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General relativity (GR) (aka general theory of relativity (GTR)) is the Geometry theory of gravitation published by Albert Einstein in 1915/16. and . It unifies special relativity, Newton's law of universal gravitation, and the insight that gravitational acceleration can be described by the curvature of space and time. General relativity further calls for the curvature of space-time to be produced by the mass-energy and momentum content of the matter in space-time. General relativity is distinguished from other metric :Category:Theories of gravitation by its use of the Einstein field equations to relate space-time content and space-time curvature.

In the mathematics of general relativity, the Einstein field equations are a system of partial differential equations whose solution represents the metric tensor (general relativity) (or the metric) of space-time, describing its "shape". Some important solutions of the Einstein field equations are the Schwarzschild solution (for the space-time surrounding a spherically symmetric uncharged and non-rotating massive object), the Reissner-Nordström black hole (for a charged spherically symmetric massive object), and the Kerr metric (for a rotating massive object). An object moving inertially in a gravitational field follows a geodesic (general relativity) that may be found using the Christoffel symbols of the metric.

General relativity is currently the most successful gravitational theory, being almost universally accepted and well-supported by observations. The first success of general relativity was in explaining the Tests of general relativity#Perihelion precession of Mercury of Mercury (planet). Then in 1919, Sir Arthur Stanley Eddington announced that observations of stars near the eclipsed Sun confirmed general relativity's prediction that massive objects bend light. Since then, other tests of general relativity have confirmed many of the #Predictions, including gravitational time dilation, the gravitational redshift of light, Shapiro delay, and gravitational radiation. In addition, numerous observations are interpreted as confirming one of general relativity's most mysterious and exotic predictions, the existence of black holes.



Justification (right)The justification for creating general relativity came from the equivalence principle, which dictates that free-falling observers are the ones in inertial motion. Roughly speaking, the principle states that the most obvious effect of gravity – things falling down – can be eliminated by making the transition to a reference frame that is in free fall, and that in such a reference frame, the laws of physics will be approximately the same as in special relativity.While the equivalence principle is still part of modern expositions of general relativity, there are some differences between the modern version and Einstein's original concept, cf. Norton 1985. A consequence of this insight is that inertial observers can accelerate with respect to each other. For example, a person in free fall in an elevator whose cable has been cut will experience weightlessness: objects will either float alongside him or her, or drift at constant speed. In this way, the experiences of an observer in free fall will be very similar to those of an observer in deep space, far away from any source of gravity, and in fact to those of the privileged ("inertial") observers in Einstein's theory of special relativity.This is described in detail in chapter 2 of Wheeler 1990. Albert Einstein realized that the close connection between weightlessness and special relativity represented a fundamental property of gravity.

Einstein's key insight was that there is no fundamental difference between the constant pull of gravity we know from everyday experience and the fictitious forces felt by an accelerating observer (in the language of physics: an observer in a non-inertial reference frame).E. g. Janssen (2005), p. 64f. Einstein himself also explains this in section XX of his non-technical book Einstein 1961. Following earlier ideas by Ernst Mach, Einstein also explored centrifugal forces and their gravitational analogue, cf. Stachel 1989. So what people standing on the surface of the Earth perceive as the 'force of gravity' is a result of their undergoing a continuous physical acceleration which could just as easily be imitated by placing an observer within a rocket accelerating at the same rate as gravity (9.81 metre per second squared).

This redefinition is incompatible with Newton's first law of motion, and cannot be accounted for in the Euclidean geometry of special relativity. To quote Einstein himself:Thus the equivalence principle led Einstein to develop a gravitational theory which involves curved space-times. Paraphrasing John Archibald Wheeler, Einstein's geometric theory of gravity can be summarized thus: spacetime tells matter how to move; matter tells spacetime how to curve.E.g. p. xi in Wheeler 1990.

Another motivating factor was the realization that relativity calls for the gravitational potential to be expressed as a symmetric rank-two tensor, and not just a scalar field as was the case in Newtonian physics (An analogy is the electromagnetic four-potential of special relativity). Thus, Einstein sought a rank-two tensor means of describing curved space-times surrounding massive objects. This effort came to fruition with the discovery of the Einstein field equations in 1915.

Fundamental principles , this (curved) geometry being interpreted as gravity. Note that the white lines do not represent the curvature of space, but instead represent the coordinate system imposed on the curved spacetime which would be rectilinear in a flat spacetimeGeneral relativity is a metric theory of gravitation. For this class of theory, the main defining feature is the concept of gravitational 'force' being replaced by spacetime geometry. Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories) are taken in general relativity to represent inertial motion within a curvature of spacetime.

General relativity (and all other metric theories of gravitation) are predicated upon several underlying assumptions. The general principle of relativity states that the laws of physics must be the same for all observers (accelerated or not). The principle of general covariance states the laws of physics must take the same form in all coordinate systems. General relativity also requires equivalence between inertial and geodesic (general relativity) because the world lines of particles unaffected by physical forces are timelike or null geodesics of spacetime. The principle of local Lorentz invariance requires that the laws of special relativity apply locally for all inertial observers. Finally there is the principle that the curvature of spacetime and its energy-momentum content are related. (As mentioned above, this relationship between curvature and spacetime content is specifically dictated by the Einstein field equations in general relativity.)

The equivalence principle, which was the starting point for the history of general relativity, ended up being a consequence of the general principle of relativity and the principle that inertial motion is geodesic motion.

Mathematical framework The requirements of the mathematics of general relativity are further modified by the other principles. Local Lorentz Invariance requires that the manifolds described in GR be 4-dimensional and Lorentzian instead of Riemannian manifold. In addition, the principle of general covariance requires that mathematics to be expressed using tensor calculus. Tensor calculus permits a manifold as mapped with a coordinate system to be equipped with a metric tensor (general relativity) which describes the incremental (spacetime) intervals between coordinates from which both the geodesic equations of motion and the curvature tensor of the spacetime can be ascertained.

Geometry Due to the expectation that spacetime is curved, non-Euclidean geometry must be used. (In particular, the geometry is described by a Pseudo-Riemannian manifold metric, or more specifically still, a Lorentzian metric.) In essence, spacetime does not adhere to the "common sense" rules of Euclidean geometry, but instead objects that were initially traveling in parallel paths through spacetime (meaning that their velocities do not differ to first order in their separation) come to travel in a non-parallel fashion. This effect is called geodesic deviation, and it is used in general relativity as an alternative to gravity. For example, two people on the Earth heading due north from different positions on the equator are initially traveling on parallel paths, yet at the north pole those paths will cross. Similarly, two balls initially at rest with respect to and above the surface of the Earth (which are parallel paths by virtue of being at rest with respect to each other) come to have a converging component of relative velocity as both accelerate towards the center of the Earth due to their subsequent free-fall.

The curvature of spacetime (caused by the presence of stress-energy) can be viewed intuitively in the following way. Placing a heavy object such as a bowling ball on a trampoline will produce a 'dent' in the trampoline. This is analogous to a large mass such as the Earth causing the local spacetime geometry to curve. This is represented by the image at the top of this article. The larger the mass, the bigger the amount of curvature. A relatively light object placed in the vicinity of the 'dent', such as a ping-pong ball, will accelerate towards the bowling ball in a manner governed by the 'dent'. Firing the ping-pong ball at some suitable combination of direction and speed towards the 'dent' will result in the ping-pong ball 'orbiting' the bowling ball. This is analogous to the Moon orbiting the Earth, for example.

Similarly, in general relativity massive objects do not directly impart a force on other massive objects as hypothesized in Newton's action at a distance idea. Instead (in a manner analogous to the ping-pong ball's response to the bowling ball's dent rather than the bowling ball itself), other massive objects respond to how the first massive object curves spacetime. Notice that the most important part of the curvature near a massive object is in the plane defined by the time and radial directions, although there is also some purely spatial curvature.

Coordinate vs. physical acceleration One of the greatest sources of confusion about general relativity comes from the need to distinguish between coordinate and physical accelerations.

