Gravitational wave

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

Although gravitational radiation has not been directly detected, there is indirect evidence for its existence.^[5] For example, the 1993 Nobel Prize in Physics was awarded for measurements of the Hulse–Taylor binary system that suggests gravitational waves are more than mathematical anomalies. Various gravitational wave detectors exist and on 17 March 2014, astronomers at the Harvard–Smithsonian Center for Astrophysics claimed that they had detected and produced "the first direct image of gravitational waves across the primordial sky" within the cosmic microwave background, providing strong evidence for inflation and the Big Bang.^{[6][7][8][9][5]} Peer review will be needed before there can be any scientific consensus about these new findings.^[10][11] On 19 June 2014, lowered confidence in confirming the cosmic inflation findings was reported;^[12][13][14] on 19 September 2014, a further reduction in confidence was reported^[15][16] and, on 30 January 2015, even less confidence yet was reported.^[17][18]

Introduction

History of the Universe - gravitational waves are hypothesized to arise from cosmic inflation, a faster-than-light expansion just after the Big Bang (17 March 2014).^[6][8][9]

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 this volume.^[5] 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 a distant observer, that observer will find spacetime distorted by the effects of strain. Distances between free objects increase and decrease rhythmically as the wave passes, at a frequency corresponding to that of the wave. This occurs despite such free objects never being subjected to an unbalanced force. The magnitude of this effect decreases inversely with distance from the source. Inspiralling binary neutron stars are predicted to be a powerful source of gravitational waves as they coalesce, due to the very large acceleration of their masses as they orbit close to one another. However, due to the astronomical distances to these sources the effects when measured on Earth are predicted to be very small, having strains of less than 1 part in 10²⁰. Scientists are attempting to demonstrate the existence of these waves with ever more sensitive detectors. The current most sensitive measurement is about one part in 5×10²² (as of 2012) provided by the LIGO and VIRGO observatories.^[19] The lack of detection in these observatories provides an upper limit on the frequency of such powerful sources.^[20][21] A space based observatory, the Laser Interferometer Space Antenna, is currently under development by ESA.

Linearly polarised gravitational wave

Gravitational waves should penetrate regions of space that electromagnetic waves cannot. It is hypothesized that they will be able to provide observers on Earth with information about black holes and other exotic objects in the distant Universe. Such systems cannot be observed with more traditional means such as optical telescopes and radio telescopes. In particular, gravitational waves could be of interest to cosmologists as they offer a possible way of observing the very early universe. This is not possible with conventional astronomy, since before recombination the universe was opaque to electromagnetic radiation.^[22] Precise measurements of gravitational waves will also allow scientists to test the general theory of relativity more thoroughly.

In principle, gravitational waves could exist at any frequency. However, very low frequency waves would be impossible to detect and there is no credible source for detectable waves of very high frequency. Stephen W. Hawking and Werner Israel list different frequency bands for gravitational waves that could be plausibly detected, ranging from 10⁻⁷ Hz up to 10¹¹ Hz.^[23]

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.

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

The oscillations depicted here in the animation are exaggerated for the purpose of discussion—in reality a gravitational wave has a very small amplitude (as formulated in linearized gravity). However they enable us to visualize the kind of oscillations associated with gravitational waves as produced for example by a pair of masses in a circular orbit. In this case the amplitude of the gravitational wave is a constant, but its plane of polarization changes or rotates at twice the orbital rate and so the time-varying gravitational wave size (or 'periodic spacetime strain') exhibits a variation as shown in the animation.^[24] If the orbit is elliptical then the gravitational wave's amplitude also varies with time according to Einstein's quadrupole formula.^[25]

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

• Amplitude: Usually denoted $h$, this is the size of the wave — the fraction of stretching or squeezing in the animation. The amplitude shown here is roughly $h=0.5$ (or 50%). Gravitational waves passing through the Earth are many billions times weaker than this — $h\approx 10^{{-20}}$. Note that this is not the quantity that would be analogous to what is usually called the amplitude of an electromagnetic wave, which would be ${\frac {{\mathrm {d}}h}{{\mathrm {d}}t}}$.
• 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 depends on the angle from the source, as we will see in the next section.

