General relativity

Overview of GR History Mathematics Resources Tests Black hole Einstein equation Equivalence principle Event horizon Exact solutions FLRW metric Gravitational lens Gravitational radiation Kerr metric Quantum gravity Schwarzschild metric Singularity

Related topics

Albert Einstein Astrophysics Gravity Cosmology Special relativity Riemannian geometry

edit Ongoing events • Abramoff-Reed gambling scandal • Al Jazeera bombing memo • Avian influenza (H5N1) outbreak • Black sites scandal • Conservative leadership race (UK) • Fuel prices • Irans nuclear program • Jilin chemical plant explosions • Kashmir earthquake • Malawi food crisis • Malaysian prisoner abuse scandal • New Delhi bombings investigation • Niger food crisis • North Indian cyclone... An expert is someone widely recognized as a reliable source of knowledge, technique, or skill whose judgment is accorded authority and status by the public or their peers. ... It has been suggested that Einsteins theory of gravitation be merged into this article or section. ... Image File history File links Download high resolution version (1024x768, 7 KB) Description: Gravitational light deflection at a neutron star Source: Gallery of Tempolimit Lichtgeschwindigkeit Date: 09. ... It has been suggested that Einsteins theory of gravitation be merged into this article or section. ... // Creation of General Relativity Early investigations The development of general relativity began in 1907 with the publication of an article by Albert Einstein on acceleration under special relativity. ... Notational point: General relativity articles using tensors will use the abstract index notation . ... // Books Popular Geroch, Robert (1981). ... Einsteins general theory of relativity was introduced in 1915. ... A black hole is a concentration of mass great enough that the force of gravity prevents anything past its event horizon from escaping it except through quantum tunnelling behaviour (known as Hawking Radiation). ... In physics, the Einstein field equation or Einstein equation is a differential equation in Einsteins theory of general relativity. ... THERE IS NO SUCH THING< MWAHAHAHAHAHAHAHA> In relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... Event Horizon is a 1997 science fiction and horror film. ... // 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. ... The Friedmann-Lemaître-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. ... This article is in need of attention from an expert on the subject. ... In general relativity, the Kerr metric describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... Quantum gravity is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... It has been suggested that Deriving the Schwarzschild solution be merged into this article or section. ... This article is in need of attention from an expert on the subject. ... Albert Einstein, photographed by Oren J. Turner in 1947. ... 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 astronomical objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... Gravity is a force of attraction that acts between bodies that have mass. ... Cosmology, as a branch of astrophysics, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. ... For a non-technical introduction to the topic, please see Introduction to Special relativity. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ...

In physics, in terms of a metric theory of gravitation, a gravitational wave is a fluctuation in the curvature of space-time which propagates as a wave. Gravitational radiation results when gravitational waves are emitted from some object or system of gravitating objects. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Curvature is the amount by which a geometric object deviates from being flat. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... A wave is a disturbance that propagates through space, often transferring energy. ...

(Gravitational waves are sometimes called gravity waves, but this term should be reserved for a completely different kind of wave encountered in hydrodynamics.) In fluid dynamics, gravity waves are those generated in a fluid medium or on an interface (e. ... Hydrodynamics is fluid dynamics applied to liquids, such as water, alcohol, oil, and blood. ...

Gravitational waves are very weak. The strongest gravitational waves we can expect to observe on Earth would be generated by very distant and ancient events in which a great deal of energy moved very violently (examples include the collision of two neutron stars, or the collision of two super massive black holes). Such a wave should cause relative changes in distance everywhere on Earth, but these changes should be on the order of at most one part in 10²¹. In the case of the arms of the LIGO gravitational wave detector, this is less than one thousandth of the "diameter" of a proton. Hence, it has proven very difficult to detect even the strongest gravitational waves. This article is about the astronomical body. ... The LIGO Hanford Control Room LIGO stands for Laser Interferometer Gravitational-Wave Observatory. ... Properties In physics, the proton (Greek proton = first) is a subatomic particle with an electric charge of one positive fundamental unit (1. ...