In classical mechanics, space is preferentially mapped with a Cartesian coordinate system. Inertial motion then occurs as one moves through this space at a constant coordinate rate with respect to time. Any change in this rate of progression must be due to a force, and therefore a physical and coordinate acceleration were in classical mechanics one and the same. It is important to note that in special relativity that same kind of Cartesian coordinate system was used, with time being added as a fourth dimension and defined for an observer using the Einstein synchronisation. As a result, physical and coordinate acceleration correspond in special relativity too, although their magnitudes may vary.

In general relativity, we abandon the unwarranted assumption that nature has provided us with a preferred set of coordinates. Instead an observer may choose a set of coordinates at his own convenience, and we only require that coordinates of a point in different coordinate systems can be expressed into each other through some smooth functional dependence. Only statements that do not depend on the arbitrary choice of a coordinate system by the observer (i.e. the description of a physical phenomenon) can be considered of physical relevance. This is the principle of general covariance of physical laws.It means for example that a quantity like acceleration, cannot simply be described asthe second derivative of the coordinate functions of a velocity, because a non zero "coordinate acceleration" may merely be an artifact of the choice of coordinates. In fact such an artifact already occurs for the description in polar coordinates of a uniformly moving particle not passing through the (chosen!) origin. In fact the definition of acceleration of a particle requires that we know how to subtract velocities measured at two different points along its track in space time. Equivalently we must be able to define in an observer and coordinate invariant way which velocity vectors are constant along such a path in space time. This is called parallel transport. It does not come for free but requires additional structure of space-time, a connection. It so happens that if there is defined a "length" of all velocity vectors (which may be negative), a Lorentzian metric , there is a natural connection , the Levi Civita connection, uniquely determined by requiring that parallel velocity vectors have constant "length" and the technical assumption of zero torsion.It was one of Einsteins great insights that this description can be applied to describe the influence of gravity. In general relativity, gravity is seen as a consequenceof the fact that parallel transport of a velocity vector may depend on the path through space time, not only on its endpoints. How metric and thereby connection and parallel transport, are determined by the transport of energy-momentum in space time by matter and radiation described with the so-called stress-energy tensor is the content of the theory of general relativity.

Einstein field equations The Einstein field equations (EFE) describe how stress-energy causes curvature of spacetime and are usually written in tensor form (using abstract index notation) as

G_{ab} = \kappa\, T_{ab}

where Gab is the Einstein tensor, Tab is the stress-energy tensor and \kappa is a constant. The Einstein tensor is related to the curvature of space-time and is a function only of the metric tensor and its first and second derivatives. The stress energy tensor, which is the source of the gravitational field, includes stress (pressure and shear), the density of momentum, and the density of energy including the energy of mass (the source for Newtonian gravity). The tensors Gab and Tab are both rank-2 symmetric tensors, that is, they can each be thought of as 4×4 matrices, each of which contains 10 independent terms.

The EFE reduce to Newton's law of gravity in the Correspondence principle#Other uses of the term of a weak-field approximation and slow-motion approximation relative to the speed of light. In fact, the value of \kappa in the EFE is determined to be \kappa = 8 \pi G / c^4 \ by making these two approximations.

The solutions of the Einstein field equations are metric tensor (general relativity). These metrics describe the structure of spacetime given the stress-energy and coordinate mapping used to obtain that solution. Being non-linear differential equations, the EFE often defy attempts to obtain an exact solutions in general relativity; however, many such solutions are known.

The EFE are the identifying feature of general relativity. Other theories built out of the same premises include additional rules and/or constraints. The result almost invariably is a theory with different field equations (such as Brans-Dicke theory, teleparallelism, Rosen's bimetric theory, and Einstein-Cartan theory).

Consequences of Einstein's theory General relativity, as laid out in the previous section, has a number of consequences; some follow directly from the theory's axioms, others have only become clear in the course of the ninety years of research that followed Einstein's initial publication.

Gravitational time dilation and frequency shift In general relativity (and, in fact, in any theory in which the equivalence principle holdsCf. and . In fact, Einstein derived these effects using the equivalence principle as early as 1907, cf. and the description in .), gravity has an immediate influence on the passage of time. Imagine two observers Alice and Bob, both of which are at rest in a stationary gravitational field, with Alice closer to the source of gravity ("deeper in the gravity well") and Bob at a greater distance. Then for light sent from Alice to Bob or vice versa, Bob will measure a lower frequency than Alice: light sent down into a gravity well is blue-shifted, light climbing out of a gravity well is redshifted. Also, Alice's clocks tick more slowly than Bob's: whenever the two are compared (either by sending light signals back and forth, or by slowly transporting clocks from one location to the other), the result will be that Bob's clocks are running faster. This effect is not restricted to clocks, but applies to all processes (the rate at which Alice and Bob age, cook five-minute eggs, or play Chopin's Minute Waltz); it is known as gravitational time dilation.; ..

The gravitational redshift was first measured in 1959 in a laboratory experiment by Pound-Rebka experimentSee , ; ; a list of further experiments is given in . and later confirmed by astronomical observations.E.g. ; the most recent and most accurate Sirius B measurements are published in . There are numerous direct measurements of gravitational time dilation using atomic clocksStarting with the Hafele-Keating experiment, and , and culminating in the Gravity Probe A experiment; an overview of experiments can be found in . while ongoing validation is provided as a side-effect of the operation of the Global Positioning System (GPS).GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistc effects, see and . Tests in stronger gravitational fields are provided by the observation of binary pulsars.Reviews are given in and . All results are in agreement with general relativity;General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; . however, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.Cf. .

Light deflection and gravitational time delay

In general relativity, light follows a special variety of straightest-possible world-line, so-called light-like or null geodesics – a generalization of the straight lines along which light travels in classical physics, and the invariance of lightspeed in special relativity.The fact that light follows null geodesics is not an independent axiom; it can be derived from Einstein's equations and the Maxwell Lagrangian using a WKB approximation, cf. . As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the Post-Newtonian expansion),A brief descriptions and pointers to the literature can be found in . several effects of gravity on the propagation of light emerge.

The best-known is the bending of light in a gravitational field: light passing a massive body is deflected towards that body. While such an effect can also be derived by extending the universality of free fall to light,See ; for the historical examples, ; in fact, Einstein published one such derivation as . Such calculations tacitly assume that the geometry of space is Euclidean, cf. . the maximal angle of deflection resulting from such heuristic calculations is only half the value given by general relativity; from the standpoint of Einstein's theory they take into account the effect of gravity on time, but not its consequences for the warping of space.E.g. . An important example of this is starlight being deflected as it passes the Sun; in consequence, the positions of stars observed in the Sun's vicinity during a solar eclipse appear shifted by up to 1.75 arc seconds. This effect was first measured by a British expedition directed by Arthur Eddington, and confirmed with significantly higher accuracy by subsequent measurements.Cf. ; for an overview of more recent measurements, see . The most precise direct modern observations measure the deflection of the light of distant quasars by the Sun, cf. .

Closely related to the bending of light is the gravitational time delay, also known as the Shapiro effect: light signals take longer to move through a gravitational field than they would in the absence of the gravitational field. This effect was discovered through the observations of radar signals sent from Earth to planets such as Venus or Mercury (planet) and thence reflected back;; a pedagogical introduction can be found in . later, much more accurate measurements utilized signals sent to space probes and sent back using active transponders.The most recent measurements are ; for an overview, see . In both the case of the planets and the probes, what was measured was the propagation of signals in the Sun's gravitational field. More recent measurements have detected the Shapiro effect in signals sent by a pulsar that is part of a binary system; in that case, the gravitational field causing the time delay is that of the other pulsar.Cf. . In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay are used to determine a parameter called \gamma that reflects the influence of gravity on the geometry of space..

Gravitational waves There are several analogies between weak-field gravity and electromagnetism. One is that, for electromagnetic waves, there are corresponding gravitational waves: ripples in spacetime that propagate at the speed of light.For an overview, see . Note, however that for gravitational waves, the dominant contribution is not the dipole, but the quadrupole cf. .