Sources of gravitational waves

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

Some more detailed examples:

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

$P={\frac {{\mathrm {d}}E}{{\mathrm {d}}t}}=-{\frac {32}{5}}\,{\frac {G^{4}}{c^{5}}}\,{\frac {(m_{1}m_{2})^{2}(m_{1}+m_{2})}{r^{5}}}$ ,^[26]

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 + 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. 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.

A more dramatic example of radiated gravitational energy is represented by two solar mass (M_☉) neutron stars orbiting at a distance from each other of 1.89×10⁸ m (only 0.63 light-seconds apart). [The Sun is 8 light minutes from the Earth.] 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. It first circularizes their orbits and then gradually shrinks their radius. As the energy of the orbit is reduced, the distance between the bodies decreases, and they rotate more rapidly. The overall angular momentum is reduced however. This reduction corresponds to the angular momentum carried off by gravitational radiation. The rate of decrease of distance between the bodies versus time is given by:^[26]

${\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 the variables are the same as in the previous equation.

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

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

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

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

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

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

Orbital lifetime limits from gravitational radiation

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

The lifetime of an orbit is given by:^[26]

$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 (that is approximately 10¹³ times larger than the age of the universe). The actual figure would be slightly less than that. The Earth will break apart from tidal forces if it orbits closer than a few radii from the Sun. This would form a ring around the Sun and instantly stop the emission of gravitational waves.

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

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

If the orbit of a neutron star binary has decayed to 1.89×10⁶m (1890 km), its remaining lifetime is 130,000 seconds or about 36 hours. The orbital frequency will vary from 1 revolution per second at the start and 918 revolutions per second when the orbit has shrunk to 20 km at merger. The gravitational radiation emitted will be at twice the orbital frequency. Just before merger, the inspiral can be observed by LIGO if the binary is close enough. LIGO has only a few minutes to observe this merger out of a total orbital lifetime that may have been billions of years. The chance of success with LIGO as initially constructed is quite low despite the large number of such mergers occurring in the universe, because the sensitivity of the instrument does not 'reach' out to enough systems to see events frequently. No mergers have been seen in the few years that initial LIGO has been in operation, and it is thought that a merger should be seen about once per several tens of years of observing time with initial LIGO.^[27] The upgraded Advanced LIGO detector, with a ten times greater sensitivity, 'reaches' out 10 times further—encompassing a volume 1000 times greater, and seeing 1000 times as many candidate sources. Thus, the expectation is that detections will be made at the rate of tens per year.

Wave amplitudes from the Earth–Sun system

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

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

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

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:

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

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

Radiation from other sources

Although the waves from the Earth-Sun system are minuscule, astronomers can point to other sources for which the radiation should be substantial. One important example is the Hulse-Taylor binary — a pair of stars, one of which is a pulsar.^[28] The characteristics of their orbit can be deduced from the Doppler shifting of radio signals given off by the pulsar. Each of the stars are about 1.4 M_☉ and the size of their orbit is about 1/75 of the Earth-Sun orbit. This means the distance between the two stars is just a few times larger than the diameter of our own Sun. The combination of greater masses and smaller separation means that the energy given off by the Hulse-Taylor binary will be far greater than the energy given off by the Earth-Sun system — roughly 10²² times as much.

The information about the orbit can be used to predict just how much energy (and angular momentum) should be given off in the form of gravitational waves. As the energy is carried off, the stars should draw closer to each other. This effect is called an inspiral, and it can be observed in the pulsar's signals. The measurements on the Hulse-Taylor system have been carried out over more than 30 years. It has been shown that the gravitational radiation predicted by general relativity allows these observations to be matched within 0.2 percent. In 1993, Russell Hulse and Joe Taylor were awarded the Nobel Prize in Physics for this work, which was the first indirect evidence for gravitational waves. The orbital lifetime of this binary system before merger is a few hundred million years.^[29]

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.^[30]

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.^[31]

Astrophysics and gravitational waves

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.

