General relativity

Key topics

Introduction to... Mathematical formulation of...

Fundamental concepts

Special relativity Equivalence principle World line · Riemannian geometry

Phenomena

Kepler problem · Lenses · Waves Frame-dragging · Geodetic effect Event horizon · Singularity Black hole For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Newton’s conception and quantification of gravitation held until the beginning of the 20th century, when Albert Einstein extended the special relativity to form the general relativity (GR) theory. ... For a less technical introduction to this topic, please see Introduction to mathematics of general relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In general relativity, the Kepler problem involves solving for the motion of a particle of negligible mass in the external gravitational field of another body of mass M. This gravitational field is described by the Schwarzschild solution to the vacuum Einstein equations of general relativity, and particle motion is described... This article or section is in need of attention from an expert on the subject. ... According to Albert Einsteins theory of general relativity, space and time get pulled out of shape near a rotating body in a phenomenon referred to as frame-dragging. ... The geodetic effect represents the effect of the curvature of spacetime, predicted by general relativity, on a spinning, moving body. ... For the science fiction film, see Event Horizon (film). ... A gravitational singularity (sometimes spacetime singularity) is, approximately, a place where quantities which are used to measure the gravitational field become infinite. ... For other uses, see Black hole (disambiguation). ...

Equations

Linearized Gravity Post-Newtonian formalism Einstein field equations

Advanced theories

Kaluza-Klein Quantum gravity

Solutions

Schwarzschild Reissner-Nordström · Gödel Kerr · Kerr-Newman Kasner · Milne · Robertson-Walker It has been suggested that Weak-field approximation be merged into this article or section. ... The parameterized post-Newtonian formalism or PPN formalism is a tool used to compare classical theories of gravitation in the limit most important for everyday gravitational experiments: the limit in which the gravitational field is weak and generated by objects moving slowly compared to the speed of light. ... The Einstein field equations (EFE) or Einsteins equations are a set of ten equations in Einsteins theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. ... Kaluza-Klein theory (or KK theory, for short) is a model which sought to unify classical gravity and electromagnetism. ... Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field. ... It has been suggested that Deriving the Schwarzschild solution be merged into this article or section. ... In physics and astronomy, a Reissner-Nordström black hole, discovered by Gunnar Nordström and Hans Reissner, is a black hole that carries electric charge , no angular momentum, and mass . ... The Gödel solution is an exact solution of the Einstein field equation in which the stress-energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant (see lambdavacuum solution). ... In general relativity, the Kerr metric (or Kerr vacuum) describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... The Kerr-Newman metric is a solution of Einsteins general relativity field equation that describes the spacetime geometry around a charged (), rotating () black hole of mass m. ... The Kasner metric is an exact solution to Einsteins theory of general relativity. ... Milnes model follows the description from special relativity of an observable universes spacetime diagram containing past and future light cones along with elsewhere in spacetime. ... // The Friedmann-Lemaître-Robertson-Walker (FLRW) metric is an exact solution of the Einstein field equations of general relativity and which describes a homogeneous, isotropic expanding/contracting universe. ...

Scientists Einstein · Minkowski · Eddington Lemaître · Schwarzschild Robertson · Kerr · Friedman Chandrasekhar · Hawking · others

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In physics, a gravitational wave is a fluctuation in the curvature of spacetime which propagates as a wave, traveling outward from a moving object or system of objects. Gravitational radiation is the energy transported by these waves. Important examples of systems which emit gravitational waves are binary star systems, where the two stars in the binary are white dwarfs, neutron stars, or black holes. “Einstein” redirects here. ... Hermann Minkowski. ... One of Sir Arthur Stanley Eddingtons papers announced Einsteins theory of general relativity to the English-speaking world. ... Monsignor Georges Lemaître, priest and scientist. ... Karl Schwarzschild (October 9, 1873 - May 11, 1916) was a noted German Jewish physicist and astronomer, father of astrophysicist Martin Schwarzschild. ... Howard Percy Robertson (January 27, 1903 - August 26, 1961) was a scientist known for contributions related to cosmology and the uncertainty principle. ... Roy Patrick Kerr (1934- ) is a New Zealand born mathematician who is best known for discovering the famous Kerr vacuum, an exact solution to the Einstein field equation of general relativity, which models the gravitational field outside an uncharged rotating massive object, or even a rotating black hole. ... Alexander Alexandrovich Friedman or Friedmann (Александр Александрович Фридман) (June 16, 1888 – September 16, 1925) was a Russian cosmologist and mathematician. ... Chandrasekhar redirects here. ... Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ... This is a partial list of persons who have made major contributions to the development of standard mainstream general relativity. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In mathematics, curvature refers to a number of loosely related concepts in different areas of geometry. ... For other uses of this term, see Spacetime (disambiguation). ... Surface waves in water This article is about waves in the most general scientific sense. ... For the band of the same name, see: Binary Star (band) Hubble image of the Sirius binary system, in which Sirius B can be clearly distinguished (lower left). ... A white dwarf is an astronomical object which is produced when a low to medium mass star dies. ... This article is about the celestial body. ... This article is about the astronomical body. ...

