by orbiting bodies

Two stars of dissimilar mass are in circular orbits. Each rotates 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. 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 $x$-$y$ 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:

$P={\frac {dE}{dt}}=-{\frac {32}{5}}\,{\frac {G^{4}}{c^{5}}}\,{\frac {(m_{1}m_{2})^{2}(m_{1}+m_{2})}{r^{5}}}$ ,

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

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 1.14×10³⁶ 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 1×10¹³ times more than the current age of the Universe to spiral onto the Sun. In only a few billion years, the Earth is predicted to be swallowed by the Sun in the red giant stage of its life.

A more dramatic example of radiated gravitational energy is represented by two solar mass neutron stars orbiting each other at a distance of 189×10⁶ m (0.63 light-seconds). Plugging their masses into the above equation shows that the gravitational radiation from them would be 1.38×10²⁸ watts, which is about 100 times more than the Sun's electromagnetic radiation.

Orbital decay from gravitational radiation

Gravitational radiation robs the orbiting bodies of energy, a process known as gravitation radiation damping. 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:

${\frac {dr}{dt}}=-{\frac {64}{5}}\,{\frac {G^{3}}{c^{5}}}\,{\frac {(m_{1}m_{2})(m_{1}+m_{2})}{r^{3}}}\$,

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

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

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

A more practical example is the orbit of a Sun-like star around a heavy black hole. Our Milky Way has a 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 1.89×10¹⁰ m (63 light-seconds). The mass of the black hole will be 4×10³⁶ kg and its gravitational radius will be 6×10⁹ m. The orbital period will be 1,000 seconds, or a little under 17 minutes. The solar-mass star will draw closer to the black hole by 7.4 meters per second or 7.4 km per orbit. A collision will not be long in coming.

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

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

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

Orbital lifetime limits from gravitational radiation

Orbital lifetime is one of the most important properties of gravitational radiation sources. It determines the average number of binary stars in the universe that are close enough to be detected. Short lifetime binaries are strong sources of gravitational radiation but are few in number. Long lifetime binaries are more plentiful but they are weak sources of 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:

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

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

Substituting the values for the mass of the Sun and Earth as well as the orbital radius gives a very large lifetime of 3.44×10³⁰ seconds or 1.09×10²³ years (which is approximately 10¹⁵ times larger than the age of the universe). The actual figure would be slightly less than that. The Earth will break apart from tidal forces if it orbits closer than a few radii from the sun. This would form a ring around the Sun and instantly stop the emission of gravity waves.

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

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

If the orbit of a neutron star binary has decayed to 1.89×10⁶m (1890 km), its remaining lifetime is 130,000 seconds or about 36 hours. The orbital frequency will vary from 1 revolution per second at the start and 918 revolutions per second when the orbit has shrunk to 20 km at merger. The gravitational radiation emitted will be at twice the orbital frequency. Just before merger, the inspiral can be observed by LIGO if the binary is close enough. LIGO has only a few minutes to observe this merger out of a total orbital lifetime that may have been billions of years. 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.

Wave amplitudes from the Earth-Sun system

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

$h_{+}=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {2m_{1}m_{2}}{r}}(1+\cos ^{2}\theta )\cos \left[2\omega (t-R)\right],$

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

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

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

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

$h_{+}=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {4m_{1}m_{2}}{r}}=-{\frac {1}{R}}\,1.7\times 10^{-10}\,\mathrm {meters} .$

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

Radiation from other sources

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

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

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.

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

Astrophysics and gravitational waves

Unsolved problem in physics:

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?

(more unsolved problems in physics)

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, 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.

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 relic gravitational waves that are theorized to be faint imprints of the Big Bang like the cosmic microwave background (see gravitational wave background). At these high frequencies it is potentially possible that the sources may be “man made” that is, gravitational waves generated and detected in the laboratory.

Energy, momentum, and angular momentum carried by gravitational waves

Waves familiar from other areas of physics such as water waves, sound waves, and electromagnetic waves are able to carry energy, momentum, and angular momentum. By carrying these away from a source, waves are able to rob that source of its energy, 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. 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. A kicked black hole can also carry a star cluster with it, forming a hyper-compact stellar system. Or it may carry gas, allowing the recoiling black hole to appear temporarily as a "naked quasar".

Gravitational wave detectors

Main article: Gravitational wave detector

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 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 h ≈ 10⁻²⁰, but generally no bigger.

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.

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. 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.

A schematic diagram of a laser interferometer.

A more sensitive class of detector uses laser interferometry to measure gravitational-wave induced motion between separated 'free' masses. This allows the masses to be separated by large distances (increasing the signal size); a further advantage is that it is sensitive to a wide range of frequencies (not just those near a resonance as is the case for Weber bars). Ground-based interferometers are now operational. Currently, the most sensitive is LIGO — the Laser Interferometer Gravitational Wave Observatory. LIGO has three detectors: one in Livingston, Louisiana; the other two (in the same vacuum tubes) at the Hanford site in Richland, Washington. Each consists of two light storage arms 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 h ≈ 5*10⁻²⁰. Upgrades to LIGO and other detectors such as Virgo, GEO 600, and TAMA 300 should increase the sensitivity still further; the next generation of instruments (Advanced LIGO and Advanced Virgo) will be more than ten times more sensitive. Another highly sensitive interferometer (LCGT) is currently in the design phase. A key point is that a tenfold increase in sensitivity (radius of 'reach') increases the volume of space accessible to the instrument by one thousand times. This increases the rate at which detectable signals should be seen from one per tens of years of observation, to tens per year.