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).^[32] At these high frequencies it is potentially possible that the sources may be "man made"^[23] that is, gravitational waves generated and detected in the laboratory.^[33][34]

Energy, momentum, and angular momentum carried by gravitational waves

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

The waves can also carry off linear momentum, a possibility that has some interesting implications for astrophysics.^[35] 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.^[36] A kicked black hole can also carry a star cluster with it, forming a hyper-compact stellar system.^[37] Or it may carry gas, allowing the recoiling black hole to appear temporarily as a "naked quasar". The quasar SDSS J092712.65+294344.0 is believed to contain a recoiling supermassive black hole.^[38]

Detecting gravitational waves

Difficulties in detection

Evidence of gravitational waves in the infant universe may have been uncovered by the BICEP2 radio telescope. The microscopic examination of the focal plane of the BICEP2 detector is shown here.^{[6][7][8][9][11]}

Gravitational waves are not easily detectable. This knowledge gap is primarily due to the massive presence of noise in the low frequencies where antennas currently operate. Gravitational waves are expected to have frequencies $10^{{-16}}\,{\mathrm {Hz}}<f<10^{4}\,{\mathrm {Hz}}$.^[39]

Ground-based interferometers

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 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.^{[citation needed]}
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.^[40]

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.^[41] 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.^[42]

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.^[43] This allows the masses to be separated by large distances (increasing the signal size); a further advantage is that it is sensitive to a wide range of frequencies (not just those near a resonance as is the case for Weber bars). Ground-based interferometers are now operational. Currently, the most sensitive is LIGO — the Laser Interferometer Gravitational Wave Observatory. LIGO has three detectors: one in Livingston, Louisiana; the other two (in the same vacuum tubes) at the Hanford site in Richland, Washington. Each consists of two light storage arms that are 2 to 4 kilometers in length. These are at 90 degree angles to each other, with the light passing through 1m diameter vacuum tubes running the entire 4 kilometers. A passing gravitational wave will slightly stretch one arm as it shortens the other. This is precisely the motion to which an interferometer is most sensitive.

Even with such long arms, the strongest gravitational waves will only change the distance between the ends of the arms by at most roughly 10⁻¹⁸ meters. LIGO should be able to detect gravitational waves as small as $h\sim 5\times 10^{{-20}}$. Upgrades to LIGO and other detectors such as Virgo, GEO 600, and TAMA 300 should increase the sensitivity still further; the next generation of instruments (Advanced LIGO and Advanced Virgo) will be more than ten times more sensitive. Another highly sensitive interferometer (LCGT) is currently in the design phase. A key point is that a tenfold increase in sensitivity (radius of 'reach') increases the volume of space accessible to the instrument by one thousand times. This increases the rate at which detectable signals should be seen from one per tens of years of observation, to tens per year.^[27]

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 $h\sim {2\times 10^{{-13}}/{\sqrt {{\mathrm {Hz}}}}}$, given as an amplitude spectral density. The INFN Genoa detector is a resonant antenna consisting of two coupled spherical superconducting harmonic oscillators a few centimeters in diameter. The oscillators are designed to have (when uncoupled) almost equal resonant frequencies. The system is currently expected to have a sensitivity to periodic spacetime strains of $h\sim {2\times 10^{{-17}}/{\sqrt {{\mathrm {Hz}}}}}$, with an expectation to reach a sensitivity of $h\sim {2\times 10^{{-20}}/{\sqrt {{\mathrm {Hz}}}}}$. The Chongqing University detector is planned to detect relic high-frequency gravitational waves with the predicted typical parameters ?_g ~10¹⁰ Hz (10 GHz) and h ~10⁻³⁰-10⁻³¹.

Using pulsar timing arrays

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.

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, aspherical neutron star would be "monochromatic"—like a pure tone in acoustics. It would not change very much in amplitude or frequency.
The Einstein@Home project is a distributed computing project similar to SETI@home intended to detect this type of simple gravitational wave. By taking data from LIGO and GEO, and sending it out in little pieces to thousands of volunteers for parallel analysis on their home computers, Einstein@Home can sift through the data far more quickly than would be possible otherwise.^[44]

Primordial gravitational waves

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

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 that — in this coordinate system — has the components we would expect for the flat space metric. For example, in these spherical coordinates, we have

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

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 linear Einstein equations can now be written^[45] 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^[46] 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:^[45]

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

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

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

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

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

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

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

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

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

Inserting this into the above equation, we arrive at

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

Finally, because we have chosen to work in coordinates for which $\nabla _{\beta }\,{\bar {h}}^{{\alpha \beta }}=0$, 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

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

In the linear case, $\tau ^{{00}}=\rho$, 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:

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

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

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

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.

Gravitational lens

From Wikipedia, the free encyclopedia

A gravitational lens refers to a distribution of matter (such as a cluster of galaxies) between a distant source and an observer, that is capable of bending the light from the source, as it travels towards the observer. This effect is known as gravitational lensing and is one of the predictions of Albert Einstein's general theory of relativity.