The simplest variety of gravitational wave can be visualized via their action on a ring of freely floating particles (see first image to the right). As a simple sine wave propagates through such a ring from out of the page towards the reader, the ring is distorted in a characteristic, rhythmic fashion (see second image to the right).Any textbook on general relativity will contain a description of these properties, e.g. . Such linearized gravitational waves are important when it comes to describing the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in distances increasing and decreasing by 10^{-21} or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposition.For example . It is, however, important to note that the linearized waves are only approximations. Generically, the non-linearity of the Einstein equations means that there will no be linear superposition for gravitational waves. Describing such more general waves is not an easy task. There are some exact solutions describing gravitational waves, for instance a wave train traveling through empty space. or so-called Gowdy universes, varieties of an expanding cosmos filled with gravitational waves,See , . while, when it comes to describing the gravitational waves produced in astrophysically relevant situations such as the merger of two black holes, numerical relativity are presently the only way to construct appropriate models.See for a brief introduction to the methods of numerical relativity, and for the connection with gravitational wave astronomy.

Orbital effects and the relativity of direction General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. The most striking one are the relativistic apsis shifts, orbital decay caused by the emission of gravitational waves, and effects that are due to the relativity of direction.

Precession of apsides In general relativity, the Apsis of orbits (the points of an orbiting body closest approach to the system's center of mass) will precess – the orbit is not an ellipse, but akin to an ellipse that rotates on its focus, resulting in a rosette-like shape (see image). Einstein himself derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body like a test particle; the result can also be obtained by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)See . or the much more general post-Newtonian formalism.See . The effect is due both to the influence of gravity on the geometry of space and to the way that self energy contributes to a body's gravity (in other words, the special kind of nonlinearity exhibited by Einstein's theory).In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms \beta and \gamma, cf. and .

An early success of general relativity was that the theory offered a straightforward explanation for an Tests of general relativity#Perihelion precession of Mercury of the planet Mercury (planet), which had been discovered by Urbain Le Verrier in 1859 but had remained mysterious.See and . This agreement between theory and experiment confirmed for Einstein that he had at last identified the correct form of the Einstein field equations. More recent observations have shown that the field equations predict the correct anomalous perihelion shift for all planets where this can be measured accurately (Mercury (planet), Venus (planet) and the Earth).The most precise measurements are VLBI measurements of planetary positions; see , , ; for an overview, .The effect has also been checked in binary pulsar systems where it is larger by five order of magnitude.See .

Orbital decay According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the solar system or for ordinary double stars, the effect is too small to be observable. Not so for a close binary pulsar, a system of two orbiting neutron stars, one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period; since the neutron stars are very compact, significant amounts of energy are emitted in the form of gravitational radiation.See and ; an accessible account can be found in .

The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Russell Alan Hulse and Joseph Hooton Taylor Jr. using binary pulsar PSR1913+16 they had discovered in 1974; it amounts to the first indirect detection of gravitational waves, rewarded with the Nobel Prize in physics in 1993.An overview can be found in ; for the pulsar discovery, see ; for the initial evidence for gravitational radiation, see . Since then, several other binary pulsars have been found, the most spectacular find being the double pulsar PSR J0737-3039 in which both stars are pulsars.Cf. .

Geodetic precession and frame-dragging Several relativistic effects are directly related to the relativity of direction.See e.g. , . One is geodetic effect: for a gyroscope in free fall in curved spacetime, the direction of its axis will change when compared, for instance, with the direction of light received from distant stars – even though its motion comes closest to keeping its axis direction constant ("parallel transport").See , . For the Moon-Earth-system, this effect has been measured with the help of lunar laser ranging;See and, for a more recent review, . more recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 1 percent.See .

Near a rotating mass, there are so-called gravitomagnetic or frame-dragging effects: for a distant observer, it will seem that objects close to the mass gets "dragged around"; this is most extreme for Kerr solution where, for an object entering a zone known as the ergosphere, rotation is inevitable.E.g. , Such effects can again be tested through their influence on the orientation of a gyroscope in free fall:E.g. , ; for a more recent review, see . somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction;E.g. , ; see the entry frame-dragging for an account of the debate. a precision measurement is the main aim of the Gravity Probe B mission, whose results are due in late 2007.A mission description can be found in ; a first post-flight evaluation is given in ; further updates will be available on the mission website .

Astrophysical applications Gravitational lensing : four images of the same astronomical object, produced by a gravitational lensThe deflection of light by gravity can have an intriguing side effect: if there is a massive object between the observer and a distant target object, it is possible for the observer to see multiple distorted images of the target. This and similar effects are known as gravitational lensingFor overviews of gravitational lensing and its applications, see and . and, depending on the configuration, scale, and mass distribution, it can result in two images, a bright ring known as an Einstein ring, or partial rings called arcs.For a simple derivation, see ; cf. .The Twin Quasar was discovered in 1979;See . since then, more than a hundred gravitational lenses have been observed.Images of all the known lenses can be found on the pages of the CASTLES project, . Images too close to be resolved can still lead to a measurable effect, namely an overall brightening of a given star or other point-like object; a number of such "microlensing events" has been observed, as well.For an overview, see .

Gravitational lensing has developed into a tool of observational astronomy. Notably, it is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data are also used to understand the structural evolution of galaxy.See .

Gravitational wave astronomy From observations of binary pulsars, there is strong indirect evidence for the existence of gravitational wave (see the section on General relativity#Orbital decay, above). However, as of yet, gravitational waves reaching us from the depths of the cosmos have not been detected directly – this is one of the major goals of current relativity-related research.For an overview, ; accessible accounts can be found in and . To this end, a number of land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (three detectors), TAMA 300 and VIRGO.An overview is given in .A joint US-European mission to launch a space-based detector, LISA (astronomy), is currently under development,See . with a precursor mission (LISA Pathfinder) due for launch in late 2009.See .

Gravitational waves promise to yield information about astronomical objects that is inaccessible by observations using electromagnetic radiation:Cf. . Terrestrial detectors are expected to yield new information about inspiral phase and mergers of binary stellar black hole and binaries consisting of one such black hole and a neutron star (of interest as a candidate mechanism for gamma ray bursts); they could also detect signals from core-collapse supernovae and from periodic sources such as rotating neutron stars with small deformation. If there is truth to speculation about certain kinds of phase transitions or kink bursts from long cosmic strings in the very early universe (at cosmic times around 10^{-25} seconds) these could also be detectable.See . Space-based detectors like LISA should detect objects such as binaries consisting of two White Dwarfs, and AM CVn stars (a White Dwarf accreting matter from its binary partner, a low-mass helium star), and also observe the mergers of supermassive black holes and the inspiral of smaller objects (between one and a thousand solar masses) into such black holes. LISA should also be able to listen to the same kind of sources from the early universe as ground-based detectors, but at even lower frequencies and with greatly increased sensitivity.See .

Black holes and other compact objects Whenever an object becomes sufficiently compact, general relativity predicts the formation of a black hole: a region of space from which nothing, not even light, can escape. In the currently accepted models of stellar evolution, neutron stars with around 1.4 solar mass and so-called stellar black holes with a few to a few dozen solar masses are thought to be the final state for the evolution of massive stars.See . Supermassive black holes with between a few million and a few 1000000000 (number) solar masses are now thought to be the rule rather than the exception in the centers of galaxies,E.g. . and their presence is thought to have played an important role in the formation of galaxies and larger cosmic structures.Cf. and the accompanying summary .

From an astronomical point of view, the most important property of compact objects such as black holes is that they provide a superbly efficient mechanism for converting gravitational into radiation energy.Cf. , Accretion, that is, the falling of material such as gas or dust onto stellar black hole or supermassive black hole black holes is thought to be responsible for some of the most spectacularly luminous astronomical objects, notably diverse kinds of Active Galactic Nucleus on galactic scales, and stellar-size objects such as Microquasars;For the basic mechanism, see ; for more about the different types of astronomical objects associated with this, cf. . in particular, it can lead to relativistic jets: focused beams of highly energetic particles that are being flung into space at almost the speed of light.For a review, see . For modelling all these phenomena, general relativity plays a central role,For stellar end states, cf. or, for more recent numerical work, ; for supernovae, there are still major problems to be solved, cf. ; for simulating accretion and the formation of jets, cf. . and relativistic lensing effects are thought to play a role for the signals received from X-ray pulsars.Cf. .

Limits on compactness from the observation of accretion-driven phenomena ("Eddington luminosity")See . observations of stellar dynamics in the center of our own Milky Way galaxy,Cf. . and indications that at least some of the compact objects in question appear to have no solid surfaceExamination of X-ray bursts for which the central compact object is either a neutron star or a black hole; cf. and, for an overview, . are strong indirect evidence for the existence of black holes; more direct evidence such as observing the "shadow" of the horizon of the Milky Way galaxy's central black hole.Cf. . is eagerly sought for.