The existence and indeed ubiquity of gravitational waves is an unambiguous prediction of Einstein's theory of General relativity. All competing theories of gravitation currently thought to be viable (apparently in agreement to the level of accuracy with all available evidence) feature predictions about the nature of gravitational radiation. In principle, these predictions are sometimes significantly different from those of general relativity, but unfortunately, at present it seems to be sufficiently challenging simply to directly confirm the existence of gravitational radiation, much less study its detailed properties. It has been suggested that Einsteins theory of gravitation be merged into this article or section. ...

Although gravitational radiation has not yet been unambiguously and directly detected, there is already significant indirect evidence for its existence. Most notably, observations of the binary pulsar PSR B1913+16, which is thought to consist of two neutron stars orbiting rather tightly and rapidly around each other, have revealed a gradual in-spiral at exactly the rate which would be predicted by general relativity. The simplest (and almost universally accepted) explanation for these observations is that general relativity must give an accurate account of gravitational radiation in such systems. Joseph H. Taylor Jr. and Russell A. Hulse shared the Nobel Prize in Physics in 1993 for this work. PSR B1913+16 is a pulsar in a binary star system, in orbit with another star around a common center of mass. ... Joseph H. Taylor, Jr. ... 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. ... Sir Edward Appletons medal Photographs of Nobel Prize Medals. ...

Overview

In Einstein's theory of General Relativity, gravitation is, essentially, identified with spacetime curvature. In the famous slogan promulgated by John Archibald Wheeler, matter tells spacetime how to curve, and spacetime tells matter how to move. For example, humans feel the ground pressing against their feet. From the viewpoint of general relativity, this means that contact with the ground is preventing them from falling freely, thereby accelerating them. Since acceleration is identified with bending of world lines, this means that the world line of a human who is not in freefall is not a geodesic. On the other hand, far from any mass-energy, spacetime is almost perfectly flat, so geodesics behave much like straight lines in familiar solid geometry, so small objects can exhibit rectilinear inertial motion. It has been suggested that Einsteins theory of gravitation be merged into this article or section. ... John Archibald Wheeler (born July 9, 1911) is an American theoretical physicist. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension —usually put as the vertical axis. ... In physics, and specifically general relativity, geodesics are the world lines of a particle free from all external force. ...

General relativity (and similar theories of gravitation) is expressed by writing down a field equation (and possibly an equation of motion, if this is not implicit from the field equation, as it is, remarkably, in the case of general relativity). That is, these are classical relativistic field theories in which the gravitational field is at least partially identified with spacetime curvature. As a more or less inevitable consequence, in these theories, the rapid motion of mass-energy in some region will generate ripples in spacetime itself which radiate outward as gravitational waves. Indeed, this is in a sense how "field updating information" is typically conveyed from place to place. A field equation is an equation in a physical theory that describes how a fundamental force (or a combination of such forces) interacts with matter. ...

Like electromagnetic radiation, in general relativity (and many other theories), gravitational radiation travels at the speed of light and is transverse (meaning that the major effects of a gravitational wave on the motion of test particles occurs in a plane orthogonal to the direction of propagation). However (roughly speaking):

• gravitational waves represent perturbations in a second rank tensor field (and, borrowing a term from quantum field theory, are said to be spin-two),
• electromagnetic waves result from perturbations in a vector field (and are said to be spin-one).
• various other waves treated in physics result from perturbations in a scalar field (and are said to be spin-zero).

In electromagnetism, certain motions of charged particles, like electrons, will radiate electromagnetic waves. Analogously, in theories of gravitation certain motions of mass or energy will radiate gravitational waves. In the quantum field theory which arises from classical electromagnetism, called quantum electrodynamics, there is a massless particle associated with electromagnetic radiation, called the photon. Attempts to create an analogous quantum field theory for classical gravity (general relativity) led to an analogous concept, a massless particle called the graviton. However, it turns out that this route to quantizing general relativity ultimately fails, which renders the possible role of the graviton somewhat problematic in gravitational physics. Probably it is best to think of this notion as one which arises in an approximation and which has some virtues, but which has not been experimentally confirmed. Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... In physics, the photon (from Greek φως, phōs, meaning light) is the quantum of the electromagnetic field; for instance, light. ... In physics, the graviton is a hypothetical elementary particle that transmits the force of gravity in most quantum gravity systems. ...

