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A simulated black hole of ten solar masses as seen from a distance of 600 kilometers with the Milky Way in the background.
General relativity
$G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}$
Einstein field equations
Introduction
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General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1915. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the four-momentum (mass-energy and linear momentum) of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.

Many predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time dilation, the gravitational redshift of light, and the gravitational time delay. General relativity's predictions have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.

Einstein's theory has important astrophysical implications. It points towards the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. There is evidence that such stellar black holes as well as more massive varieties of black hole are responsible for the intense radiation emitted by certain types of astronomical objects such as active galactic nuclei or microquasars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, where multiple images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of gravitational waves, which have since been measured indirectly; a direct measurement is the aim of projects such as LIGO. In addition, general relativity is the basis of current cosmological models of a consistently expanding universe.

## History

First page from Einstein's manuscript explaining general relativity

Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall, he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the November, 1915 presentation to the Prussian Academy of Science of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter is present, and form the core of Einstein's general theory of relativity.[1]

The Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the so-called Schwarzschild metric. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, which eventually resulted in the Reissner-Nordström solution, now associated with charged black holes.[2] In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to reproduce that "observation".[3] By 1929, however, the work of Hubble and others had shown that our universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the big bang models, in which our universe has evolved from an extremely hot and dense earlier state.[4] Einstein later declared the cosmological constant the biggest blunder of his life.[5]

During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to Newtonian gravity, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein himself had shown in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters ("fudge factors").[6] Similarly, a 1919 expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of May 29, 1919,[7] making Einstein instantly famous.[8] Yet the theory entered the mainstream of theoretical physics and astrophysics only with the developments between approximately 1960 and 1975, now known as the Golden age of general relativity. Physicists began to understand the concept of a black hole, and to identify these objects' astrophysical manifestation as quasars.[9] Ever more precise solar system tests confirmed the theory's predictive power,[10] and relativistic cosmology, too, became amenable to direct observational tests.[11]

## From classical mechanics to general relativity

General relativity is best understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit of a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity.[12]

### Geometry of Newtonian gravity

At the base of classical mechanics is the notion that a body's motion can be described as a combination of free (or inertial) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second law of motion, which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration.[13] The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics, straight world lines in spacetime.[14]

Ball falling to the floor in an accelerating rocket (left), and on Earth (right)

Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction), can be used to define the geometry of space, as well as a time coordinate. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment), there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties.[15] A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in a small enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard an accelerated rocket.[16]

Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry. However, spacetime as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable. From this, one can deduce that spacetime is curved. The result is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired coordinate system.[17] In this geometric description, tidal effects—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.[18]

### Relativistic generalization

As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics.[19] In the language of symmetry: where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group which also includes translations and rotations.) The differences between the two become significant when we are dealing with speeds approaching the speed of light, and with high-energy phenomena.[20]

With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see the image on the left). The light-cones define a causal structure: for each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer-independent.[21] In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the space-time's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure.[22]

Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.[23]

A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. below). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.[24] The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing special-relativistic physics to include gravity.[25]

The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).[26]

### Einstein's equations

Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the energy-momentum tensor, which includes both energy and momentum densities as well as stress (that is, pressure and shear).[27] Using the equivalence principle, this tensor is readily generalized to curved space-time. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy-momentum corresponds to the statement that the energy-momentum tensor is divergence-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of the energy-momentum tensor, and hence of whatever is on the other side of the equation, is zero— the simplest set of equations are what are called Einstein's (field) equations:

$R_{ab} - {\textstyle 1 \over 2}R\,g_{ab} = \kappa T_{ab}.\,$

On the left-hand side is the Einstein tensor, a specific divergence-free combination of the Ricci tensor Rab and the metric. In particular,

$R=R_{cd}g^{cd}\,$

is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as

$\quad R_{ab}={R^d}_{adb}.\,$

On the right-hand side, Tab is the energy-momentum tensor. All tensors are written in abstract index notation.[28] Matching the theory's prediction to observational results for planetary orbits (or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics), the proportionality constant can be fixed as κ = 8πG/c4, with G the gravitational constant and c the speed of light.[29] When there is no matter present, so that the energy-momentum tensor vanishes, the result are the vacuum Einstein equations,

$R_{ab}=0.\,$

There are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Brans-Dicke theory, teleparallelism, and Einstein-Cartan theory.[30]

## Definition and basic applications

The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for model-building.