Black holes are also sought-after targets in the search for gravitational waves (see the section General_relativity#Gravitational_waves, above): merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and reliable simulations of such mergers are one of the main goals of current research in numerical relativity;Cf. . the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events, and hence as a probe of cosmic expansion at large distances;Cf. . the gravitational waves produced as a stellar black hole plunges into a supermassive one should serve as a probe of the supermassive black hole's geometry.E.g. .

Cosmology Each solution of Einstein's equations describes a whole universe, so it should come as no surprise that there are solutions that provide useful models for cosmology, the study of the universe as a whole. The current models are based on an extension of the original form of Einstein's equations which include the cosmological constant \Lambda, an additional term that has an important influence on the large-scale dynamics of the cosmos, G_{ab} + \Lambda\ g_{ab} = \kappa\, T_{ab} where gab is the metric tensor (general relativity).Originally ; cf. the description in .

On the basis of isotropic and homogeneous solutions of these enhanced equations, the so-called Friedmann-Lemaître-Robertson-Walker metric,See . are built the Physical cosmology in which the universe has evolved over the past 14 1000000000 (number) years from a hot, early Big bang phase.See ; use of these models is justified by the fact that, at large scales of around hundred million light-years and more, our own universe indeed appears to be isotropic and homogeneous, cf. . Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,E.g. with WMAP data, see . further observational data can be used to put the models to the test: successful predictions include the initial abundance of chemical elements formed in a period of Big bang nucleosynthesis,See ; for a recent account of predictions, see ; an accessible account can be found in . which is in good agreement with astronomical observations;See , , , and . the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation,Cf. and, for a pedagogical introduction, see ; for the initial detection, see , andfor precision measurements by satellite observatories see (COBE) and (WMAP). and the large-scale distribution of galaxies.A review can be found in .

The status of the resulting models is mixed. On the one hand, the standard models of cosmology have been very successful: to date, they have passed all observational tests,See, e.g., fig. 2 in . and they have proven a sound basis to explaining the evolution of the universe's large-scale structure.For a review, see ; more recent results can be found in . On the other hand, there are a number of important open questions. The determination of cosmological parameters (in line with other astronomical observationsThese additional observations involve the dynamics of galaxies and galaxy clusters cf. chapter 18 of , evidence from gravitational lensing, cf. , and simulations of large-scale structure formation, see .) suggests that about 90 percent of all matter in the universe is in the form of so-called dark matter, which has mass (and hence gravitational influence), but does not interact electromagnetically (and hence cannot be observed directly); there is currently no generally accepted description of this new kind of matter within the framework of particle physicsSee , and ; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual elementary particles ("non-baryonic matter"), cf. . or otherwise.Namely, some physicists have questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity cf. the overview in . A similar open question is that of dark energy. Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual equation of state, namely dark energy;See ; an accessible overview is given in . the nature of this new form of energy remains unclear.Here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models, cf. ; aforementioned modifications need not be modifications of general relativity, they could, for example, be modifications in the way we treat the inhomogeneities in the universe, cf. .

A number of further problems of the classical cosmological models (such as "why is the cosmic background radiation so highly homogeneous")More precisely, these are the flatness problem, the horizon problem, and the monopole problem; a pedagogical introduction can be found in , see also . have led to the introduction of an additional phase of strongly accelerated expansion at cosmic times of around 10^{-} seconds, known as an cosmic inflation.A good introduction is ; for a more recent review, see . While recent measurements of the cosmic background radiation have resulted in first evidence for this scenario,See . problems remain. The re is a bewildering variety of possible inflationary scenarios not restricted by current observations.More concretely, the potential function that is crucial in determining the dynamics of the inflaton is simply postulated, but not derived from an underlying physical theory. Also, the question remains what happened in the earliest universe, close to where the classical models predict the big bang singularity; an authoritative answer would require a complete theory of quantum gravity, which does not exist at the momentSee . (cf. the section #, below).

Advanced concepts Causal structure and global geometry universeIn general relativity, no material body can catch up with or over take a light pulse; no influence from an event A can reach any other location before light sent out at A does so. Hence an exploration of all light worldlines (Geodesic (general relativity)s) yields key information about the spacetime's causal structure. This structure can be displayed using so-called Penrose diagram in which infinitely large regions of space and infinite time intervals are shrunk ("compactification") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.See , , and

Aware of the importance of causal structure, Roger Penrose and others developed important techniques that are nowadays known as global geometry. In global geometry, the object of study is not one particular Solutions of the Einstein field equations (or family of solutions) to Einstein's equations; rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, are utilized in conjunction with non-specific assumptions about the nature of matter (usually in the form of so-called energy conditions) to derive general results.E.g. and

Cosmic segregation: Horizons One of the most striking conclusions that can be drawn from studies of global geometry is the existence of boundaries called event horizon, which segregate one spacetime region from the rest of the world. The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space, one can define a surface that separates the inside from the outside world. No light from the inside can escape to the outside, and since, in general relativity, no object can overtake a light pulse, all inside matter is imprisoned, as well. The resulting object is known as a black hole, and the surface in question as the black hole's horizon.For an account of the evolution of this concept, see . A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and apparent horizons cf. or ; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. . The hoop conjecture states when a black hole is expected to form: With every mass M, one can associate a length known as the Schwarzschild radius,

\mathcal{R}=\frac{2GM}{c^2},

where G is the gravitational constant and c the speed of light. Imagine a circular hoop with the circumference 2\pi\mathcal{R}. A mass small enough to fit through that hoop regardless of their relative orientation, then it is compact enough to form a black hole.See ; for an account of more recent numerical studies, see .

The first studies of black holes relied on simplified model universes (namely explicit exact solution of Einstein's equation, in particular the spherically-symmetric Schwarzschild solution, which turns out to describe a static black hole, and the axisymmetric Kerr solution which describes a rotating stationary black hole). Subsequent studies using global geometry have revealed more general properties of black holes. In the long run, they are rather simple objects, characterized by eleven parameters specifying: energy, linear momentum, angular momentum, location at a specified time, and electric charge. This is the result of what are called the no hair theorem: "black holes have no hair", that is, no distinguishing marks akin to the differing hairstyles of humans. However complex an object that might collapse to form a black hole; in the long term (having emitted gravitational waves), the resulting object is very simple.For first steps, cf. ; see or for a derivation, and as well as as overviews of more recent results.

Even more remarkable, there is a general set of laws known as black hole mechanics, analogous to the laws of thermodynamics. For example, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, just as the entropy of a thermodynamic system; among other things, this law sets a limit to the energy that can be extracted from a rotating black hole (e.g. by the Penrose process).The laws of black hole mechanics were first described in ; a more pedagogical presentation can be found in ; for a more recent review, see chapter 2 of . A thorough, book-length introduction including an introduction to the necessary mathematics . For the Penrose process, see . In fact, there is strong evidence that the laws of black hole mechanics are indeed a special case of the laws of thermodynamics, and that the black hole area does indeed denote its entropy:See , . semi-classical calculations indicate that black holes do emit thermal radiation, with the surface gravity playing the role of temperature in Planck's law. This radiation is known as Hawking radiation, and we will come back to it in the section on general relativity and quantum theory, below.The fact that black holes radiate, quantum mechanically, was first derived in ; a more thorough derivation can be found in . A review is given in chapter 3 of .

Horizons also play a role for other kinds of solutions. In an expanding universe, some regions of the past can be unobservable ("particle horizon"), and some regions of the future cannot by any means be influenced (event horizon); in both cases, the location of the horizon in spacetime depends on the event under study.Cf. . Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizonsCf. (associated with a semi-classical radiation known as Unruh effect)., cf. .