Although Orest Chwolson (1924) or Frantisek Klin (1936) are sometimes credited as being the first ones to discuss the effect in print, the effect is more commonly associated with Einstein, who published a more famous article on the subject in 1936.

Fritz Zwicky posited in 1937 that the effect could allow galaxy clusters to act as gravitational lenses. It was not until 1979 that this effect was confirmed by observation of the so-called "Twin QSO" SBS 0957+561.

Description

Unlike an optical lens, maximum 'bending' occurs closest to, and minimum 'bending' furthest from, the center of a gravitational lens. Consequently, a gravitational lens has no single focal point, but a focal line instead. If the (light) source, the massive lensing object, and the observer lie in a straight line, the original light source will appear as a ring around the massive lensing object. If there is any misalignment the observer will see an arc segment instead. This phenomenon was first mentioned in 1924 by the St. Petersburg physicist Orest Chwolson,^[1] and quantified by Albert Einstein in 1936. It is usually referred to in the literature as an Einstein ring, since Chwolson did not concern himself with the flux or radius of the ring image. More commonly, where the lensing mass is complex (such as a galaxy group or cluster) and does not cause a spherical distortion of space–time, the source will resemble partial arcs scattered around the lens. The observer may then see multiple distorted images of the same source; the number and shape of these depending upon the relative positions of the source, lens, and observer, and the shape of the gravitational well of the lensing object.^[2]

Play media

Gravitational lensing - intervening galaxy modifies appearance of a galaxy far behind it (video; artist's concept).

There are three classes of gravitational lensing:^[3]

1. Strong lensing: where there are easily visible distortions such as the formation of Einstein rings, arcs, and multiple images.

2. Weak lensing: where the distortions of background sources are much smaller and can only be detected by analyzing large numbers of sources to find coherent distortions of only a few percent. The lensing shows up statistically as a preferred stretching of the background objects perpendicular to the direction to the center of the lens. By measuring the shapes and orientations of large numbers of distant galaxies, their orientations can be averaged to measure the shear of the lensing field in any region. This, in turn, can be used to reconstruct the mass distribution in the area: in particular, the background distribution of dark matter can be reconstructed. Since galaxies are intrinsically elliptical and the weak gravitational lensing signal is small, a very large number of galaxies must be used in these surveys. These weak lensing surveys must carefully avoid a number of important sources of systematic error: the intrinsic shape of galaxies, the tendency of a camera's point spread function to distort the shape of a galaxy and the tendency of atmospheric seeing to distort images must be understood and carefully accounted for. The results of these surveys are important for cosmological parameter estimation, to better understand and improve upon the Lambda-CDM model, and to provide a consistency check on other cosmological observations. They may also provide an important future constraint on dark energy.

3. Microlensing: where no distortion in shape can be seen but the amount of light received from a background object changes in time. The lensing object may be stars in the Milky Way in one typical case, with the background source being stars in a remote galaxy, or, in another case, an even more distant quasar. The effect is small, such that (in the case of strong lensing) even a galaxy with a mass more than 100 billion times that of the Sun will produce multiple images separated by only a few arcseconds. Galaxy clusters can produce separations of several arcminutes. In both cases the galaxies and sources are quite distant, many hundreds of megaparsecs away from our Galaxy.

Gravitational lenses act equally on all kinds of electromagnetic radiation, not just visible light. Weak lensing effects are being studied for the cosmic microwave background as well as galaxy surveys. Strong lenses have been observed in radio and x-ray regimes as well. If a strong lens produces multiple images, there will be a relative time delay between two paths: that is, in one image the lensed object will be observed before the other image.

History

Bending light around a massive object from a distant source. The orange arrows show the apparent position of the background source. The white arrows show the path of the light from the true position of the source.

Spacetime around a massive object (such as a galaxy cluster or a black hole) is curved, and as a result light rays from a background source (such as a galaxy) propagating through spacetime are bent. The lensing effect can magnify and distort the image of the background source.

According to general relativity, mass "warps" space–time to create gravitational fields and therefore bend light as a result. This theory was confirmed in 1919 during a solar eclipse, when Arthur Eddington and Frank Watson Dyson observed the light from stars passing close to the Sun was slightly bent, so that stars appeared slightly out of position.^[4]

In the formation known as Einstein's Cross, four images of the same distant quasar appear around a foreground galaxy due to strong gravitational lensing.