### Definition and basic properties

General relativity is a metric theory of gravitation. At its core are Einstein's equations, which describe the relation between the geometry of a four-dimensional, semi-Riemannian manifold representing spacetime on the one hand, and the energy-momentum contained in that spacetime on the other.[31] Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow.[32] The curvature is, in turn, caused by the energy-momentum of matter. Paraphrasing the relativist John Archibald Wheeler, spacetime tells matter how to move; matter tells spacetime how to curve.[33]

While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases. For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.[34]

As it is constructed using tensors, general relativity exhibits general covariance: its laws—and further laws formulated within the general relativistic framework—take on the same form in all coordinate systems.[35] Furthermore, the theory does not contain any invariant geometric background structures. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers.[36] Locally, as expressed in the equivalence principle, spacetime is Minkowskian, and the laws of physics exhibit local Lorentz invariance.[37]

### Model-building

The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy-momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.[38]

Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.[39] Nevertheless, a number of exact solutions are known, although only a few have direct physical applications.[40] The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner-Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe,[41] and the Friedmann-Lemaître-Robertson-Walker and de Sitter universes, each describing an expanding cosmos.[42] Exact solutions of great theoretical interest include the Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub-NUT solution (a model universe that is homogeneous, but anisotropic), and Anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture).[43]

Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions. In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes.[44] In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities. Approximate solutions may also be found by perturbation theories such as linearized gravity[45] and its generalization, the post-Newtonian expansion, both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity.[46] An extension of this expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.[47]

## Consequences of Einstein's theory

General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of the ninety years of research that followed Einstein's initial publication.

### Gravitational time dilation and frequency shift

Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body

Assuming that the equivalence principle holds,[48] gravity influences the passage of time. Light sent down into a gravity well is blueshifted, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.[49]

Gravitational redshift has been measured in the laboratory[50] and using astronomical observations.[51] Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks,[52] while ongoing validation is provided as a side-effect of the operation of the Global Positioning System (GPS).[53] Tests in stronger gravitational fields are provided by the observation of binary pulsars.[54] All results are in agreement with general relativity.[55] However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.[56]

### Light deflection and gravitational time delay

General relativity predicts that the path of light is bent in a gravitational field; light passing a massive body is deflected towards that body. This effect has been confirmed by observing the light of stars or distant quasars being deflected as it passes the Sun.[57]

Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray)

This and related predictions follow from the fact that light follows what is called a light-like or null geodesic—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity.[58] As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion),[59] several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light,[60] the angle of deflection resulting from such calculations is only half the value given by general relativity.[61]

Closely related to light deflection is the gravitational time delay (or Shapiro effect), the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction.[62] In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.[63]

### Gravitational waves

Ring of test particles floating in space
Ring of test particles influenced by gravitational wave

One of several analogies between weak-field gravity and electromagnetism is that, analogous to electromagnetic waves, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light.[64] The simplest type of such a wave can be visualized by its action on a ring of freely floating particles (upper image to the right). A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (lower, animated image to the right).[65] Since Einstein's equations are non-linear, arbitrarily strong gravitational waves do not obey linear superposition, making their description difficult. However, for weak fields, a linear approximation can be made. Such linearized gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative 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 decomposed.[66]

Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space[67] or so-called Gowdy universes, varieties of an expanding cosmos filled with gravitational waves.[68] But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.[69]

### Orbital effects and the relativity of direction

General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation (precession) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction.

#### Precession of apsides

Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star

In general relativity, the apsides of any orbit (the point of the orbiting body's 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 rose curve-like shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a test particle. For him, the fact that his theory gave a straightforward explanation of the anomalous perihelion shift of the planet Mercury, discovered earlier by Urbain Le Verrier in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.[70]

The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)[71] or the much more general post-Newtonian formalism.[72] It is due to the influence of gravity on the geometry of space and to the contribution of self-energy to a body's gravity (encoded in the nonlinearity of Einstein's equations).[73] Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus and the Earth),[74] as well as in binary pulsar systems, where it is larger by five orders of magnitude.[75]

#### Orbital decay

Orbital decay for PSR1913+16: time shift in seconds, tracked over three decades.[76]

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.[77]

The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor, using the binary pulsar PSR1913+16 they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993 Nobel Prize in physics.[78] Since then, several other binary pulsars have been found, in particular the double pulsar PSR J0737-3039, in which both stars are pulsars.[79]

#### Geodetic precession and frame-dragging

Several relativistic effects are directly related to the relativity of direction.[80] One is geodetic precession: the axis direction of a gyroscope in free fall in curved spacetime will change when compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible ("parallel transport").[81] For the Moon-Earth-system, this effect has been measured with the help of lunar laser ranging.[82] More recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 1 percent.[83]

Near a rotating mass, there are so-called gravitomagnetic or frame-dragging effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for rotating black holes where, for any object entering a zone known as the ergosphere, rotation is inevitable.[84] Such effects can again be tested through their influence on the orientation of gyroscopes in free fall.[85] Somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction.[86]. Also the Mars Global Surveyor probe around Mars has been used [87] [88]; see the entry frame-dragging for an account of the debate. A precision measurement is the main aim of the Gravity Probe B mission, with the results expected in September 2008.[89]