Just as an electromagnetic wave has electric and magnetic components, so too does a gravitational wave have gravitoelectric and gravitomagnetic components. A "+" wave has an "X" gravitomagnetic component, and vice versa.

The Nature of Gravitational Waves

Gravitational waves represent fluctuations in the metric of space-time. That is, they alter the relative distance between particles. It follows that to directly detect a gravitational wave, you should in essence look for tiny relative motions between two objects. In the case of the LIGO detectors, this is essentially relative motion between two suspended mirrors, and as we saw above the motion to be detected is far smaller than the size of an atom, in fact smaller than the "size" of an atomic nucleus. Since thermal motion in each mirror is far larger than this, understanding why anyone would expect LIGO to work takes some explaining! (See LIGO.) 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. ... The LIGO Hanford Control Room LIGO stands for Laser Interferometer Gravitational-Wave Observatory. ...

Imagine a perfectly flat region of spacetime, with a group of mutually motionless test particles constrained to a single plane. Along comes a monochromatic linearly polarized gravitational wave. What happens to the test particles? Roughly speaking, they will oscillate in a cruciform manner, orthogonal to the direction of motion:

• first, East/West separated particles draw together while North/South separated particles draw apart,
• next, East/West separated particles draw apart while North/South separated particles draw together,

and so forth. (Diagonally separated particles exhibit a relative motion which is more difficult to describe verbally, but which is more or less implied by this description.) The cross-sectional of a small box of test particles is invariant under these changes, and there is negligible motion in the direction of propagation (at least, neglecting gravitomagnetic effects; that is, we are tacitly assuming that the relative motion of our test particles is not very rapid).

A monochromatic circularly polarized gravitational wave induces a similar cruciform oscillation, except the crucifix rotates with the same frequency as the above described cruciform oscillation.

Interestingly enough, after the wave has passed, there may be some residual "secular" relative motion of the test particles. There are also some interesting optical effects. If, before the wave arrives, we look through the oncoming wavefronts at objects behind these wavefronts, we can see no optical distortion (if we could, of course, we would have advance notice of its impending arrival, in violation of the principle of causality). But if, after the wave has passed by, we turn and look through the departing wavefronts at objects which the wave has not yet reached, we will see optical distortions in the images of small shapes such as galaxies. Unfortunately, this is an utterly impractical method of detecting the very weak waves we can expect to occur in the vicinity of the solar system.

Sources of Gravitational Waves

Gravitational waves are caused by certain motions of mass or energy. The type of motion required is different from electromagnetism in one very important respect however: the strongest type of electromagnetic radiation is dipole radiation, while the strongest type of gravitational radiation is quadrupole radiation. [1]

According to general relativity, the quadrupole moment (or some higher moment) of an isolated system must be time-varying in order for it to emit gravitational radiation. Here are some examples which illustrate when we should (assuming general relativity gives accurate predictions) expect a system to emit gravitational radiation: Quadrupole magnet(four-pole), focus particle beams in a particle accelerator. ...