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 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 a detection may be considered a true gravitational wave event.

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

Einstein@Home

Main article: Einstein@Home

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.

Mathematics

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

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

$G_{\mu \nu }={\frac {8\pi G_{N}}{c^{4}}}T_{\mu \nu }\,$ ,

where $G_{N}\,$ is Newton's gravitational constant, and $c$ is the speed of light. We assume geometrized units, so $G_{N}=1=c\,$.

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

$\eta _{\mu \nu }={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&r^{2}&0\\0&0&0&r^{2}\sin ^{2}\theta \end{bmatrix}}\,$ .

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 $g_{\mu \nu }\,$ as a matrix, and find its determinant, $\det \ g\,$. Finally, we define a quantity

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

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

$\nabla _{\beta }\,{\bar {h}}^{\alpha \beta }=0\,$ ,

where $\nabla$ represents the flat-space derivative operator. These equations say that the divergence of the field is zero. The full, nonlinear Einstein equations can now be written as

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

where $\Box =-\partial _{t}^{2}+\Delta \,$ represents the flat-space d'Alembertian operator, and $\tau ^{\alpha \beta }\,$ represents the stress-energy tensor plus quadratic terms involving ${\bar {h}}^{\alpha \beta }\,$. This is just a wave equation for the field with a source, despite the fact that the source involves terms quadratic in the field itself. That is, it can be shown that solutions to this equation are waves traveling with velocity 1 in these coordinates.

Linear approximation

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

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

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

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

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

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

where we now see the origin of the two polarizations.

Relation to the source

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

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

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

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

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

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

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

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

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

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

where $r=|{\vec {x}}|$ .

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

$\int \,\tau ^{ij}(t-r,{\vec {x}}')\,d^{3}x'=\int \,x'^{i}x'^{j}\nabla _{k}\nabla _{l}\tau ^{kl}(t-r,{\vec {x}}')\,d^{3}x'$ ,

Inserting this into the above equation, we arrive at

${\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,\int \,x'^{i}x'^{j}\nabla _{k}\nabla _{l}\tau ^{kl}(t-r,{\vec {x}}')\,d^{3}x'$ ,

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

$\nabla _{0}\nabla _{0}\tau ^{00}=\nabla _{j}\nabla _{k}\tau ^{jk}$ .

With this relation, the expression for the radiated field is

${\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {d^{2}}{dt^{2}}}\,\int \,x'^{i}x'^{j}\tau ^{00}(t-r,{\vec {x}}')\,d^{3}x'$ .

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

To a very good approximation, the density of a simple binary can be described by a pair of delta-functions, which eliminates the integral. Explicitly, if the masses of the two objects are $M_{1}$ and $M_{2}$, and the positions are ${\vec {x}}_{1}$ and ${\vec {x}}_{2}$, then

$\rho (t-r,{\vec {x}}')=M_{1}\delta ^{3}({\vec {x}}'-{\vec {x}}_{1}(t-r))+M_{2}\delta ^{3}({\vec {x}}'-{\vec {x}}_{2}(t-r))$ .

We can use this expression to do the integral above:

${\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {d^{2}}{dt^{2}}}\,\left\{M_{1}x_{1}^{i}(t-r)x_{1}^{j}(t-r)+M_{2}x_{2}^{i}(t-r)x_{2}^{j}(t-r)\right\}$ .

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

${\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {M_{1}M_{2}}{R}}\,n^{i}(t-r)n^{j}(t-r)$ ,

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

See also

• General Relativity
• Graviton
• Linearised Einstein field equations
• Gravitomagnetism
• Graviton astronomy
• LIGO, VIRGO, GEO 600, and TAMA 300 — Gravitational wave detectors
• LISA the proposed Laser Interferometer Space Antenna
• DECIGO "Deci-hertz Interferometer Gravitational wave Observatory", the planned laser Interferometric detector in space
• Big Bang Observer (BBO), proposed successor to LISA
• Sticky bead argument, for a physical way to see that gravitational radiation should carry energy
• pp-wave spacetime, for an important class of exact solutions modeling gravitational radiation
• Hawking Radiation, for gravitationally induced electromagnetic radiation from black holes

• Spin-flip, a consequence of gravitational wave emission from binary supermassive black holes
• Gravitational field

References

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

Bibliography

• Berry, Michael, Principles of cosmology and gravitation (Adam Hilger, Philadelphia, 1989). ISBN 0-85274-037-9
• Collins, Harry, Gravity's Shadow: the search for gravitational waves, University of Chicago Press, 2004.
• P. J. E. Peebles, Principles of Physical Cosmology (Princeton University Press, Princeton, 1993). ISBN 0691019339.
• Wheeler, John Archibald and Ciufolini, Ignazio, Gravitation and Inertia (Princeton University Press, Princeton, 1995). ISBN 0-691-03323-4.
• Woolf, Harry, ed., Some Strangeness in the Proportion (Addison–Wesley, Reading, Massachusetts, 1980). ISBN 0201099241.

External links

Media related to Gravitational waves at Wikimedia Commons

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