## Astrophysical applications

### Gravitational lensing

Einstein cross: four images of the same astronomical object, produced by a gravitational lens

The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass and relative distances, the astronomer will see multiple distorted images of the target. Such effects are known as gravitational lensing.[90] Depending on the configuration, scale, and mass distribution, there can be two or more images, a bright ring known as an Einstein ring, or partial rings called arcs.[91] The earliest example was discovered in 1979;[92] since then, more than a hundred gravitational lenses have been observed.[93] Even if the multiple images are too close to each other to be resolved, the effect can still be measured, e.g., as an overall brightening of the target object; a number of such "microlensing events" have been observed.[94]

Gravitational lensing has developed into a tool of observational astronomy. It is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and to obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data provide valuable insight into the structural evolution of galaxies.[95]

### Gravitational wave astronomy

Artist's impression of the space-borne gravitational wave detector LISA

Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see Orbital decay, above). However, gravitational waves reaching us from the depths of the cosmos have not been detected directly, which is a major goal of current relativity-related research.[96] Several land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (three detectors), TAMA 300 and VIRGO.[97] A joint US-European space-based detector, LISA, is currently under development,[98] with a precursor mission (LISA Pathfinder) due for launch in 2012.[99]

Observations of gravitational waves promise to complement observations in the electromagnetic spectrum.[100] They are expected to yield information about black holes and other dense objects such as neutron stars and white dwarfs, about certain kinds of supernova implosions, and about processes in the very early universe, including the signature of certain types of hypothetical cosmic string.[101]

### Black holes and other compact objects

Whenever the ratio of an object's mass to its radius becomes sufficiently large, 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 of around 1.4 solar masses, and 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.[102] Usually a galaxy has one supermassive black hole with a few million to a few billion solar masses in its center,[103] and its presence is thought to have played an important role in the formation of the galaxy and larger cosmic structures.[104]

Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves

Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.[105] Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as microquasars.[106] In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that are being flung into space at almost light speed.[107] General relativity plays a central role in modelling all these phenomena,[108] and observations provide strong evidence for the existence of black holes with the properties predicted by the theory.[109]

Black holes are also sought-after targets in the search for gravitational waves (cf. Gravitational waves, above). Merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events–and hence serve as a probe of cosmic expansion at large distances.[110] The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about supermassive black hole's geometry.[111]

### Cosmology

The current models of cosmology are based on Einstein's equations including cosmological constant Λ, which has important influence on the large-scale dynamics of the cosmos,

$R_{ab} - {\textstyle 1 \over 2}R\,g_{ab} + \Lambda\ g_{ab} = \kappa\, T_{ab}$

where gab is the spacetime metric.[112] Isotropic and homogeneous solutions of these enhanced equations, the Friedmann-Lemaître-Robertson-Walker solutions,[113] allow physicists to model a universe that has evolved over the past 14 billion years from a hot, early Big Bang phase.[114] Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,[115] further observational data can be used to put the models to the test.[116] Predictions, all successful, include the initial abundance of chemical elements formed in a period of primordial nucleosynthesis,[117] the large-scale structure of the universe,[118] and the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation.[119]

Image of radiation emitted no more than a few hundred thousand years after the big bang, captured with the satellite telescope WMAP

Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. About 90 percent of all matter appears to be so-called dark matter, which has mass (or, equivalently, gravitational influence), but does not interact electromagnetically and, hence, cannot be observed directly.[120] There is no generally accepted description of this new kind of matter, within the framework of known particle physics[121] or otherwise.[122] Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation also 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, known as dark energy, the nature of which remains unclear.[123]

A so-called inflationary phase,[124] an additional phase of strongly accelerated expansion at cosmic times of around 10 − 33 seconds, was hypothesized in 1980 to account for several puzzling observations that were unexplained by classical cosmological models, such as the nearly perfect homogeneity of the cosmic background radiation.[125] Recent measurements of the cosmic background radiation have resulted in the first evidence for this scenario.[126] However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations.[127] An even larger question is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang singularity. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed[128] (cf. the section on quantum gravity, below).

### Causal structure and global geometry

Penrose diagram of an infinite Minkowski universe

In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event A can reach any other location X before light sent out at A to X. In consequence, an exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. This structure can be displayed using Penrose-Carter diagrams in which infinitely large regions of space and infinite time intervals are shrunk ("compactified") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.[129]

Aware of the importance of causal structure, Roger Penrose and others developed what is known as global geometry. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, and additional non-specific assumptions about the nature of matter (usually in the form of so-called energy conditions) are used to derive general results.[130]

### Horizons

Using global geometry, some spacetimes can be shown to contain boundaries called horizons, which demarcate one region from the rest of spacetime. The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the hoop conjecture, the relevant length scale is the Schwarzschild radius[131]), no light from inside can escape to the outside. Since no object can overtake a light pulse, all interior matter is imprisoned as well. Passage from the exterior to the interior is still possible, showing that the boundary, the black hole's horizon, is not a physical barrier.[132]