• An isolated object in approximately "rectilinear" motion will not radiate. (Needless to say, this motion is with respect to some observer and can be only approximately rectilinear. Technically, this entails defining a weakly gravitating system possessing a time varying dipole moment but stationary quadrupole moment, with all moments being taken with respect to the origin.) This can be regarded as a consequence of the principle of conservation of linear momentum. (Caveat: this example is trickier than it looks, and in the case of a small object falling toward a large one, say, it leads to one of the most vexed questions in general relativity, the problem of treating radiation reaction).
• A spherically pulsating spherical star (nonzero and non-stationary monopole moment or mass, but vanishing and hence stationary quadrupole moment) will not radiate, in agreement with Birkhoff's theorem.
• A spinning disk (nonzero but stationary monopole and quadrupole moments) will not radiate. This can be regarded as a consequence of the principle of conservation of angular momentum. (Caveat: in general relativity, unlike Newtonian gravitation, a spinning disk will not generate an external field identical to the field of an equivalent but non-spinning disk, due to gravitomagnetic effects, but this does not contradict the absence of radiation. Roughly speaking, the field is generated as we concentrate matter, and if that matter has some angular momentum, but we end up with a stationary external gravitational field; that field will exhibit gravitomagnetism but not radiation.)
• Two objects mounted on the endpoints of an isolated extensible curtain rod, which is provided with some kind of engine and which oscillates long/short/long with frequency ω, gives a system with time-varying quadrupole moment, so this system will radiate. Observers far from the rod and in the equatorial plane of the rod will observe linearly polarized radiation (aligned with the rod) with frequency ω. Observers lying on the axis of symmetry of the rod will observe no radiation, however.
• A spinning non-axisymmetric planetoid (say with a large bump or dimple on the equator) will define a system with a time-varying quadrupole moment, so this system will radiate. As an idealization, one can study an isolated uniform mass curtain rod which is spinning with angular frequency ω about a rotation axis orthogonal to the rod, but passing through some point other than the centroid of the rod. This gives a system with time varying quadrupole moment, so the system will radiate. Observers far from the system and lying in the plane of rotation will observe linearly polarized radiation with frequency 2ω. Observers far from the system and near its axis of symmetry will observe circularly polarized radiation with frequency ω.
• Two objects orbiting each other with angular frequency ω in a quasi-Keplerian planar orbit, gives a system with time-varying quadrupole moment, so this system will radiate. Observers far from the system and in its equatorial plane will observe linearly polarized radiation (aligned with the rod) with frequency 2ω. Observers far from the system and lying on its axis of symmetry will observe circularly polarized radiation with frequency ω.

The last three examples illustrate a general rule-of-thumb: far from a radiating system, projection of the system on the "viewing plane" affords a rough and ready indication of what kind of radiation will be observed. Mass is a property of a physical object that quantifies the amount of matter it contains. ... In general relativity, Birkhoffs theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. ...

These examples (and others) are most commonly studied using a simplified version of general relativity, sometimes called linearized general relativity, which gives indistinguishable results in the case of weak gravitational fields. (The external field of our Sun would be considered "weak" in this terminology.) Similar conclusions hold for the fully nonlinear theory, but it is much more difficult to obtain analytic results outside the domain of the linearized theory. This is one reason why so much work on phenomena such as the collision and merger of two black holes currently requires numerical analysis. This article is in need of attention from an expert on the subject. ...

Gravitational radiation carries energy away from a radiating system. Consequently, in the case of the quasi-Keplerian system discussed above, the two objects will gradually spiral in towards one another, becoming more tightly bound to compensate for this loss of energy. The predicted rate of this in-spiral can also be computed, using the linearized approximation, and the result gives excellent agreement for observed binary pulsars (this is the theoretical basis for the Nobel Prize awarded to Hulse and Taylor). In the late stages of the inspiral of two neutron stars or black holes, however, the linearized theory is no longer adequate, so one must resort to more complicated approximations, and eventually to numerical simulations. Sir Edward Appletons medal Photographs of Nobel Prize Medals. ...

Similarly, in the case of the eccentric rotating rod, the frequency will decrease as the radiation gradually carries off energy from the system.

We stress that some theories of gravitation give significantly different predictions concerning the nature and generation of gravitational radiation, while others give predictions which are almost identical to those of general relativity. All currently known theories other than general relativity are either in disagreement with observation, or in some sense more complicated than general relativity (see for example Brans-Dicke theory for an example illustrating the latter possibility). In mathematical physics, the Brans-Dicke theory of gravitation (sometimes called the Jordan/Brans/Dicke theory) is a well-known competitor of Einsteins theory of general relativity. ...