Singularities Another general – and quite disturbing – feature of general relativity is the appearance of space-time boundaries known as singularities. Ordinary spacetime can be explored by following up on all possible ways that light and particles in free fall can travel (that is, all time-like and light-like geodesics). But there are spacetimes which fulfill all the requirements of Einstein's theory, yet have "ragged edges" – regions where the paths of l General relativity (GR) (aka general theory of relativity (GTR)) is the Geometry theory of gravitation published by Albert Einstein in 1915/16. and . It unifies special relativity, Newton's law of universal gravitation, and the insight that gravitational acceleration can be described by the curvature of space and time. General relativity further calls for the curvature of space-time to be produced by the mass-energy and momentum content of the matter in space-time. General relativity is distinguished from other metric :Category:Theories of gravitation by its use of the Einstein field equations to relate space-time content and space-time curvature.

In the mathematics of general relativity, the Einstein field equations are a system of partial differential equations whose solution represents the metric tensor (general relativity) (or the metric) of space-time, describing its "shape". Some important solutions of the Einstein field equations are the Schwarzschild solution (for the space-time surrounding a spherically symmetric uncharged and non-rotating massive object), the Reissner-Nordström black hole (for a charged spherically symmetric massive object), and the Kerr metric (for a rotating massive object). An object moving inertially in a gravitational field follows a geodesic (general relativity) that may be found using the Christoffel symbols of the metric.

General relativity is currently the most successful gravitational theory, being almost universally accepted and well-supported by observations. The first success of general relativity was in explaining the Tests of general relativity#Perihelion precession of Mercury of Mercury (planet). Then in 1919, Sir Arthur Stanley Eddington announced that observations of stars near the eclipsed Sun confirmed general relativity's prediction that massive objects bend light. Since then, other tests of general relativity have confirmed many of the #Predictions, including gravitational time dilation, the gravitational redshift of light, Shapiro delay, and gravitational radiation. In addition, numerous observations are interpreted as confirming one of general relativity's most mysterious and exotic predictions, the existence of black holes.



Justification (right)The justification for creating general relativity came from the equivalence principle, which dictates that free-falling observers are the ones in inertial motion. Roughly speaking, the principle states that the most obvious effect of gravity – things falling down – can be eliminated by making the transition to a reference frame that is in free fall, and that in such a reference frame, the laws of physics will be approximately the same as in special relativity.While the equivalence principle is still part of modern expositions of general relativity, there are some differences between the modern version and Einstein's original concept, cf. Norton 1985. A consequence of this insight is that inertial observers can accelerate with respect to each other. For example, a person in free fall in an elevator whose cable has been cut will experience weightlessness: objects will either float alongside him or her, or drift at constant speed. In this way, the experiences of an observer in free fall will be very similar to those of an observer in deep space, far away from any source of gravity, and in fact to those of the privileged ("inertial") observers in Einstein's theory of special relativity.This is described in detail in chapter 2 of Wheeler 1990. Albert Einstein realized that the close connection between weightlessness and special relativity represented a fundamental property of gravity.

Einstein's key insight was that there is no fundamental difference between the constant pull of gravity we know from everyday experience and the fictitious forces felt by an accelerating observer (in the language of physics: an observer in a non-inertial reference frame).E. g. Janssen (2005), p. 64f. Einstein himself also explains this in section XX of his non-technical book Einstein 1961. Following earlier ideas by Ernst Mach, Einstein also explored centrifugal forces and their gravitational analogue, cf. Stachel 1989. So what people standing on the surface of the Earth perceive as the 'force of gravity' is a result of their undergoing a continuous physical acceleration which could just as easily be imitated by placing an observer within a rocket accelerating at the same rate as gravity (9.81 metre per second squared).

This redefinition is incompatible with Newton's first law of motion, and cannot be accounted for in the Euclidean geometry of special relativity. To quote Einstein himself:Thus the equivalence principle led Einstein to develop a gravitational theory which involves curved space-times. Paraphrasing John Archibald Wheeler, Einstein's geometric theory of gravity can be summarized thus: spacetime tells matter how to move; matter tells spacetime how to curve.E.g. p. xi in Wheeler 1990.

Another motivating factor was the realization that relativity calls for the gravitational potential to be expressed as a symmetric rank-two tensor, and not just a scalar field as was the case in Newtonian physics (An analogy is the electromagnetic four-potential of special relativity). Thus, Einstein sought a rank-two tensor means of describing curved space-times surrounding massive objects. This effort came to fruition with the discovery of the Einstein field equations in 1915.

Fundamental principles , this (curved) geometry being interpreted as gravity. Note that the white lines do not represent the curvature of space, but instead represent the coordinate system imposed on the curved spacetime which would be rectilinear in a flat spacetimeGeneral relativity is a metric theory of gravitation. For this class of theory, the main defining feature is the concept of gravitational 'force' being replaced by spacetime geometry. Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories) are taken in general relativity to represent inertial motion within a curvature of spacetime.

General relativity (and all other metric theories of gravitation) are predicated upon several underlying assumptions. The general principle of relativity states that the laws of physics must be the same for all observers (accelerated or not). The principle of general covariance states the laws of physics must take the same form in all coordinate systems. General relativity also requires equivalence between inertial and geodesic (general relativity) because the world lines of particles unaffected by physical forces are timelike or null geodesics of spacetime. The principle of local Lorentz invariance requires that the laws of special relativity apply locally for all inertial observers. Finally there is the principle that the curvature of spacetime and its energy-momentum content are related. (As mentioned above, this relationship between curvature and spacetime content is specifically dictated by the Einstein field equations in general relativity.)

The equivalence principle, which was the starting point for the history of general relativity, ended up being a consequence of the general principle of relativity and the principle that inertial motion is geodesic motion.

Mathematical framework The requirements of the mathematics of general relativity are further modified by the other principles. Local Lorentz Invariance requires that the manifolds described in GR be 4-dimensional and Lorentzian instead of Riemannian manifold. In addition, the principle of general covariance requires that mathematics to be expressed using tensor calculus. Tensor calculus permits a manifold as mapped with a coordinate system to be equipped with a metric tensor (general relativity) which describes the incremental (spacetime) intervals between coordinates from which both the geodesic equations of motion and the curvature tensor of the spacetime can be ascertained.

Geometry Due to the expectation that spacetime is curved, non-Euclidean geometry must be used. (In particular, the geometry is described by a Pseudo-Riemannian manifold metric, or more specifically still, a Lorentzian metric.) In essence, spacetime does not adhere to the "common sense" rules of Euclidean geometry, but instead objects that were initially traveling in parallel paths through spacetime (meaning that their velocities do not differ to first order in their separation) come to travel in a non-parallel fashion. This effect is called geodesic deviation, and it is used in general relativity as an alternative to gravity. For example, two people on the Earth heading due north from different positions on the equator are initially traveling on parallel paths, yet at the north pole those paths will cross. Similarly, two balls initially at rest with respect to and above the surface of the Earth (which are parallel paths by virtue of being at rest with respect to each other) come to have a converging component of relative velocity as both accelerate towards the center of the Earth due to their subsequent free-fall.

The curvature of spacetime (caused by the presence of stress-energy) can be viewed intuitively in the following way. Placing a heavy object such as a bowling ball on a trampoline will produce a 'dent' in the trampoline. This is analogous to a large mass such as the Earth causing the local spacetime geometry to curve. This is represented by the image at the top of this article. The larger the mass, the bigger the amount of curvature. A relatively light object placed in the vicinity of the 'dent', such as a ping-pong ball, will accelerate towards the bowling ball in a manner governed by the 'dent'. Firing the ping-pong ball at some suitable combination of direction and speed towards the 'dent' will result in the ping-pong ball 'orbiting' the bowling ball. This is analogous to the Moon orbiting the Earth, for example.

Similarly, in general relativity massive objects do not directly impart a force on other massive objects as hypothesized in Newton's action at a distance idea. Instead (in a manner analogous to the ping-pong ball's response to the bowling ball's dent rather than the bowling ball itself), other massive objects respond to how the first massive object curves spacetime. Notice that the most important part of the curvature near a massive object is in the plane defined by the time and radial directions, although there is also some purely spatial curvature.

Coordinate vs. physical acceleration One of the greatest sources of confusion about general relativity comes from the need to distinguish between coordinate and physical accelerations.

In classical mechanics, space is preferentially mapped with a Cartesian coordinate system. Inertial motion then occurs as one moves through this space at a constant coordinate rate with respect to time. Any change in this rate of progression must be due to a force, and therefore a physical and coordinate acceleration were in classical mechanics one and the same. It is important to note that in special relativity that same kind of Cartesian coordinate system was used, with time being added as a fourth dimension and defined for an observer using the Einstein synchronisation. As a result, physical and coordinate acceleration correspond in special relativity too, although their magnitudes may vary.