The ergosphere of a rotating black hole, which plays a key role when it comes to extracting energy from such a black hole

Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution (used to describe a static black hole) and the axisymmetric Kerr solution (used to describe a rotating, stationary black hole, and introducing interesting features such as the ergosphere). Using global geometry, later studies 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 stated by the black hole uniqueness theorems: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans. Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.[133]

Even more remarkably, there is a general set of laws known as black hole mechanics, which is analogous to the laws of thermodynamics. For instance, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, analogous to the entropy of a thermodynamic system. This limits the energy that can be extracted by classical means from a rotating black hole (e.g. by the Penrose process).[134] There is strong evidence that the laws of black hole mechanics are, in fact, a subset of the laws of thermodynamics, and that the black hole area is proportional to its entropy.[135] This leads to a modification of the original laws of black hole mechanics: for instance, as the second law of black hole mechanics becomes part of the second law of thermodynamics, it is possible for black hole area to decrease—as long as other processes ensure that, overall, entropy increases. As thermodynamical objects with non-zero temperature, black holes should emit thermal radiation. Semi-classical calculations indicate that indeed they do, with the surface gravity playing the role of temperature in Planck's law. This radiation is known as Hawking radiation (cf. the quantum theory section, below).[136]

There are other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed ("particle horizon"), and some regions of the future cannot be influenced (event horizon).[137] Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons associated with a semi-classical radiation known as Unruh radiation.[138]

### Singularities

Another general—and quite disturbing—feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations have "ragged edges"—regions known as spacetime singularities, where the paths of light and falling particles come to an abrupt end, and geometry becomes ill-defined. In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the Ricci scalar, take on infinite values.[139] Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a singularity inside an eternal static black hole,[140] or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole.[141] The Friedmann-Lemaître-Robertson-Walker solutions, and other spacetimes describing universes, have past singularities on which worldlines begin, namely big bang singularities, and some have future singularities (big crunch) as well.[142]

Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artefact of idealization. The famous singularity theorems, proved using the methods of global geometry, say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage[143] and also at the beginning of a wide class of expanding universes.[144] However, the theorems say little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized e.g. by the so-called BKL conjecture).[145] The cosmic censorship hypothesis states that all realistic future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity.[146]

### Evolution equations

Each solution of Einstein's equation encompasses the whole history of a universe — it is not just some snapshot of how things are, but a whole, possibly matter-filled, spacetime. It describes the state of matter and geometry everywhere and at every moment in that particular universe. Due to its general covariance, Einstein's theory is not sufficient by itself to determine the time evolution of the metric tensor. It must be combined with a coordinate condition, which is analogous to gauge fixing in other field theories.[147]

To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in so-called "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example is the ADM formalism.[148] These decompositions show that the spacetime evolution equations of general relativity are well-behaved: solutions always exist, and are uniquely defined, once suitable initial conditions have been specified.[149] Such formulations of Einstein's field equations are the basis of numerical relativity.[150]

### Global and quasi-local quantities

The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy.[151]

Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass)[152] or suitable symmetries (Komar mass).[153] If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the so-called Bondi mass at null infinity.[154] Just as in classical physics, it can be shown that these masses are positive.[155] Corresponding global definitions exist for momentum and angular momentum.[156] There have also been a number of attempts to define quasi-local quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about isolated systems, such as a more precise formulation of the hoop conjecture.[157]

## Relationship with quantum theory

If general relativity is considered one of the two pillars of modern physics, quantum theory, the basis of understanding matter from elementary particles to solid state physics, is the other.[158] However, it is still an open question as to how the concepts of quantum theory can be reconciled with those of general relativity.

### Quantum field theory in curved spacetime

Ordinary quantum field theories, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth.[159] In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on classical general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.[160] Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as Hawking radiation, leading to the possibility that they evaporate over time.[161] As briefly mentioned above, this radiation plays an important role for the thermodynamics of black holes.[162]

### Quantum gravity

The demand for consistency between a quantum description of matter and a geometric description of spacetime,[163] as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.[164] Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.[165]

Projection of a Calabi-Yau manifold, one of the ways of compactifying the extra dimensions posited by string theory

Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems. At low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity.[166] At very high energies, however, the result are models devoid of all predictive power ("non-renormalizability").[167]

Simple spin network of the type used in loop quantum gravity

One attempt to overcome these limitations is string theory, a quantum theory not of point particles, but of minute one-dimensional extended objects.[168] The theory promises to be a unified description of all particles and interactions, including gravity;[169] the price to pay is unusual features such as six extra dimensions of space in addition to the usual three.[170] In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity[171] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.[172]