Although gravitational radiation has not yet been directly detected, it has been indirectly shown to exist. This was the basis for the 1993 Nobel Prize in Physics, awarded for measurements of the Hulse-Taylor binary system. Year 1993 (MCMXCIII) was a common year starting on Friday (link will display full 1993 Gregorian calendar). ... Hannes Alfvén (1908–1995) accepting the Nobel Prize for his work on magnetohydrodynamics [1]. List of Nobel Prize laureates in Physics from 1901 to the present day. ... PSR B1913+16 (also known as J1915+1606) is a pulsar in a binary star system, in orbit with another star around a common center of mass. ...

(Gravitational waves are sometimes called gravity waves, but this term is generally reserved for a completely different kind of wave encountered in hydrodynamics.) Ocean wave Wave clouds over Theresa, Wisconsin, USA Atmospheric gravity waves as seen from space. ... Hydrodynamics is fluid dynamics applied to liquids, such as water, alcohol, oil, and blood. ...

Introduction

In Einstein's theory of general relativity, the force of gravity is due to curvature of spacetime. This curvature is caused by the presence of massive objects. Roughly speaking, the more massive the object is, the greater the curvature it causes, and hence the more intense the gravity. As massive objects move around in spacetime, the curvature will change. If the objects move around in a certain way, ripples in spacetime can spread outward like ripples on the surface of a pond. These ripples are gravitational waves. For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... For other uses of this term, see Spacetime (disambiguation). ...

The simplest example of a strong source of gravitational waves is a spinning neutron star with a small mountain on its surface. The mountain's mass will cause curvature of the spacetime. Its movement will "stir up" spacetime, much like a paddle stirring up water. The waves will spread out through the Universe at the speed of light, never stopping or slowing down. For the story by Larry Niven, see Neutron Star (story). ...

As these waves pass a distant observer, that observer will find spacetime distorted in a very particular way. Distances between objects will increase and decrease rhythmically as the wave passes. The magnitude of this effect will decrease the farther the observer is from the source. Any gravitational waves expected to be seen on Earth will be quite small; the change in size of any object will never be much more than 1 in 10²⁰. Still, scientists are attempting to measure the effects of these waves using extraordinarily precise experiments.

By measuring these waves, astrophysicists hope to learn about systems they could not observe with more traditional means such as optical telescopes, radio telescopes, etc. Gravitational waves can penetrate regions that the more familiar waves cannot, providing us with a view of black holes and other mysterious objects in the distant Universe. Using precise measurements of gravitational waves in this way will also allow us to test the general theory of relativity more thoroughly. Eight Inch refracting telescope. ... The 64 meter radio telescope at Parkes Observatory A radio telescope is a form of directional radio antenna used in radio astronomy and in tracking and collecting data from satellites and space probes. ...

In principle, gravitational waves could exist at any frequency. However, very low frequency waves would be impossible to detect, and very high frequency waves have no credible source able to generate detectable waves. 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.^[1] Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ... Werner Israel, OC, FRSC, FRS (born October 4, 1931) is a Canadian physicist. ...

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. Then, a weak gravitational wave arrives, passing through the particles along a line perpendicular to the plane of the particles. What happens to the test particles? Roughly speaking, they 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. In the animation at the right, the wave would be passing from you, through the screen, and out the back. Image File history File links GravitationalWave_PlusPolarization. ... Image File history File links GravitationalWave_PlusPolarization. ... Image File history File links GravitationalWave_CrossPolarization. ... Image File history File links GravitationalWave_CrossPolarization. ... This page is a candidate to be moved to Wiktionary. ...