Einstein realized that it was also possible for astronomical objects to bend light, and that under the correct conditions, one would observe multiple images of a single source, called a gravitational lens or sometimes a gravitational mirage.

However, as he only considered gravitational lensing by single stars, he concluded that the phenomenon would most likely remain unobserved for the foreseeable future. In 1937, Fritz Zwicky first considered the case where a galaxy could act as a source, something that according to his calculations should be well within the reach of observations.

It was not until 1979 that the first gravitational lens would be discovered. It became known as the "Twin QSO" since it initially looked like two identical quasistellar objects; it is officially named SBS 0957+561. This gravitational lens was discovered by Dennis Walsh, Bob Carswell, and Ray Weymann using the Kitt Peak National Observatory 2.1 meter telescope.^[5]

In the 1980s, astronomers realized that the combination of CCD imagers and computers would allow the brightness of millions of stars to be measured each night. In a dense field, such as the galactic center or the Magellanic clouds, many microlensing events per year could potentially be found. This led to efforts such as Optical Gravitational Lensing Experiment, or OGLE, that have characterized hundreds of such events.

Explanation in terms of space–time curvature

Simulated gravitational lensing (black hole going past a background galaxy).

In general relativity, light follows the curvature of spacetime, hence when light passes around a massive object, it is bent. This means that the light from an object on the other side will be bent towards an observer's eye, just like an ordinary lens. Since light always moves at a constant speed, lensing changes the direction of the velocity of the light, but not the magnitude.

Light rays are the boundary between the future, the spacelike, and the past regions. The gravitational attraction can be viewed as the motion of undisturbed objects in a background curved geometry or alternatively as the response of objects to a force in a flat geometry. The angle of deflection is:

$\theta ={\frac {4GM}{rc^{2}}}$

toward the mass M at a distance r from the affected radiation, where G is the universal constant of gravitation and c is the speed of light in a vacuum.
Since the Schwarzschild radius $r_{{\mathrm {s}}}$ is defined as $r_{{\mathrm {s}}}={2Gm}/{c^{2}}$, this can also be expressed in simple form as

$\theta =2{\frac {r_{{\mathrm {s}}}}{r}}$

Search for gravitational lenses

Most of the gravitational lenses in the past have been discovered accidentally. A search for gravitational lenses in the northern hemisphere (Cosmic Lens All Sky Survey, CLASS), done in radio frequencies using the Very Large Array (VLA) in New Mexico, led to the discovery of 22 new lensing systems, a major milestone. This has opened a whole new avenue for research ranging from finding very distant objects to finding values for cosmological parameters so we can understand the universe better.

This image from the NASA/ESA Hubble Space Telescope shows the galaxy cluster MACS J1206.

A similar search in the southern hemisphere would be a very good step towards complementing the northern hemisphere search as well as obtaining other objectives for study. If such a search is done using well-calibrated and well-parameterized instrument and data, a result similar to the northern survey can be expected. The use of the Australia Telescope 20 GHz (AT20G) Survey data collected using the Australia Telescope Compact Array (ATCA) stands to be such a collection of data. As the data were collected using the same instrument maintaining a very stringent quality of data we should expect to obtain good results from the search. The AT20G survey is a blind survey at 20 GHz frequency in the radio domain of the electromagnetic spectrum. Due to the high frequency used, the chances finding gravitational lenses increases as the relative number of compact core objects (e.g. Quasars) are higher (Sadler et al. 2006). This is important as the lensing is easier to detect and identify in simple objects compared to objects with complexity in them. This search involves the use of interferometric methods to identify candidates and follow them up at higher resolution to identify them. Full detail of the project is currently under works for publication.