Another approach starts with the canonical quantization procedures of quantum theory. Using the initial-value-formulation of general relativity (cf. the section on evolution equations, above), the result is the Wheeler-deWitt equation (an analogue of the Schrödinger equation) which, regrettably, turns out to be ill-defined.[173] However, with the introduction of what are now known as Ashtekar variables,[174] this leads to a promising model known as loop quantum gravity. Space is represented by a web-like structure called a spin network, evolving over time in discrete steps.[175]

Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,[176] there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being dynamical triangulations,[177] causal sets,[178] twistor models[179] or the path-integral based models of quantum cosmology.[180]

All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.[181]

## Current status

General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed every unambiguous observational and experimental test. Even so, there are strong indications the theory is incomplete.[182] The problem of quantum gravity and the question of the reality of spacetime singularities remain open.[183] Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics,[184] and while the so-called Pioneer anomaly might yet admit of a conventional explanation, it, too, could be a harbinger of new physics.[185] Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations,[186] and increasingly powerful computer simulations (such as those describing merging black holes) are run.[187] The race for the first direct detection of gravitational waves continues apace,[188] in the hope of creating opportunities to test the theory's validity for much stronger gravitational fields than has been possible to date.[189] More than ninety years after its publication, general relativity remains a highly active area of research.[190]

 Eötvös experiment Contributors to general relativity Einstein–Hilbert action General relativity resources, an annotated reading list giving bibliographic information on some of the most cited resources