In general relativity, we abandon the unwarranted assumption that nature has provided us with a preferred set of coordinates. Instead an observer may choose a set of coordinates at his own convenience, and we only require that coordinates of a point in different coordinate systems can be expressed into each other through some smooth functional dependence. Only statements that do not depend on the arbitrary choice of a coordinate system by the observer (i.e. the description of a physical phenomenon) can be considered of physical relevance. This is the principle of general covariance of physical laws.It means for example that a quantity like acceleration, cannot simply be described asthe second derivative of the coordinate functions of a velocity, because a non zero "coordinate acceleration" may merely be an artifact of the choice of coordinates. In fact such an artifact already occurs for the description in polar coordinates of a uniformly moving particle not passing through the (chosen!) origin. In fact the definition of acceleration of a particle requires that we know how to subtract velocities measured at two different points along its track in space time. Equivalently we must be able to define in an observer and coordinate invariant way which velocity vectors are constant along such a path in space time. This is called parallel transport. It does not come for free but requires additional structure of space-time, a connection. It so happens that if there is defined a "length" of all velocity vectors (which may be negative), a Lorentzian metric , there is a natural connection , the Levi Civita connection, uniquely determined by requiring that parallel velocity vectors have constant "length" and the technical assumption of zero torsion.It was one of Einsteins great insights that this description can be applied to describe the influence of gravity. In general relativity, gravity is seen as a consequenceof the fact that parallel transport of a velocity vector may depend on the path through space time, not only on its endpoints. How metric and thereby connection and parallel transport, are determined by the transport of energy-momentum in space time by matter and radiation described with the so-called stress-energy tensor is the content of the theory of general relativity.

Einstein field equations The Einstein field equations (EFE) describe how stress-energy causes curvature of spacetime and are usually written in tensor form (using abstract index notation) as

G_{ab} = \kappa\, T_{ab}

where Gab is the Einstein tensor, Tab is the stress-energy tensor and \kappa is a constant. The Einstein tensor is related to the curvature of space-time and is a function only of the metric tensor and its first and second derivatives. The stress energy tensor, which is the source of the gravitational field, includes stress (pressure and shear), the density of momentum, and the density of energy including the energy of mass (the source for Newtonian gravity). The tensors Gab and Tab are both rank-2 symmetric tensors, that is, they can each be thought of as 4×4 matrices, each of which contains 10 independent terms.

The EFE reduce to Newton's law of gravity in the Correspondence principle#Other uses of the term of a weak-field approximation and slow-motion approximation relative to the speed of light. In fact, the value of \kappa in the EFE is determined to be \kappa = 8 \pi G / c^4 \ by making these two approximations.

The solutions of the Einstein field equations are metric tensor (general relativity). These metrics describe the structure of spacetime given the stress-energy and coordinate mapping used to obtain that solution. Being non-linear differential equations, the EFE often defy attempts to obtain an exact solutions in general relativity; however, many such solutions are known.

The EFE are the identifying feature of general relativity. Other theories built out of the same premises include additional rules and/or constraints. The result almost invariably is a theory with different field equations (such as Brans-Dicke theory, teleparallelism, Rosen's bimetric theory, and Einstein-Cartan theory).

Consequences of Einstein's theory General relativity, as laid out in the previous section, has a number of consequences; some follow directly from the theory's axioms, others have only become clear in the course of the ninety years of research that followed Einstein's initial publication.

Gravitational time dilation and frequency shift In general relativity (and, in fact, in any theory in which the equivalence principle holdsCf. and . In fact, Einstein derived these effects using the equivalence principle as early as 1907, cf. and the description in .), gravity has an immediate influence on the passage of time. Imagine two observers Alice and Bob, both of which are at rest in a stationary gravitational field, with Alice closer to the source of gravity ("deeper in the gravity well") and Bob at a greater distance. Then for light sent from Alice to Bob or vice versa, Bob will measure a lower frequency than Alice: light sent down into a gravity well is blue-shifted, light climbing out of a gravity well is redshifted. Also, Alice's clocks tick more slowly than Bob's: whenever the two are compared (either by sending light signals back and forth, or by slowly transporting clocks from one location to the other), the result will be that Bob's clocks are running faster. This effect is not restricted to clocks, but applies to all processes (the rate at which Alice and Bob age, cook five-minute eggs, or play Chopin's Minute Waltz); it is known as gravitational time dilation.; ..

The gravitational redshift was first measured in 1959 in a laboratory experiment by Pound-Rebka experimentSee , ; ; a list of further experiments is given in . and later confirmed by astronomical observations.E.g. ; the most recent and most accurate Sirius B measurements are published in . There are numerous direct measurements of gravitational time dilation using atomic clocksStarting with the Hafele-Keating experiment, and , and culminating in the Gravity Probe A experiment; an overview of experiments can be found in . while ongoing validation is provided as a side-effect of the operation of the Global Positioning System (GPS).GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistc effects, see and . Tests in stronger gravitational fields are provided by the observation of binary pulsars.Reviews are given in and . All results are in agreement with general relativity;General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; . however, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.Cf. .

Light deflection and gravitational time delay

In general relativity, light follows a special variety of straightest-possible world-line, so-called light-like or null geodesics – a generalization of the straight lines along which light travels in classical physics, and the invariance of lightspeed in special relativity.The fact that light follows null geodesics is not an independent axiom; it can be derived from Einstein's equations and the Maxwell Lagrangian using a WKB approximation, cf. . As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the Post-Newtonian expansion),A brief descriptions and pointers to the literature can be found in . several effects of gravity on the propagation of light emerge.

The best-known is the bending of light in a gravitational field: light passing a massive body is deflected towards that body. While such an effect can also be derived by extending the universality of free fall to light,See ; for the historical examples, ; in fact, Einstein published one such derivation as . Such calculations tacitly assume that the geometry of space is Euclidean, cf. . the maximal angle of deflection resulting from such heuristic calculations is only half the value given by general relativity; from the standpoint of Einstein's theory they take into account the effect of gravity on time, but not its consequences for the warping of space.E.g. . An important example of this is starlight being deflected as it passes the Sun; in consequence, the positions of stars observed in the Sun's vicinity during a solar eclipse appear shifted by up to 1.75 arc seconds. This effect was first measured by a British expedition directed by Arthur Eddington, and confirmed with significantly higher accuracy by subsequent measurements.Cf. ; for an overview of more recent measurements, see . The most precise direct modern observations measure the deflection of the light of distant quasars by the Sun, cf. .

Closely related to the bending of light is the gravitational time delay, also known as the Shapiro effect: light signals take longer to move through a gravitational field than they would in the absence of the gravitational field. This effect was discovered through the observations of radar signals sent from Earth to planets such as Venus or Mercury (planet) and thence reflected back;; a pedagogical introduction can be found in . later, much more accurate measurements utilized signals sent to space probes and sent back using active transponders.The most recent measurements are ; for an overview, see . In both the case of the planets and the probes, what was measured was the propagation of signals in the Sun's gravitational field. More recent measurements have detected the Shapiro effect in signals sent by a pulsar that is part of a binary system; in that case, the gravitational field causing the time delay is that of the other pulsar.Cf. . In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay are used to determine a parameter called \gamma that reflects the influence of gravity on the geometry of space..

Gravitational waves There are several analogies between weak-field gravity and electromagnetism. One is that, for electromagnetic waves, there are corresponding gravitational waves: ripples in spacetime that propagate at the speed of light.For an overview, see . Note, however that for gravitational waves, the dominant contribution is not the dipole, but the quadrupole cf. .