If two spinning black holes were to collide, they could emit an enormous amount of gravitational radiation and lose energy in the process.

Detection

Russell Alan Hulse and Joseph Hooton Taylor Jr. were awarded the Nobel Prize in Physics in 1993 for their observations of a remarkable binary pulsar, PSR B1913+16. According to general relativity, this system should emit gravitational radiation which carries off energy at a specific rate, which should in turn cause the orbit to decay at a rate of roughly 7 mm per day. This prediction agrees with the observations of Hulse and Taylor. 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. ... 1993 (MCMXCIII) was a common year starting on Friday of the Gregorian calendar and marked the Beginning of the International Decade to Combat Racism and Racial Discrimination (1993-2003). ... PSR B1913+16 is a pulsar in a binary star system, in orbit with another star around a common center of mass. ...

But to directly detect gravitational waves you would have to look for any motion they cause. Typically you would look for the expansion and contraction oscillations caused by the gravitational wave. A simple version of this setup is called a Weber bar -- a large, solid piece of metal with electronics attached to detect any vibrations. Unfortunately, Weber bars are not likely to be sensitive enough to detect anything but very powerful gravitational waves. A more sensitive version is the Interferometer, with test masses placed as many as four kilometers apart. Ground-based interferometers such as LIGO are now coming on line. The motion to be detected would be very slight — a small fraction of the width of an atom, over a distance of four kilometers. A number of teams are working on making more sensitive and selective gravitational wave detectors and analysing their results. Space-based interferometers, such as LISA, are also being developed. 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. ... The LIGO Hanford Control Room LIGO stands for Laser Interferometer Gravitational-Wave Observatory. ... The LISA is the Laser Interferometer Space Antenna experiment. ...

One reason for the lack of direct detection so far is that the gravitational waves that we expect to be produced in nature are very weak, so that the signals for gravitational waves, if they exist, are buried under noise generated from other sources. Reportedly, ordinary terrestrial sources would be undetectable, despite their closeness, because of the great relative weakness of the gravitational force. This is a Root page. ... This article covers the physics of gravitation. ...

A commonly used technique to reduce the effects of noise is to use coincidence detection to filter out events that do not register on both detectors. There are two common types of detectors used in these experiments: Walther Bothes coincidence circuit was the first AND logic gate. ...

• laser interferometers, which use long light paths, such as GEO, LIGO, TAMA, VIRGO, ACIGA and the space-based LISA;
• resonant mass gravitational wave detectors which use large masses at very low temperatures such as AURIGA, ALLEGRO, EXPLORER and NAUTILUS.

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. Lasers range in size from microscopic diode lasers (top) with numerous applications, to football field sized neodymium glass lasers (bottom) used for inertial confinement fusion, nuclear weapons research and other physics experiments. ... 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. ... The LIGO Hanford Control Room LIGO stands for Laser Interferometer Gravitational-Wave Observatory. ... Virgo (Latin for virgin, symbol , Unicode ♍) is a constellation of the zodiac. ... The LISA is the Laser Interferometer Space Antenna experiment. ... MiniGRAIL is a gravitational wave telescope based at Leiden University, the Netherlands. ... Leiden University in the city of Leiden, is the oldest university in the Netherlands. ...

Einstein@Home

Bruce Allen of University of Wisconsin-Milwaukee's LIGO Scientific Collaboration (LSC) group is leading the development of the Einstein@Home project, developed to search data for signals coming from selected, extremely dense, rapidly rotating stars observed from LIGO in the US and the GEO 600 gravitational wave observatory in Germany . Such sources are believed to be either quark stars or neutron stars; a subclass of these stars are already observed by conventional means and are known as pulsars, electromagnetic wave-emitting celestial bodies. If some of these stars are not quite near-perfectly spherical, they should emit gravitational waves, which LIGO and GEO 600 may begin to detect. Mitchell Hall at the University of Wisconsin-Milwaukee The University of Wisconsin–Milwaukee is a public university located in Milwaukee, Wisconsin. ... A screenshot of the Einstein@home graphics. ... Wikipedia does not yet have an article with this exact name. ... This article is about the celestial body. ... Composite Optical/X-ray image of the Crab Nebula pulsar, showing surrounding nebular gases stirred by the pulsars magnetic field and radiation. ...