## Notes

1. ^ This development is traced in chapters 9 through 15 of Pais 1982 and in Janssen 2005; an up-to-date collection of current research, including reprints of many of the original articles, is Renn 2007; an accessible overview can be found in Renn 2005, p. 110ff.. An early key article is Einstein 1907, cf. Pais 1982, ch. 9. The publication featuring the field equations is Einstein 1915, cf. Pais 1982, ch. 11–15.
2. ^ See Schwarzschild 1916a, Schwarzschild 1916b and Reissner 1916 (later complemented in Nordström 1918).
3. ^ Einstein 1917, cf. Pais 1982, ch. 15e.
4. ^ Hubble's original article is Hubble 1929; an accessible overview is given in Singh 2004, ch. 2–4.
5. ^ As reported in Gamow 1970. Einstein's condemnation would prove to be premature, cf. the sectionCosmology, below.
6. ^ Cf. Pais 1982, p. 253–254.
7. ^ Cf. Kennefick 2005 and Kennefick 2007.
8. ^ Cf. Pais 1982, ch. 16.
9. ^ Cf. Israel 1987, ch. 7.8–7.10 and Thorne 1994, ch. 3–9.
10. ^ Cf. the sections Orbital effects and the relativity of direction, Gravitational time dilation and frequency shift and Light deflection and gravitational time delay, and references therein.
11. ^ Cf. the section Cosmology and references therein; the historical development is traced in Overbye 1999.
12. ^ The following exposition re-traces that of Ehlers 1973, section 1.
13. ^ See, for instance, Arnold 1989, chapter 1.
14. ^ See Ehlers 1973, pp. 5f..
15. ^ See Will 1993, section 2.4 or Will 2006, section 2.
16. ^ Cf. Wheeler 1990, chapter 2; similar accounts can be found in most other popular-science books on general relativity.
17. ^ See Ehlers 1973, section 1.2, Havas 1964, and Künzle 1972. The simple thought experiment in question was first described in Heckmann & Schücking 1959.
18. ^ See Ehlers 1973, pp. 10f..
19. ^ Good introductions are, in order of increasing presupposed knowledge of mathematics, Giulini 2005, Mermin 2005, and Rindler 1991; for accounts of precision experiments, cf. part IV of Ehlers & Lämmerzahl 2006.
20. ^ An in-depth comparison between the two symmetry groups can be found in Giulini 2006a.
21. ^ For instance Rindler 1991, section 22; a thorough treatment can be found in Synge 1972, ch. 1 and 2.
22. ^ E.g. Ehlers 1973, sec. 2.3.
23. ^ Cf. Ehlers 1973, sec. 1.4. and Schutz 1985, sec. 5.1.
24. ^ See Ehlers 1973, p. 17ff.; a derivation can be found e.g. in Mermin 2005, ch. 12. For the experimental evidence, cf. the section Gravitational time dilation and frequency shift, below.
25. ^ Cf. Rindler 2001, sec. 1.13; for an elementary account, see chapter 2 of Wheeler 1990; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. Norton 1985.
26. ^ Ehlers 1973, sec. 1.4. for the experimental evidence, see once more section Gravitational time dilation and frequency shift. Choosing a different connection with non-zero torsion leads to a modified theory known as Einstein-Cartan theory.
27. ^ Cf. Ehlers 1973, p. 16; Kenyon 1990, sec. 7.2; Weinberg 1972, sec. 2.8.
28. ^ See Ehlers 1973, pp. 19–22; for similar derivations, see sections 1 and 2 of ch. 7 in Weinberg 1972. The Einstein tensor is the only divergence-free tensor that is a function of the metric coefficients, their first and second derivatives at most, and allows the spacetime of special relativity as a solution in the absence of sources of gravity, cf. Lovelock 1972. The tensors on both side are of second rank, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations. The fact that, as a consequence of geometric relations known as Bianchi identities, the Einstein tensor satisfies a further four identities reduces these to six independent equations, e.g. Schutz 1985, sec. 8.3.
29. ^ E.g. Kenyon 1990, sec. 7.4.
30. ^ Cf. Brans & Dicke 1961 and section 3 in ch. 7 of Weinberg 1972, Goenner 2004, sec. 7.2, and Trautman 2006, respectively.
31. ^ E.g. Wald 1984, ch. 4, Weinberg 1972, ch. 7 or, in fact, any other text-book on general relativity.
32. ^ At least approximately, cf. Poisson 2004.
33. ^ E.g. p. xi in Wheeler 1990.
34. ^ E.g. Wald 1984, sec. 4.4.
35. ^ E.g. in Wald 1984, sec. 4.1.
36. ^ For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see Giulini 2006b.
37. ^ E.g. section 5 in ch. 12 of Weinberg 1972.
38. ^ Cf. the introductory chapters of Stephani et al. 2003.
39. ^ A review showing Einstein's equation in the broader context of other PDEs with physical significance is Geroch 1996.
40. ^ For background information and a list of solutions, cf. Stephani et al. 2003; a more recent review can be found in MacCallum 2006.
41. ^ E.g. chapters 3, 5, and 6 of Chandrasekhar 1983.
42. ^ E.g. ch. 4 and sec. 3.3. in Narlikar 1993.
43. ^ Brief descriptions of these and further interesting solutions can be found in Hawking & Ellis 1973, ch. 5.
44. ^ See Lehner 2002 for an overview.
45. ^ For instance Wald 1984, sec. 4.4.
46. ^ E.g. Will 1993, sec. 4.1 and 4.2.
47. ^ Cf. section 3.2 of Will 2006 as well as Will 1993, ch. 4.
48. ^ Cf. Rindler 2001, pp. 24–26 vs. pp. 236–237 and Ohanian & Ruffini 1994, pp. 164–172. In fact, Einstein derived these effects using the equivalence principle as early as 1907, cf. Einstein 1907 and the description in Pais 1982, pp. 196–198.
49. ^ Rindler 2001, pp. 24–26; Misner, Thorne & Wheeler 1973, § 38.5.
50. ^ Pound-Rebka experiment, see Pound & Rebka 1959, Pound & Rebka 1960; Pound & Snider 1964; a list of further experiments is given in Ohanian & Ruffini 1994, table 4.1 on p. 186.