The foregoing animation is the result of a pair of masses that orbit about each other (e.g., black holes) on a circular orbit or a rotating rod or dumbbell. In this case the amplitude, A, of the gravitational wave is a constant, but its plane of polarization changes or rotates (at twice the orbital or rotating-rod rate) and so the time-varying gravitational wave size or periodic spacetime strain h, exhibits a variation as shown in the animation.^[2] If the orbit is elliptical or the rotating rod’s centrifugal-force change varies during rotation, then the gravitational wave’s amplitude (that is, the amplitude of the periodic spacetime h), A, actually also varies with time according to an equation called the “quadrupole”.^[3]

Like other waves, there are a few useful numbers describing a gravitational wave: Surface waves in water This article is about waves in the most general scientific sense. ...

• 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 — .
• Frequency: Usually denoted ν, this is the frequency with which the wave oscillates (1 divided by the amount of time between maximum stretch or squeeze)
• Wavelength: Usually denoted λ, this is the distance along the wave between points of maximum stretch or squeeze.
• Speed: This is the speed at which a point on the wave (for example, a point of maximum stretch or squeeze) travels. For gravitational waves with small amplitudes, this is equal to the speed of light, c.

The frequency, wavelength, and speed are related by the equation c = λ ν, 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. The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ...

In the example just discussed, we actually assume something special about the wave. We have assumed that the wave is linearly polarized, with a "plus" polarization, written . Polarization of a gravitational wave is just like polarization of a light wave, except that the polarizations of a gravitational wave are at 45 degrees, as opposed to 90 degrees. In particular, if we had a "cross"-polarized gravitational wave, , the effect on the test particles would be basically the same, but rotated by 45 degrees, as shown in the second animation. Just as with light polarization, the polarizations of gravitational waves may also be expressed in terms of circularly polarized waves. Gravitational waves are polarized because of the nature of their sources. The polarization of a wave actually depends on the angle from the source, as we will see in the next section. In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. ... In electrodynamics, circular polarization of electromagnetic radiation is a polarization such that the tip of the electric field vector, at a fixed point in space, describes a circle as time progresses. ...

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). Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

A simple example is the spinning dumbbell. Set upon one end, so that one side of the dumbell is on the ground and the other end is pointing up, the dumbbell will not radiate when it spins around its vertical axis but will radiate if it tumbles end-over-end. 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 it 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. On the other hand, this system 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 second 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. For other uses, see Supernova (disambiguation). ... This article is about momentum in physics. ... This gyroscope remains upright while spinning due to its angular momentum. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... For other uses, see Mass (disambiguation). ... In general relativity, Birkhoffs theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. ... Quadrupole magnet(four-pole), focus particle beams in a particle accelerator. ... This article is in need of attention from an expert on the subject. ... This article is in need of attention from an expert on the subject. ...

Power radiated by the Earth-Sun system

We imagine a simple system of two masses — such as the Earth-Sun system — moving slowly compared to the speed of light. 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. Two bodies with a slight difference in mass orbiting around a common barycenter. ...

Suppose that the two masses are M₁ and M₂, and they are separated by a distance R. The power given off (radiated) by this system is

where G is the Gravitational Constant and E_kin,1, T and v are the kinetic energy, the period time and velocity of the first mass. This is derived from Einstein's quadrupole equation.^[2] For a system like the Earth and the Sun, R is very large (about 1.5×10¹¹ m) and M₁ and M₂ are relatively very small (about 2×10³⁰ and 6×10²⁴ kg respectively). Substituting these values into the above equation gives about 313 Watts of power. According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ...

Thus the total power radiated by the Earth-Sun system in the form of gravitational waves is about 300 Watts (i.e. about five 60 Watt light bulbs). This is tiny compared to the total electromagnetic radiation given off by the Sun (about 3.86×10²⁶ Watts) or even the amount of solar radiation reaching the Earth (about 1.74×10¹⁷ Watts). Solar irradiance spectrum at top of atmosphere. ...

Wave amplitudes from the Earth-Sun system

We can also think in terms of the amplitude of the wave. Suppose that an observer is positioned at a distance r from the center of mass of the system, at spherical coordinates (r,θ,φ). If the observer is well outside the system (in fact, we need r > c / Ω), the two polarizations of the wave will be This article describes some of the common coordinate systems that appear in elementary mathematics. ...