The simplest variety of gravitational wave can be visualized via their action on a ring of freely floating particles (see first image to the right). As a simple sine wave propagates through such a ring from out of the page towards the reader, the ring is distorted in a characteristic, rhythmic fashion (see second image to the right).Any textbook on general relativity will contain a description of these properties, e.g. . Such linearized gravitational waves are important when it comes to describing the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in distances increasing and decreasing by 10^{-21} or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposition.For example . It is, however, important to note that the linearized waves are only approximations. Generically, the non-linearity of the Einstein equations means that there will no be linear superposition for gravitational waves. Describing such more general waves is not an easy task. There are some exact solutions describing gravitational waves, for instance a wave train traveling through empty space. or so-called Gowdy universes, varieties of an expanding cosmos filled with gravitational waves,See , . while, when it comes to describing the gravitational waves produced in astrophysically relevant situations such as the merger of two black holes, numerical relativity are presently the only way to construct appropriate models.See for a brief introduction to the methods of numerical relativity, and for the connection with gravitational wave astronomy.

Orbital effects and the relativity of direction General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. The most striking one are the relativistic apsis shifts, orbital decay caused by the emission of gravitational waves, and effects that are due to the relativity of direction.

Precession of apsides In general relativity, the Apsis of orbits (the points of an orbiting body closest approach to the system's center of mass) will precess – the orbit is not an ellipse, but akin to an ellipse that rotates on its focus, resulting in a rosette-like shape (see image). Einstein himself derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body like a test particle; the result can also be obtained by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)See . or the much more general post-Newtonian formalism.See . The effect is due both to the influence of gravity on the geometry of space and to the way that self energy contributes to a body's gravity (in other words, the special kind of nonlinearity exhibited by Einstein's theory).In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms \beta and \gamma, cf. and .

An early success of general relativity was that the theory offered a straightforward explanation for an Tests of general relativity#Perihelion precession of Mercury of the planet Mercury (planet), which had been discovered by Urbain Le Verrier in 1859 but had remained mysterious.See and . This agreement between theory and experiment confirmed for Einstein that he had at last identified the correct form of the Einstein field equations. More recent observations have shown that the field equations predict the correct anomalous perihelion shift for all planets where this can be measured accurately (Mercury (planet), Venus (planet) and the Earth).The most precise measurements are VLBI measurements of planetary positions; see , , ; for an overview, .The effect has also been checked in binary pulsar systems where it is larger by five order of magnitude.See .

Orbital decay According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the solar system or for ordinary double stars, the effect is too small to be observable. Not so for a close binary pulsar, a system of two orbiting neutron stars, one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period; since the neutron stars are very compact, significant amounts of energy are emitted in the form of gravitational radiation.See and ; an accessible account can be found in .

The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Russell Alan Hulse and Joseph Hooton Taylor Jr. using binary pulsar PSR1913+16 they had discovered in 1974; it amounts to the first indirect detection of gravitational waves, rewarded with the Nobel Prize in physics in 1993.An overview can be found in ; for the pulsar discovery, see ; for the initial evidence for gravitational radiation, see . Since then, several other binary pulsars have been found, the most spectacular find being the double pulsar PSR J0737-3039 in which both stars are pulsars.Cf. .

Geodetic precession and frame-dragging Several relativistic effects are directly related to the relativity of direction.See e.g. , . One is geodetic effect: for a gyroscope in free fall in curved spacetime, the direction of its axis will change when compared, for instance, with the direction of light received from distant stars – even though its motion comes closest to keeping its axis direction constant ("parallel transport").See , . For the Moon-Earth-system, this effect has been measured with the help of lunar laser ranging;See and, for a more recent review, . more recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 1 percent.See .

Near a rotating mass, there are so-called gravitomagnetic or frame-dragging effects: for a distant observer, it will seem that objects close to the mass gets "dragged around"; this is most extreme for Kerr solution where, for an object entering a zone known as the ergosphere, rotation is inevitable.E.g. , Such effects can again be tested through their influence on the orientation of a gyroscope in free fall:E.g. , ; for a more recent review, see . somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction;E.g. , ; see the entry frame-dragging for an account of the debate. a precision measurement is the main aim of the Gravity Probe B mission, whose results are due in late 2007.A mission description can be found in ; a first post-flight evaluation is given in ; further updates will be available on the mission website .

Astrophysical applications Gravitational lensing : four images of the same astronomical object, produced by a gravitational lensThe deflection of light by gravity can have an intriguing side effect: if there is a massive object between the observer and a distant target object, it is possible for the observer to see multiple distorted images of the target. This and similar effects are known as gravitational lensingFor overviews of gravitational lensing and its applications, see and . and, depending on the configuration, scale, and mass distribution, it can result in two images, a bright ring known as an Einstein ring, or partial rings called arcs.For a simple derivation, see ; cf. .The Twin Quasar was discovered in 1979;See . since then, more than a hundred gravitational lenses have been observed.Images of all the known lenses can be found on the pages of the CASTLES project, . Images too close to be resolved can still lead to a measurable effect, namely an overall brightening of a given star or other point-like object; a number of such "microlensing events" has been observed, as well.For an overview, see .

Gravitational lensing has developed into a tool of observational astronomy. Notably, it is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data are also used to understand the structural evolution of galaxy.See .

Gravitational wave astronomy From observations of binary pulsars, there is strong indirect evidence for the existence of gravitational wave (see the section on General relativity#Orbital decay, above). However, as of yet, gravitational waves reaching us from the depths of the cosmos have not been detected directly – this is one of the major goals of current relativity-related research.For an overview, ; accessible accounts can be found in and . To this end, a number of land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (three detectors), TAMA 300 and VIRGO.An overview is given in .A joint US-European mission to launch a space-based detector, LISA (astronomy), is currently under development,See . with a precursor mission (LISA Pathfinder) due for launch in late 2009.See .

Gravitational waves promise to yield information about astronomical objects that is inaccessible by observations using electromagnetic radiation:Cf. . Terrestrial detectors are expected to yield new information about inspiral phase and mergers of binary stellar black hole and binaries consisting of one such black hole and a neutron star (of interest as a candidate mechanism for gamma ray bursts); they could also detect signals from core-collapse supernovae and from periodic sources such as rotating neutron stars with small deformation. If there is truth to speculation about certain kinds of phase transitions or kink bursts from long cosmic strings in the very early universe (at cosmic times around 10^{-25} seconds) these could also be detectable.See . Space-based detectors like LISA should detect objects such as binaries consisting of two White Dwarfs, and AM CVn stars (a White Dwarf accreting matter from its binary partner, a low-mass helium star), and also observe the mergers of supermassive black holes and the inspiral of smaller objects (between one and a thousand solar masses) into such black holes. LISA should also be able to listen to the same kind of sources from the early universe as ground-based detectors, but at even lower frequencies and with greatly increased sensitivity.See .

Black holes and other compact objects Whenever an object becomes sufficiently compact, general relativity predicts the formation of a black hole: a region of space from which nothing, not even light, can escape. In the currently accepted models of stellar evolution, neutron stars with around 1.4 solar mass and so-called stellar black holes with a few to a few dozen solar masses are thought to be the final state for the evolution of massive stars.See . Supermassive black holes with between a few million and a few 1000000000 (number) solar masses are now thought to be the rule rather than the exception in the centers of galaxies,E.g. . and their presence is thought to have played an important role in the formation of galaxies and larger cosmic structures.Cf. and the accompanying summary .

From an astronomical point of view, the most important property of compact objects such as black holes is that they provide a superbly efficient mechanism for converting gravitational into radiation energy.Cf. , Accretion, that is, the falling of material such as gas or dust onto stellar black hole or supermassive black hole black holes is thought to be responsible for some of the most spectacularly luminous astronomical objects, notably diverse kinds of Active Galactic Nucleus on galactic scales, and stellar-size objects such as Microquasars;For the basic mechanism, see ; for more about the different types of astronomical objects associated with this, cf. . in particular, it can lead to relativistic jets: focused beams of highly energetic particles that are being flung into space at almost the speed of light.For a review, see . For modelling all these phenomena, general relativity plays a central role,For stellar end states, cf. or, for more recent numerical work, ; for supernovae, there are still major problems to be solved, cf. ; for simulating accretion and the formation of jets, cf. . and relativistic lensing effects are thought to play a role for the signals received from X-ray pulsars.Cf. .

Limits on compactness from the observation of accretion-driven phenomena ("Eddington luminosity")See . observations of stellar dynamics in the center of our own Milky Way galaxy,Cf. . and indications that at least some of the compact objects in question appear to have no solid surfaceExamination of X-ray bursts for which the central compact object is either a neutron star or a black hole; cf. and, for an overview, . are strong indirect evidence for the existence of black holes; more direct evidence such as observing the "shadow" of the horizon of the Milky Way galaxy's central black hole.Cf. . is eagerly sought for.