In a recent article on Science Daily (Jan. 21 2009) a team of scientists led by a cosmologist from the U.S. Department of Energy's Lawrence Berkeley National Laboratory has made major progress in extending the use of gravitational lensing to the study of much older and smaller structures than was previously possible by stating that weak gravitational lensing improves measurements of distant galaxies.^[6]

Astronomers from the Max Planck Institute for Astronomy in Heidelberg, Germany, the results of which are accepted for publication on Oct 21, 2013 in the Astrophysical Journal Letters (arXiv.org), discovered what at the time was the most distant gravitational lens galaxy termed as J1000+0221 using NASA’s Hubble Space Telescope.^[7][8] While it remains the most distant quad-image lensing galaxy known, an even more distant two-image lensing galaxy was subsequently discovered by an international team of astronomers using a combination of Hubble Space Telescope and Keck telescope imaging and spectroscopy. The discovery and analysis of the IRC 0218 lens was published in the Astrophysical Journal Letters on June 23, 2014.^[9]

A research published Sep 30, 2013 in the online edition of Physical Review Letters, led by McGill University in Montreal, Canada, has discovered the B-modes, that are formed due to gravitational lensing effect, using National Science Foundation's South Pole Telescope and with help from the Herschel space observatory. This discovery would open the possibilities of testing the theories of how our universe originated.^[10][11]

Abell 2744 galaxy cluster - extremely distant galaxies revealed by gravitational lensing (16 October 2014).^[12][13]

Solar gravitational lens

Albert Einstein predicted in 1936 that rays of light from the same direction that skirt the edges of the Sun would converge to a focal point approximately 542 AU from the Sun.^[14] This distance is far beyond the progress of space probes such as Voyager 1, and the known planets and dwarf planets, though over thousands of years 90377 Sedna will move further away on its highly elliptical orbit. The high gain for potentially detecting signals through this lens, such as microwaves at the 21-cm hydrogen line, led to the suggestion by Frank Drake in the early days of SETI that a probe could be sent to this distance. A multipurpose probe "SETISAIL" and later "FOCAL" was proposed to the ESA in 1993, but is expected to be a difficult task. If a probe does pass 542 AU, the gain and image-forming capabilities of the lens will continue to improve at further distances as the rays that come to a focus at these distances pass further away from the distortions of the Sun's corona.^[15]

Measuring weak lensing

Kaiser et al. (1995),^[16] Luppino & Kaiser (1997)^[17] and Hoekstra et al. (1998) prescribed a method to invert the effects of the Point Spread Function (PSF) smearing and shearing, recovering a shear estimator uncontaminated by the systematic distortion of the PSF. This method (KSB+) is the most widely used method in current weak lensing shear measurements.

Galaxies have random rotations and inclinations. As a result, the shear effects in weak lensing need to be determined by statistically preferred orientations. The primary source of error in lensing measurement is due to the convolution of the PSF with the lensed image. The KSB method measures the ellipticity of a galaxy image. The shear is proportional to the ellipticity. The objects in lensed images are parameterized according to their weighted quadrupole moments. For a perfect ellipse, the weighted quadrupole moments are related to the weighted ellipticity. KSB calculate how a weighted ellipticity measure is related to the shear and use the same formalism to remove the effects of the PSF.

KSB’s primary advantages are its mathematical ease and relatively simple implementation. However, KSB is based on a key assumption that the PSF is circular with an anisotropic distortion. It’s fine for current cosmic shear surveys, but the next generation of surveys (e.g. LSST) may need much better accuracy than KSB can provide. Because during that time, the statistical errors from the data are negligible, the systematic errors will dominate.

Gallery

• 

  "Smiley" image of galaxy cluster (SDSS J1038+4849) & gravitational lensing (an Einstein ring) (HST).^[18]
• 

  Abell 1689 - actual gravitational lensing effects (Hubble Space Telescope).
• 

  Dark matter distribution - weak gravitational lensing (Hubble Space Telescope).
• 

  Gravitational lens discovered at redshift z = 1.53.^[19]

Gravitationally-lensed distant star-forming galaxy.^[20]

Historical papers and references

• Chwolson, O (1924). "Über eine mögliche Form fiktiver Doppelsterne". Astronomische Nachrichten 221 (20): 329. Bibcode:1924AN....221..329C. doi:10.1002/asna.19242212003. 
• Einstein, Albert (1936). "Lens-like Action of a Star by the Deviation of Light in the Gravitational Field". Science 84 (2188): 506–7. Bibcode:1936Sci....84..506E. doi:10.1126/science.84.2188.506. JSTOR 1663250. PMID 17769014. 
• Renn, Jürgen; Tilman Sauer; John Stachel (1997). "The Origin of Gravitational Lensing: A Postscript to Einstein's 1936 Science paper". Science 275 (5297): 184–6. Bibcode:1997Sci...275..184R. doi:10.1126/science.275.5297.184. PMID 8985006.