Here, we use the constant angular velocity of a circular orbit in Newtonian physics. Note that the polarization depends on the angle to the system. For example, if the observer is in the x-y plane, θ = π / 2, and cosθ = 0, so the polarization is always zero. We also see that the frequency of the wave given off is ν = 2Ω / 2π = Ω / π. If we put in numbers for the Earth-Sun system, we find Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ...

In this case, the minimum distance to find waves is r = 1 light year, so typical amplitudes will be . That is, a ring of particles would stretch or squeeze by just 10 ^{− 24} percent. A light-year or lightyear (symbol: ly) is a unit of measurement of length, specifically the distance light travels in vacuum in one year. ...

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. Also, their orbit is about 75 times smaller than the distance between the Earth and Sun — which means the distance between the two stars is just a few times larger than the diameter of our own Sun. This combination of greater masses and smaller separation means that the energy given off by the Hulse-Taylor binary will be far greater than the energy given off by the Earth-Sun system — roughly 10²² times as much. PSR B1913+16 (also known as J1915+1606) is a pulsar in a binary star system, in orbit with another star around a common center of mass. ... It has been suggested that Radio pulsar be merged into this article or section. ... The Doppler effect is the apparent change in frequency or wavelength of a wave that is perceived by an observer moving relative to the source of the waves. ...

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 orbit will change; the stars will draw closer to each other. This effect of drawing closer is called an inspiral, and it can be observed in the pulsar's signals. The measurements on this system were carried out over several decades, and it was shown that the changes predicted by gravitational radiation in general relativity matched the observations very well. In 1993, Russell Hulse and Joe Taylor were awarded the Nobel Prize in Physics for this experiment, which was the first experimental evidence for gravitational waves. Year 1993 (MCMXCIII) was a common year starting on Friday (link will display full 1993 Gregorian calendar). ... Russell Alan Hulse (born November 28, 1950) is an American physicist and winner of the Nobel Prize in Physics, shared with his thesis advisor Joseph Hooton Taylor Jr. ... Joseph H. Taylor, Jr. ... Hannes Alfvén (1908–1995) accepting the Nobel Prize for his work on magnetohydrodynamics [1]. List of Nobel Prize laureates in Physics from 1901 to the present day. ...

Inspirals are very important sources of gravitational waves. Any time two compact objects (white dwarfs, neutron stars, or black holes) come close to each other, they send out intense gravitational waves. As the objects come closer and closer to each other (that is, as R becomes smaller and smaller), the gravitational waves become more and more intense. At some point these waves should become so intense that they can be directly detected by their effect on objects on the Earth. This direct detection is the goal of several large experiments around the world.

The only difficulty is that 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 . There are some sources, however, that astrophysicists expect to find with the somewhat larger amplitudes of .

Astrophysics and gravitational waves

Unsolved problems 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?

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. Image File history File links No higher resolution available. ... This is a list of some of the unsolved problems in physics. ... For the story by Larry Niven, see Neutron Star (story). ... Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... For other uses, see Astronomy (disambiguation). ... The optical spectrum (light or visible spectrum) is the portion of the electromagnetic spectrum that is visible to the human eye. ... Galileo redirects here. ... Legend γ = Gamma rays HX = Hard X-rays SX = Soft X-Rays EUV = Extreme ultraviolet NUV = Near ultraviolet Visible light NIR = Near infrared MIR = Moderate infrared FIR = Far infrared Radio waves EHF = Extremely high frequency (Microwaves) SHF = Super high frequency (Microwaves) UHF = Ultra high frequency VHF = Very high frequency HF = High... In contrast to an ordinary telescope, which produces visible light images, a radio telescope sees radio waves emitted by radio sources, typically by means of a large parabolic (dish) antenna, or arrays of them. ... Composite Optical/X-ray image of the Crab Nebula pulsar, showing surrounding nebular gases stirred by the pulsars magnetic field and radiation. ... This view, taken with infrared light, is a false-color image of a quasar-starburst tandem with the most luminous starburst ever seen in such a combination. ... This article is about the type of Electromagnetic radiation. ... CMB redirects here. ... For other uses, see Big Bang (disambiguation). ... Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ... This article is about electromagnetic radiation. ... In the NATO phonetic alphabet, X-ray represents the letter X. An X-ray picture (radiograph) taken by Röntgen An X-ray is a form of electromagnetic radiation with a wavelength approximately in the range of 5 pm to 10 nanometers (corresponding to frequencies in the range 30 PHz... Note: Ultraviolet is also the name of a 1998 UK television miniseries about vampires. ... Image of a small dog taken in mid-infrared (thermal) light (false color) Infrared (IR) radiation is electromagnetic radiation of a wavelength longer than visible light, but shorter than microwave radiation. ...