Einstein@Home is a small part of the LSC scientific program. It has been set up and released as a distributed computing project similar to SETI@home. That is, it relies on computer time donated by private computer users to process data generated by LIGO's and GEO 600's search for gravity waves. Distributed computing is an aspect of computer science that deals with the coordination of multiple computers in remote physical locations in order to accomplish a common objective or task. ... SETI@home (SETI at home) is a grid computing (distributed computing in the projects own terminology) project using Internet-connected computers, hosted by the Space Sciences Laboratory, at the University of California, Berkeley, in the United States. ...

Prospects

Unsolved problems in physics: Is our universe filled with gravitational radiation from the big bang? From astrophysical sources, such as inspiralling neutron stars? What can this tell us about quantum gravity and general relativity?

Scientists are eager to directly measure gravitational waves from astronomical sources, as they can probe phenomena that are difficult or impossible to study with electromagnetic radiation. For instance, although a black hole emits no visible radiation in the way that a regular star does, gravitational waves can be emitted when an object falls into a black hole, or when two black holes collide. If the inspiraling mass is significantly smaller than the central black hole, the emitted gravitational waves may, at least in some circumstances, allow physicists to directly probe the spacetime geometry around the event horizon (such observations are a primary goal of the LISA mission). Also, because gravitational waves are so weak (and thus difficult to detect), objects opaque to light are often transparent to gravitational radiation. In particular, gravitational waves could propagate while the universe was still opaque to light (i.e., at times before recombination). In this way, gravitational waves could help reveal information about the very structure of the universe. Image File history File links Question_dropshade. ... This is an incomplete list of some of the unsolved problems in physics. ... Neutron stars are one of the few possible endpoints of stellar evolution. ... Quantum gravity is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... Electromagnetic radiation can be conceptualized as a self propagating transverse oscillating wave of electric and magnetic fields. ... A black hole is a concentration of mass great enough that the force of gravity prevents anything past its event horizon from escaping it except through quantum tunnelling behaviour (known as Hawking Radiation). ... Event Horizon is a 1997 science fiction and horror film. ... The LISA is the Laser Interferometer Space Antenna experiment. ... A substance or object that is opaque is neither transparent nor translucent. ... Recombination usually denotes a genetic event that occurs during the formation of sperm and egg cells (especially in areas of study of biology topics). ... The deepest visible-light image of the cosmos, the Hubble Ultra Deep Field. ...

In contrast to electromagnetic radiation, it is not fully understood what difference the presence of gravitational radiation would make for the workings of the universe. A sufficiently strong sea of primordial gravitational radiation, with an energy density exceeding that of the big bang electromagnetic radiation by a few orders of magnitude, would shorten the life of the universe, violating existing data that show it is at least 13 billion years old. More promising is the hope to detect waves emitted by sources on astronomic size scales, such as: According to the Big Bang theory, the universe emerged from an extremely dense and hot state (bottom). ...

• supernovas or gamma ray bursts;
• "chirps" from inspiraling coalescing binary stars;
• periodic signals from spherically asymmetric neutron stars or quark stars;
• stochastic gravitational wave background sources.

Multiwavelength X-ray image of the remnant of Keplers Supernova, SN 1604. ... Optical afterglow of gamma ray burst GRB-990123 (the bright dot within the white square and in the enlarged cutout) on 23 January 1999. ... A chirp is a signal in which the frequency increases (up-chirp) or decreases (down-chirp) with time. ... A binary star system consists of two stars both orbiting around their barycenter. ... Neutron stars are one of the few possible endpoints of stellar evolution. ... A strange star or quark star is a hypothetical type of star composed of strange matter. ... Stochastic, from the Greek stochos or goal, means of, relating to, or characterized by conjecture; conjectural; random. ... A possible target of gravitational wave detection experiments is a stochastic background of gravitational waves of cosmological origin. ...