Black holes are also sought-after targets in the search for gravitational waves (see the section General_relativity#Gravitational_waves, above): merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and reliable simulations of such mergers are one of the main goals of current research in numerical relativity;Cf. . the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events, and hence as a probe of cosmic expansion at large distances;Cf. . the gravitational waves produced as a stellar black hole plunges into a supermassive one should serve as a probe of the supermassive black hole's geometry.E.g. .

Cosmology Each solution of Einstein's equations describes a whole universe, so it should come as no surprise that there are solutions that provide useful models for cosmology, the study of the universe as a whole. The current models are based on an extension of the original form of Einstein's equations which include the cosmological constant \Lambda, an additional term that has an important influence on the large-scale dynamics of the cosmos, G_{ab} + \Lambda\ g_{ab} = \kappa\, T_{ab} where gab is the metric tensor (general relativity).Originally ; cf. the description in .

On the basis of isotropic and homogeneous solutions of these enhanced equations, the so-called Friedmann-Lemaître-Robertson-Walker metric,See . are built the Physical cosmology in which the universe has evolved over the past 14 1000000000 (number) years from a hot, early Big bang phase.See ; use of these models is justified by the fact that, at large scales of around hundred million light-years and more, our own universe indeed appears to be isotropic and homogeneous, cf. . Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,E.g. with WMAP data, see . further observational data can be used to put the models to the test: successful predictions include the initial abundance of chemical elements formed in a period of Big bang nucleosynthesis,See ; for a recent account of predictions, see ; an accessible account can be found in . which is in good agreement with astronomical observations;See , , , and . the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation,Cf. and, for a pedagogical introduction, see ; for the initial detection, see , andfor precision measurements by satellite observatories see (COBE) and (WMAP). and the large-scale distribution of galaxies.A review can be found in .

The status of the resulting models is mixed. On the one hand, the standard models of cosmology have been very successful: to date, they have passed all observational tests,See, e.g., fig. 2 in . and they have proven a sound basis to explaining the evolution of the universe's large-scale structure.For a review, see ; more recent results can be found in . On the other hand, there are a number of important open questions. The determination of cosmological parameters (in line with other astronomical observationsThese additional observations involve the dynamics of galaxies and galaxy clusters cf. chapter 18 of , evidence from gravitational lensing, cf. , and simulations of large-scale structure formation, see .) suggests that about 90 percent of all matter in the universe is in the form of so-called dark matter, which has mass (and hence gravitational influence), but does not interact electromagnetically (and hence cannot be observed directly); there is currently no generally accepted description of this new kind of matter within the framework of particle physicsSee , and ; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual elementary particles ("non-baryonic matter"), cf. . or otherwise.Namely, some physicists have questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity cf. the overview in . A similar open question is that of dark energy. Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual equation of state, namely dark energy;See ; an accessible overview is given in . the nature of this new form of energy remains unclear.Here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models, cf. ; aforementioned modifications need not be modifications of general relativity, they could, for example, be modifications in the way we treat the inhomogeneities in the universe, cf. .

A number of further problems of the classical cosmological models (such as "why is the cosmic background radiation so highly homogeneous")More precisely, these are the flatness problem, the horizon problem, and the monopole problem; a pedagogical introduction can be found in , see also . have led to the introduction of an additional phase of strongly accelerated expansion at cosmic times of around 10^{-} seconds, known as an cosmic inflation.A good introduction is ; for a more recent review, see . While recent measurements of the cosmic background radiation have resulted in first evidence for this scenario,See . problems remain. The re is a bewildering variety of possible inflationary scenarios not restricted by current observations.More concretely, the potential function that is crucial in determining the dynamics of the inflaton is simply postulated, but not derived from an underlying physical theory. Also, the question remains what happened in the earliest universe, close to where the classical models predict the big bang singularity; an authoritative answer would require a complete theory of quantum gravity, which does not exist at the momentSee . (cf. the section #, below).

Advanced concepts Causal structure and global geometry universeIn general relativity, no material body can catch up with or over take a light pulse; no influence from an event A can reach any other location before light sent out at A does so. Hence an exploration of all light worldlines (Geodesic (general relativity)s) yields key information about the spacetime's causal structure. This structure can be displayed using so-called Penrose diagram in which infinitely large regions of space and infinite time intervals are shrunk ("compactification") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.See , , and

Aware of the importance of causal structure, Roger Penrose and others developed important techniques that are nowadays known as global geometry. In global geometry, the object of study is not one particular Solutions of the Einstein field equations (or family of solutions) to Einstein's equations; rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, are utilized in conjunction with non-specific assumptions about the nature of matter (usually in the form of so-called energy conditions) to derive general results.E.g. and

Cosmic segregation: Horizons One of the most striking conclusions that can be drawn from studies of global geometry is the existence of boundaries called event horizon, which segregate one spacetime region from the rest of the world. The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space, one can define a surface that separates the inside from the outside world. No light from the inside can escape to the outside, and since, in general relativity, no object can overtake a light pulse, all inside matter is imprisoned, as well. The resulting object is known as a black hole, and the surface in question as the black hole's horizon.For an account of the evolution of this concept, see . A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and apparent horizons cf. or ; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. . The hoop conjecture states when a black hole is expected to form: With every mass M, one can associate a length known as the Schwarzschild radius,

\mathcal{R}=\frac{2GM}{c^2},

where G is the gravitational constant and c the speed of light. Imagine a circular hoop with the circumference 2\pi\mathcal{R}. A mass small enough to fit through that hoop regardless of their relative orientation, then it is compact enough to form a black hole.See ; for an account of more recent numerical studies, see .

The first studies of black holes relied on simplified model universes (namely explicit exact solution of Einstein's equation, in particular the spherically-symmetric Schwarzschild solution, which turns out to describe a static black hole, and the axisymmetric Kerr solution which describes a rotating stationary black hole). Subsequent studies using global geometry have revealed more general properties of black holes. In the long run, they are rather simple objects, characterized by eleven parameters specifying: energy, linear momentum, angular momentum, location at a specified time, and electric charge. This is the result of what are called the no hair theorem: "black holes have no hair", that is, no distinguishing marks akin to the differing hairstyles of humans. However complex an object that might collapse to form a black hole; in the long term (having emitted gravitational waves), the resulting object is very simple.For first steps, cf. ; see or for a derivation, and as well as as overviews of more recent results.

Even more remarkable, there is a general set of laws known as black hole mechanics, analogous to the laws of thermodynamics. For example, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, just as the entropy of a thermodynamic system; among other things, this law sets a limit to the energy that can be extracted from a rotating black hole (e.g. by the Penrose process).The laws of black hole mechanics were first described in ; a more pedagogical presentation can be found in ; for a more recent review, see chapter 2 of . A thorough, book-length introduction including an introduction to the necessary mathematics . For the Penrose process, see . In fact, there is strong evidence that the laws of black hole mechanics are indeed a special case of the laws of thermodynamics, and that the black hole area does indeed denote its entropy:See , . semi-classical calculations indicate that black holes do emit thermal radiation, with the surface gravity playing the role of temperature in Planck's law. This radiation is known as Hawking radiation, and we will come back to it in the section on general relativity and quantum theory, below.The fact that black holes radiate, quantum mechanically, was first derived in ; a more thorough derivation can be found in . A review is given in chapter 3 of .

Horizons also play a role for other kinds of solutions. In an expanding universe, some regions of the past can be unobservable ("particle horizon"), and some regions of the future cannot by any means be influenced (event horizon); in both cases, the location of the horizon in spacetime depends on the event under study.Cf. . Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizonsCf. (associated with a semi-classical radiation known as Unruh effect)., cf. .

Singularities Another general – and quite disturbing – feature of general relativity is the appearance of space-time boundaries known as singularities. Ordinary spacetime can be explored by following up on all possible ways that light and particles in free fall can travel (that is, all time-like and light-like geodesics). But there are spacetimes which fulfill all the requirements of Einstein's theory, yet have "ragged edges" – regions where the paths of l

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General Relativity, MT 2003

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