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. Whereas light from distant stars may be blocked out by interstellar dust, for example, gravitational waves will pass through unimpeded. These two features allow gravitational waves to carry information about astronomical phenomena never before observed by humans. Interstellar cloud is the generic name given to accumulations of gas and dust in our galaxy. ...

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.^[4] At these high frequencies it is potentially possible that the sources may be “man made”^[1] that is, gravitational waves generated and detected in the laboratory.^[5][6]

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. Consider for instance a cluster of stars with a binary black hole system in the center. The holes orbit each other, but their center of mass doesn't move with respect to the cluster at first. However, as the binary inspirals, the radiated gravitational waves carry away linear momentum in some direction. In keeping with Newton's third law of motion, the binary will gain some linear momentum in the opposite direction. Thus, it may be shot out of the cluster. This article is about momentum in physics. ... This gyroscope remains upright while spinning due to its angular momentum. ... Spiral Galaxy ESO 269-57 Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature, and chemical composition) of celestial objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

Gravitational wave detectors

Though the Hulse-Taylor observations were very important, they were only indirect evidence for gravitational waves. A more interesting observation would be a direct measurement of the effect of a passing gravitational wave. Not only would a direct measurement of gravitational waves rule out other possible (however unlikely) reasons for changes to the orbit of an inspiraling system, it would also provide us more information on the system. Perhaps more importantly, such a detection could give us information about things we can't see with radio or light waves — such as black holes. This would provide us with a rigorous test for Einstein's theory of general relativity. For other uses, see Black hole (disambiguation). ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ...

The great challenge of this type of detection, though, is the extraordinarily small effect the waves would produce on a detector. The amplitude of any 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 , but generally no bigger. For an object 1 meter in length, this means that its ends would move by 10 ^{− 20} meters relative to each other. This distance is about 1 billionth of the width of a typical atom, and roughly one one-hundred-thousandth the width of a proton. Properties For other meanings of Atom, see Atom (disambiguation). ... For other uses, see Proton (disambiguation). ...

A simple device to detect this motion is called a Weber bar — a large, solid piece of metal with electronics attached to detect any vibrations. This type of instrument was the first type of gravitational wave detector. The idea is to wait for a passing gravitational wave to "ring up" a bar at its resonant frequency, which would basically amplify the wave naturally. Alternatively, a nearby supernova might be strong enough to be seen without resonant amplification. Modern forms of the Weber bar are still operated, cryogenically cooled, with superconducting quantum interference devices to detect the motion. Unfortunately, Weber bars are not sensitive enough to detect anything but extremely powerful gravitational waves. For a review of early experiments using Weber bars, see J. Levine, "Early Gravity-Wave Detection Experiments, 1960-1975, Physics in Perspective (Birkhäuser Basel) Volume 6, Number 1 Pages 42-75 April, 2004. A Weber bar is a device used in the detection of gravitational waves first devised and constructed by physicist Joseph Weber at the University of Maryland. ... This article is about resonance in physics. ... Cryogenics is the study of very low temperatures or the production of the same, and is often confused with cryobiology, the study of the effect of low temperatures on organisms, or the study of cryopreservation. ... For other uses, see Squid (disambiguation). ...

A schematic diagram of a laser interferometer.

A more sensitive version is the laser interferometer, with separate masses placed many hundreds of meters to several kilometers apart acting as two ends of a bar. Ground-based interferometers are now operating, and taking data. Currently, the most sensitive is LIGO — the Laser Interferometer Gravitational Wave Observatory. This is actually a set of three devices: one in Livingston, Louisiana; the other two (in the same vacuum tubes) in Hanford, 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, and consist of large vacuum tubes running the entire 4 kilometers. A passing gravitational wave will then slightly stretch one arm as it shortens the other. This is precisely the motion to which an interferometer is most sensitive. Image File history File links Ligo. ... Image File history File links Ligo. ... Interferometry is the applied science of combining two or more input points of a particular data type, such as optical measurements, to form a greater picture based on the combination of the two sources. ... LIGO stands for Lesser Inner Greater Outer. ... Livingston is a town located in Livingston Parish, Louisiana. ... Hanford was a small agricultural community in Benton County, Washington. ... In optics, a Fabry-Perot interferometer or etalon is typically made of a transparent plate with two reflecting surfaces, or two parallel highly-reflecting mirrors. ...