Derivation

Perturbation of Flat Space-time

Consider that the full metric g is nearly the flat metric η plus some small perturbation h.

g_μν = η_μν + h_μν

The Einstein equation in vacuum is

Where R is the Ricci curvature. We will expand R in perturbatively in powers of h. In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ...

The zeroth order term can only be a function of the flat metric and therefore is identically zero. As the perturbation is to be small, we will solve only for the first order term and ignore all higher orders.

Where δR_μν is the deviation from the flat (and thus zero) Ricci curvature that depends linearly on the perturbation h.

Now we need the formula for the Ricci curvature.

Where Γ are the Christoffel symbols and is shorthand for . Only first two terms which are linear in Γ will contribute to the first order correction. In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

Next we need the formula for the Christoffel symbols.

Seeing as the flat metric is constant, the only first order terms will involve derivatives of the perturbation.

The linearized Einstien equation now becomes

Where V_α substitutes the expression and is the d'Alembertian or 4-Laplacian. Raising and lowering indices can be tricky. To first order you only use the flat metric. Also note the inverse metric has a negative perturbation plus higher order terms.

Next we choose a particular coordinate system where V_α is identically zero. Some proof is necessary to make sure this is possible, but it is. We are left with a wave equation and our gauge condition.

From experience with simpler wave equations we can guess the general form of the solution.

Where is a null vector. The wave equation is now satisfied, but what choices of A will satisfy the gauge condition we used.

If we don't want transformations to disturb our choice of gauge, then we better make the wave traceless, , and transverse, .

For a wave traveling in the z direction, k = (1,0,0,1), the perturbation will take the following form.

Thus the oscillations are transverse spacial distortions. The wave is called spin-2 because there are 2 different polarizations. Light only has one! A ₊ is called the plus polarization and is called the cross polarization.

Perturbation with Sources

The Einstein equation is the relationship between space-time curvature and the matter or source of that curvature.

Our first order contributions to the curvature have previously been determined.

We stick to the choice of Lorentz gauge, which will now be written in a very suggestive form.

The right hand side is the divergence of the trace-reversed h_αβ. The traced reversed perturbation will be abbreviated as from now on.

We can now combine these equations into the linearized Einstein equation.

This is a long solved problem from electricity and magnetism analogous to electromagnetic waves with sources. It is solved via retarded Green's functions.

Where is the retarded time.

Far from Source Approximation

If we want to study metric perturbations far from the source then we can envoke a very useful approximation.

Where r is the approximate distance to the source.

We now invoke the local conservation of energy-momentum (to first order) to find useful interrelationships in the stress-energy tensor.

We now take this relationship and massage it into the form of our original integral and see what new information it gives us.

We wanted to multiply the right hand side with the two powers of x so that we can integrate by parts twice and get down to a regular volume integral.

Assuming the stress-energy tensor takes the simple form

T_αβ = ρu_αu_β

Where ρ is the mass density and u_α is the 4-velocity. If the source is nonrelativistic, then the energy density will be dominated by the mass density, T^tt = ρ

Here we see something very similar to the moment of inertia, we call it the second mass moment.

We now have our final expression that relates the gravitational waves with their source.

Perturbative versus Exact

Gravitational waves differ markedly from electromagnetic waves in that electromagnetic waves can be derived exactly from Maxwell's equations whereas gravitational waves, interpreted as linear spin-2 waves, are only perturbations to certain space-time geometries. In other words, classically there are always linear, spin-1 E&M waves, but there are never linear, spin-2 gravitational waves. There are still wave-like fluctuations, but in general things are nonlinear, as is always the case in general relativity. This provides one argument against the existence of gravitons. It has been suggested that Einsteins theory of gravitation be merged into this article or section. ...