51. ^ E.g. Greenstein, Oke & Shipman 1971; the most recent and most accurate Sirius B measurements are published in Barstow, Bond & Holberg 2005.
52. ^ Starting with the Hafele-Keating experiment, Hafele & Keating 1972a and Hafele & Keating 1972b, and culminating in the Gravity Probe A experiment; an overview of experiments can be found in Ohanian & Ruffini 1994, table 4.1 on p. 186.
53. ^ GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see Ashby 2002 and Ashby 2003.
54. ^ Reviews are given in Stairs 2003 and Kramer 2004.
55. ^ General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; Ohanian & Ruffini 1994, section 4.2.
56. ^ Cf. Ohanian & Ruffini 1994, pp. 164–172.
57. ^ Cf. Kennefick 2005 for the classic early measurements by the Eddington expeditions; for an overview of more recent measurements, see Ohanian & Ruffini 1994, chapter 4.3. For the most precise direct modern observations using quasars, cf. Shapiro et al. 2004.
58. ^ This is not an independent axiom; it can be derived from Einstein's equations and the Maxwell Lagrangian using a WKB approximation, cf. Ehlers 1973, section 5.
59. ^ A brief descriptions and pointers to the literature can be found in Blanchet 2006, section 1.3.
60. ^ See Rindler 2001, section 1.16; for the historical examples, Israel 1987, p. 202–204.; in fact, Einstein published one such derivation as Einstein 1907. Such calculations tacitly assume that the geometry of space is Euclidean, cf. Ehlers & Rindler 1997.
61. ^ From the standpoint of Einstein's theory, these derivations take into account the effect of gravity on time, but not its consequences for the warping of space, cf. Rindler 2001, sec. 11.11.
62. ^ For the Sun's gravitational field using radar signals reflected from planets such as Venus and Mercury, cf. Shapiro 1964, with a pedagogical introduction to be found in Weinberg 1972, ch. 8, sec. 7; for signals actively sent back by space probes (transponder measurements), cf. Bertotti, Iess & Tortora 2003; for an overview, see Ohanian & Ruffini 1994, table 4.4 on p. 200; for more recent measurements using signals received from a pulsar that is part of a binary system, the gravitational field causing the time delay being that of the other pulsar, cf. Stairs 2003, section 4.4.
63. ^ Will 1993, sec. 7.1 and 7.2.
64. ^ These have been indirectly observed through the loss of energy in binary pulsar systems such as the Hulse-Taylor binary, the subject of the 1993 Nobel Prize in physics. A number of projects are underway to attempt to observe directly the effects of gravitational waves. For an overview, see Misner, Thorne & Wheeler 1973, part VIII. Unlike electromagnetic waves, the dominant contribution for gravitational waves is not the dipole, but the quadrupole; see Schutz 2001.
65. ^ Most advanced textbooks on general relativity contain a description of these properties, e.g. Schutz 1985, ch. 9.
66. ^ For example Jaranowski & Królak 2005.
67. ^ Rindler 2001, ch. 13.
68. ^ See Gowdy 1971, Gowdy 1974.
69. ^ See Lehner 2002 for a brief introduction to the methods of numerical relativity, and Seidel 1998 for the connection with gravitational wave astronomy.
70. ^ See Schutz 2003, pp. 48–49 and Pais 1982, pp. 253–254.
71. ^ See Rindler 2001, sec. 11.9.
72. ^ See Will 1993, pp. 177–181.
73. ^ In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf. Will 2006, sec. 3.5 and Will 1993, sec. 7.3.
74. ^ The most precise measurements are VLBI measurements of planetary positions; see Will 1993, chapter 5, Will 2006, section 3.5, Anderson et al. 1992; for an overview, Ohanian & Ruffini 1994, pp. 406–407.
75. ^ See Kramer et al. 2006.
76. ^ A figure that includes error bars is figure 7, in section 5.1, of Will 2006.
77. ^ See Stairs 2003 and Schutz 2003, pp. 317–321; an accessible account can be found in Bartusiak 2000, pp. 70–86.
78. ^ An overview can be found in Weisberg & Taylor 2003; for the pulsar discovery, see Hulse & Taylor 1975; for the initial evidence for gravitational radiation, see Taylor 1994.
79. ^ Cf. Kramer 2004.
80. ^ See e.g. Penrose 2004, §14.5, Misner, Thorne & Wheeler 1973, sec. §11.4.
81. ^ See Weinberg 1972, sec. 9.6, Ohanian & Ruffini 1994, sec. 7.8.
82. ^ See Bertotti, Ciufolini & Bender 1987 and, for a more recent review, Nordtvedt 2003.
83. ^ See Kahn 2007.
84. ^ E.g. Townsend 1997, sec. 4.2.1, Ohanian & Ruffini 1994, pp. 469–471.
85. ^ E.g. Ohanian & Ruffini 1994, sec. 4.7, Weinberg 1972, sec. 9.7; for a more recent review, see Schäfer 2004.
86. ^ E.g. Ciufolini & Pavlis 2004, Ciufolini, Pavlis & Peron 2006, Iorio 2009
87. ^ Iorio L. (2006). "COMMENTS, REPLIES AND NOTES: A note on the evidence of the gravitomagnetic field of Mars". Classical Quantum Gravity 23 (17): 5451–5454. doi:10.1088/0264-9381/23/17/N01.
88. ^ Iorio L. (2009). "On the Lense-Thirring test with the Mars Global Surveyor in the gravitational field of Mars". Central European Journal of Physics. doi:10.2478/s11534-009-0117-6.
89. ^ A mission description can be found in Everitt et al. 2001; a first post-flight evaluation is given in Everitt et al. 2007; further updates will be available on the mission website Kahn 1996–2008.
90. ^ For overviews of gravitational lensing and its applications, see Ehlers, Falco & Schneider 1992 and Wambsganss 1998.