Even with such long arms, a gravitational wave will only change the distance between the ends of the arms by about 10 ^{− 17} meters at most. This is still only a fraction of the width of a proton. Nonetheless, LIGO's interferometers are now running routinely at an even better sensitivity level. LIGO's should be able to detect gravitational waves as small as , but needs to wait until a gravitational wave with at least that amplitude passes. Upgrades to LIGO and other detectors such as VIRGO, GEO, and TAMA should increase the sensitivity still further — by a factor of up to 100. Another highly sensitive interferometer (LCGT) is currently in the design phase. Virgo (Latin for virgin, symbol , Unicode ♍) is a constellation of the zodiac. ... GEO or Geo may refer to any of the following: Geo or gio, is a creek (inlet) or gulley in the Orkney and Shetland Islands GEO (newspaper), a popular scientific magazine Geo (microformat), a microformat for marking up WSG84 geographical coordinates in (X)HTML Geo (automobile), a brand of entry... Tama may mean: Person or being: Sam Fatu, a professional wrestler, uses this name amongst others Hazel Tama a Bosnian actress. ... There are very few or no other articles that link to this one. ...

All detectors are limited at high frequencies by shot noise, which occurs because the laser beams are made up of photons. If there are not enough photons arriving in a given time interval (that is, if the laser is not intense enough), it will be impossible to tell whether a measurement is due to real data, or just random fluctuations in the number of photons. Photon noise simulation. ...

All ground-based detectors are also limited at low frequencies by seismic noise, and must be very well isolated from seismic disturbances. Passing cars and trains, falling trees, earthquakes, and even waves crashing on the shore hundreds of miles away are all very significant sources of noise in real interferometers. Seismology (from the Greek seismos = earthquake and logos = word) is the scientific study of earthquakes and the movement of waves through the Earth. ...

Space-based interferometers, such as LISA, are also being developed. LISA's design calls for test masses to be placed five million kilometers apart, in separate spacecraft with lasers running between them. Because of the distance between the spacecraft, it will be impossible to create light storage arms. Also, the arms will be at 60 degree angles to each other, rather than 90 degree angles. Still, the principle will be the same. Although LISA will not be affected by seismic noise, it will be affected by other noise sources, including noise from cosmic rays and solar wind and — of course — shot noise. The LISA is the Laser Interferometer Space Antenna experiment. ... Cosmic rays can loosely be defined as energetic particles originating outside of the Earth. ... The plasma in the solar wind meeting the heliopause The solar wind is a stream of charged particles (i. ...

There are other prospects such as MiniGRAIL, a spherical gravitational wave antenna based at Leiden University. Some scientists even want to use the moon as a giant gravitational wave detector. The moon should be somewhat pliable to the contortions caused by gravitational waves, and the hope is that the motion of the moon caused by these waves will be detectable, much like the motion of a Weber bar. MiniGRAIL is a gravitational wave telescope based at Leiden University, the Netherlands. ... Leiden University, located in the city of Leiden, is the oldest university in the Netherlands[1]. It is a member of the Coimbra Group, the Europaeum and the League of European Research Universities. ... A Weber bar is a device used in the detection of gravitational waves first devised and constructed by physicist Joseph Weber at the University of Maryland. ...