Gravitational waves transmit energy

Within parts of the scientific community there was initially some confusion as to whether gravitational waves could transmit energy as electromagnetic waves can. This confusion came from the fact that gravitational waves have no local energy density - no contribution to the stress-energy tensor. Unlike Newtonian gravity, Einstein gravity is not a force theory. Gravity is not a force in general relativity; it is geometry. Therefore the gravitational field was thought not to contain energy, as would a gravitational potential. But the field can most certainly carry energy as it can do mechanical work at a distance. And this has been proven using stress-energy pseudo tensors that transport energy as well as seeing how radiation can carry energy out to infinity.^{[citation needed]}

See also

• Gravitoelectromagnetism
• LIGO, an American gravitational wave detector.
• VIRGO and GEO 600, two European detectors.
• TAMA, a Japanese detector.
• LISA the proposed Laser Interferometer Space Antenna to be launched in 2015.
• Sticky bead argument, for Feynman's way to see that gravitational radiation should carry energy.
• pp-wave spacetime, for an important class of exact solutions modeling gravitational radiation (possibly accompanied by electromagnetic or other radiation) with planar wavefronts.

This article is in need of attention from an expert on the subject. ... The LIGO Hanford Control Room LIGO stands for Laser Interferometer Gravitational-Wave Observatory. ... Virgo (Latin for virgin, symbol , Unicode ♍) is a constellation of the zodiac. ... Geo 600 is a gravitational wave detector located in Hannover, Germany. ... The LISA is the Laser Interferometer Space Antenna experiment. ... 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. ... // Introduction In general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutions of Einsteins field equation. ...

External links

• Laser Interferometer Gravitational Wave Observatory. LIGO Laboratory, California Institute of Technology.
• Columbia Experimental Gravity
• Einstein's Messengers - The LIGO Movie by NSF
• Discussion of gravitational radiation on the USENET physics FAQ
• Table of gravitational wave detectors
• Info page for "Einstein@Home," a distributed computing project processing raw data from LIGO Laboratory, at CalTech searching for gravity waves
• Home page for Einstein@Home project
• The Italian researchers' paper analyzing data from EXPLORER and NAUTILUS
• Center for Gravitational Wave Physics. National Science Foundation [PHY 01- 14375].
• Australian International Gravitational Research Center. University of Western Australia.
• TAMA project. Developing advanced techniques for km-sized interferometer.
• Could superconductors transmute electromagnetic radiation into gravitational waves? -- Scientific American article
• Science to ride gravitational waves, BBC news (Nov 2005 announcement of science run of LIGO and GEO 600 gravitational wave detectors).
• Black hole mergers modelled in 3D, BBC news (April 2006 announcement of simulations on a supercomputer that has allowed to understand the pattern of gravitational waves produced by merging black holes).
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The California Institute of Technology (commonly referred to as Caltech) is a private, coeducational university located in Pasadena, California, in the United States. ... NSF is an abbreviation. ... The University of Western Australia (UWA) is Western Australias oldest university, established in February 1911. ... A kilometre (American spelling: kilometer), symbol: km is a unit of length in the metric system equal to 1000 metres (from the Greek words χίλια (khilia) = thousand and μέτρο (metro) = count/measure). ... 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. ... The LIGO Hanford Control Room LIGO stands for Laser Interferometer Gravitational-Wave Observatory. ... Geo 600 is a gravitational wave detector located in Hannover, Germany. ...

References

• B. Allen, et al., Observational Limit on Gravitational Waves from Binary Neutron Stars in the Galaxy. The American Physical Society, March 31, 1999.
• Davis, Warren F.,Gravitational Radiation. Davis Associates, Inc., Newton, MA
• Amos, Jonathan, Gravity wave detector all set. BBC, February 28, 2003.
• Rickyjames, Doing the (Gravity) Wave. SciScoop, December 8, 2003.
• Will, Clifford M., The Confrontation between General Relativity and Experiment. Living Rev. Relativity 9 (2006) 3.
• Chakrabarty, Indrajit, "Gravitational Waves: An Introduction". arXiv:physics/9908041 v1, Aug 21, 1999.