The detectors of gravitational waves described in the foregoing are in the low-frequency end of the gravitational-wave spectrum (10^-7 to 10⁵ Hz). There are currently two fabricated high-frequency gravitational wave (HFGW) detectors: one at Birmingham University, England and the other at INFN Genoa, Italy. In addition there are some under development at Chongqing University, China. The Birmingham HFGW high-frequency gravitational wave detector measures changes in the polarization state of a microwave beam (caused by the presence of a gravitational wave or GW) circulating in a closed loop about one meter across. Two have been fabricated and they are currently expected to be sensitive to HFGWs having periodic spacetime strains of , where Hz is the GW frequency in Hertz. The INFN Genoa detector, or resonant antenna, consists of two coupled, superconducting, spherical, harmonic oscillators a few centimeters in diameter. The oscillators are designed to have (when uncoupled) almost equal resonant frequencies. In theory the system is currently expected to have a sensitivity to HFGWs having periodic spacetime strains of with an expectation to reach a sensitivity of . This article is about the type of Electromagnetic radiation. ... Superconductivity is a phenomenon occurring in certain materials at low temperatures, characterised by the complete absence of electrical resistance and the damping of the interior magnetic field (the Meissner effect. ...

The Chongqing University detectors are based upon the GW theory first put forth by Gertsenshtein in 1962. These Chinese detectors involve the selection and detection of relic high-frequency gravitational waves with the predicted typical parameters ν_g ~ 10¹⁰ Hz (10 GHz) and h ~ 10^-30-10^-31. The usual Gertsenshtein effect ^[7] involves the generation of GWs by electromagnetic (EM) waves under the influence of a strong static magnetic field. In this case the GWs produced will have the same frequency as the EM waves producing them. The inverse Gertsenshtein effect, which is alluded to at the end of the paper involves what is termed a synchro-resonance condition. Thus in the Chinese detectors a strong EM beam (essentially a focused microwave, so called “Gaussian beam”) at the expected frequency, phase and bandwidth of the relic GWs is passed through a strong static magnetic field. There are then produced EM “detection” photons due to the GWs. These detection photons move off at right angles to the EM beam and to the direction of the magnetic field. According to theory they move off in both directions and are intercepted by two very sensitive microwave receivers or detectors. Noise is suppressed by keeping the detector at very low temperature (less than 0.048 K) and including superconductor interior baffles and a superconductor enclosure called a “Faraday Cage,” both composed of a mosaic of high-temperature superconductor tiles (e.g., YBCO). Similar to the Advanced LIGO, LISA, et al. detectors, these Chinese detectors are in the design phase. In optics, a Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity (irradiance) distributions are described by Gaussian functions. ... Entrance to a Faraday room A Faraday cage or Faraday shield is an enclosure formed by conducting material, or by a mesh of such material. ... Unsolved problems in physics: Why do certain materials exhibit superconductivity at temperatures much higher than 50 kelvins? The term high-temperature superconductor was initially employed to designate the new family of cuprate-perovskite ceramic materials discovered by J.G. Bednorz and K.A. Müller in 1986. ... Yttrium barium copper oxide, or YBCO, chemical formula YBa2Cu3O7-δ, is a high-temperature superconductor with a superconducting temperature of 94K. Its discovery by C.W. Chu in 1987 launched the era of high-temperature superconductors. ...

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 their waves 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. Einstein@Home is a distributed computing project running on the Berkeley Open Infrastructure for Network Computing (BOINC) software platform. ... Something which is monochromatic has a single color. ... Pure tone is a single frequency tone with no harmonic content (no overtones). ... Acoustics is the branch of physics concerned with the study of sound (mechanical waves in gases, liquids, and solids). ...

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 analysis on their home computers, Einstein@Home can sift through the data far more quickly than would be possible otherwise. Searches for gravitational waves from other types of systems require large supercomputers running for long periods. Einstein@Home is a distributed computing project running on the Berkeley Open Infrastructure for Network Computing (BOINC) software platform. ... Distributed computing is a method of computer processing in which different parts of a program are run simultaneously on two or more computers that are communicating with each other over a network. ... SETI@home logo SETI@home (SETI at home) is a distributed computing project using Internet-connected computers, hosted by the Space Sciences Laboratory, at the University of California, Berkeley, in the United States. ...

A simple computer program can be downloaded to any home computer, and acts as a screen saver to use computer time that would otherwise be wasted. The program automatically downloads the data, analyzes it while the screen saver is running, and sends the final results back to a central computer.

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_μν. 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. For other topics related to Einstein see Einstein (disambig) Introduction In physics, the Einstein field equation or Einstein equation is a tensor equation in the Einsteins theory of general relativity. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...

Spacetime curvature is also expressed with respect to a covariant derivative, , in the form of the Einstein tensor — G_μν. This curvature is related to the stress-energy tensor — T_μν — by the key equation In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ... Definition In differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds. ... This article is in need of attention from an expert on the subject. ...

where G_N is Newton's gravitational constant, and c is the speed of light. We assume geometrized units, so G_N = 1 = c. According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... In physics, especially in the general theory of relativity, geometrized units or sometimes geometric units, is a physical unit system in which all physical quantities are expressed in the unit of length: meter. ...

With some simple assumptions, Einstein's equations can be rewritten to show explicitly that they are just wave equations. To begin with, we adopt some coordinate system, like (t,r,θ,φ). We define the "flat-space metric" η_μν 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 The wave equation is an important partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. ...

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_μν as a matrix, and find its determinant, . Finally, we define a quantity In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

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): This article is in need of attention from an expert on the subject. ...

where 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^[8] as In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

where represents the flat-space d'Alembertian operator, and τ^αβ represents the stress-energy tensor plus quadratic terms involving . This is just a wave equation for the field with a source, despite the fact that the source involves terms quadratic in the field itself. That is, it can be shown that solutions to this equation are waves traveling with velocity 1 in these coordinates. In special relativity, electromagnetism and wave theory, the dAlembert operator, also called dAlembertian, is the Laplace operator of Minkowski space. ...

Linear approximation

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

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

Now, this is the usual homogeneous wave equation — one for each component of . Solutions to this equation are well known. For a wave moving away from a point source, the radiated part (meaning the part that dies off as 1 / r far from the source) can always be written in the form A(t − r,θ,φ) / r, where A is just some function. It can be shown^[9] that — to a linear approximation — it is always possible to make the field traceless. Now, if we further assume that the source is positioned at r = 0, the general solution to the wave equation in spherical coordinates is

where we now see the origin of the two polarizations.

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

we can write the solution in terms of the tensorial Green's function for the d'Alembertian operator:^[8] In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ...

Though it is possible to expand the Green's function in tensor spherical harmonics, it is easier to simply use the form In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ...

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

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

where .

Now, because we will eventually only be interested in the spatial components of this equation (time components can be set to zero with a coordinate transformation), and we are integrating this quantity — presumably over a region of which there is no boundary — we can put this in a different form. Ignoring divergences with the help of Stokes' theorem and an empty boundary, we can see that Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

Inserting this into the above equation, we arrive at

Finally, because we have chosen to work in coordinates for which , we know that . With a few simple manipulations, we can use this to prove that

With this relation, the expression for the radiated field is

In the linear case, τ⁰⁰ = ρ, 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 M₁ and M₂, and the positions are and , then

We can use this expression to do the integral above:

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

where . Plugging in the known values of , we obtain the expressions given above for the radiation from a simple binary.

See also

For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... This article is in need of attention from an expert on the subject. ... This article is in need of attention from an expert on the subject. ... LIGO stands for Lesser Inner Greater Outer. ... Virgo (Latin for virgin, symbol , Unicode ♍) is a constellation of the zodiac. ... Geo 600 is a gravitational wave detector located in Hannover, Germany. ... TAMA 300 is a gravitational wave detector located in Japan. ... The LISA is the Laser Interferometer Space Antenna experiment. ... The Big Bang Explorer is a proposed succesor to LISA. The primary scientific goal will be the observation of gravitational waves from the time shortly after the Big Bang, but it will also be able to detect younger sources of gravitational radiation. ... In general relativity, the sticky bead argument is a simple thought experiment designed to show that gravitational radiation is indeed predicted by general relativity, and can have physical effects. ... In general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutions of Einsteins field equation. ... In physics, Hawking radiation (also known as Bekenstein-Hawking radiation) is a thermal radiation thought to be emitted by black holes due to quantum effects. ...

References

• Chakrabarty, Indrajit, "Gravitational Waves: An Introduction". arXiv:physics/9908041 v1, Aug 21, 1999.
• Landau, L.D. and E.M. Lifshitz, The Classical Theory of Fields (Pergamon Press),(1987).
• Will, Clifford M., The Confrontation between General Relativity and Experiment. Living Rev. Relativity 9 (2006) 3.

External links

• 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)
• GraveWave Home Page - Information and a literature survey of high-frequency gravitational waves.
• Laser Interferometer Gravitational Wave Observatory. LIGO Laboratory, California Institute of Technology.
• Interferometric Detection of Gravitational Waves: 5 Needed Breakthroughs (2Physics.com).
• 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