91. ^ For a simple derivation, see Schutz 2003, ch. 23; cf. Narayan & Bartelmann 1997, sec. 3.
92. ^ See Walsh, Carswell & Weymann 1979.
93. ^ Images of all the known lenses can be found on the pages of the CASTLES project, Kochanek et al. 2007.
94. ^ For an overview, see Roulet & Mollerach 1997.
95. ^ See Narayan & Bartelmann 1997, sec. 3.7.
96. ^ For an overview, Barish 2005; accessible accounts can be found in Bartusiak 2000 and Blair & McNamara 1997.
97. ^ An overview is given in Hough & Rowan 2000.
98. ^ See Danzmann & Rüdiger 2003.
99. ^ See Landgraf, Hechler & Kemble 2005.
100. ^ Cf. Thorne 1995.
101. ^ Cf. Cutler & Thorne 2002.
102. ^ See Miller 2002, lectures 19 and 21.
103. ^ E.g. Celotti, Miller & Sciama 1999, sec. 3.
104. ^ Cf. Springel et al. 2005 and the accompanying summary Gnedin 2005.
105. ^ Cf. Blandford 1987, section 8.2.4
106. ^ For the basic mechanism, see Carroll & Ostlie 1996, sec. 17.2; for more about the different types of astronomical objects associated with this, cf. Robson 1996.
107. ^ For a review, see Begelman, Blandford & Rees 1984. Interestingly, to a distant observer, some of these jets even appear to move faster than light; this, however, can be explained as an optical illusion that does not violate the tenets of relativity, see Rees 1966.
108. ^ For stellar end states, cf. Oppenheimer & Snyder 1939 or, for more recent numerical work, Font 2003, sec. 4.1; for supernovae, there are still major problems to be solved, cf. Buras et al. 2003; for simulating accretion and the formation of jets, cf. Font 2003, sec. 4.2. Also, relativistic lensing effects are thought to play a role for the signals received from X-ray pulsars, cf. Kraus 1998.
109. ^ The evidence includes limits on compactness from the observation of accretion-driven phenomena ("Eddington luminosity"), see Celotti, Miller & Sciama 1999, observations of stellar dynamics in the center of our own Milky Way galaxy, cf. Schödel et al. 2003, and indications that at least some of the compact objects in question appear to have no solid surface, which can be deduced from the examination of X-ray bursts for which the central compact object is either a neutron star or a black hole; cf. Remillard et al. 2006 for an overview, Narayan 2006, sec. 5. Observations of the "shadow" of the Milky Way galaxy's central black hole horizon are eagerly sought for, cf. Falcke, Melia & Agol 2000.
110. ^ Cf. Dalal et al. 2006.
111. ^ E.g. Barack & Cutler 2004.
112. ^ Originally Einstein 1917; cf. the description in Pais 1982, pp. 285–288.
113. ^ See Carroll 2001, ch. 2.
114. ^ See Bergström & Goobar 2003, ch. 9–11; 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. Peebles et al. 1991.
115. ^ E.g. with WMAP data, see Spergel et al. 2003.
116. ^ These tests involve the separate observations detailed further on, see, e.g., fig. 2 in Bridle et al. 2003.
117. ^ See Peebles 1966; for a recent account of predictions, see Coc et al. 2004; an accessible account can be found in Weiss 2006; compare with the observations in Olive & Skillman 2004, Bania, Rood & Balser 2002, O'Meara et al. 2001, and Charbonnel & Primas 2005.
118. ^ A review can be found in Lahav & Suto 2004 and Bertschinger 1998; for more recent results, see Springel et al. 2005.
119. ^ Cf. Alpher & Herman 1948 and, for a pedagogical introduction, see Bergström & Goobar 2003, ch. 11; for the initial detection, see Penzias & Wilson 1965 and, for precision measurements by satellite observatories, Mather et al. 1994 (COBE) and Bennett et al. 2003 (WMAP). Future measurements could also reveal evidence about gravitational waves in the early universe; this additional information is contained in the background radiation's polarization, cf. Kamionkowski, Kosowsky & Stebbins 1997 and Seljak & Zaldarriaga 1997.
120. ^ Evidence for this comes from the determination of cosmological parameters and additional observations involving the dynamics of galaxies and galaxy clusters cf. chapter 18 of Peebles 1993, evidence from gravitational lensing, cf. Peacock 1999, sec. 4.6, and simulations of large-scale structure formation, see Springel et al. 2005.
121. ^ See Peacock 1999, ch. 12, and Peskin 2007; 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. Peacock 1999, ch. 12.
122. ^ 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 Mannheim 2006, sec. 9.
123. ^ See Carroll 2001; an accessible overview is given in Caldwell 2004. 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. Mannheim 2006, sec. 10; 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. Buchert 2007.
124. ^ A good introduction is Linde 1990; for a more recent review, see Linde 2005.
125. ^ More precisely, these are the flatness problem, the horizon problem, and the monopole problem; a pedagogical introduction can be found in Narlikar 1993, sec. 6.4, see also Börner 1993, sec. 9.1.
126. ^ See Spergel et al. 2007, sec. 5 & 6.
127. ^ More concretely, the potential function that is crucial to determining the dynamics of the inflaton is simply postulated, but not derived from an underlying physical theory.
128. ^ See Brandenberger 2007, sec. 2.
129. ^ See Frauendiener 2004, Wald 1984, section 11.1, and Hawking & Ellis 1973, section 6.8 & 6.9
130. ^ E.g. Wald 1984, sec. 9.2–9.4 and Hawking & Ellis 1973, ch. 6.
131. ^ See Thorne 1972; for an account of more recent numerical studies, see Berger 2002, sec. 2.1.
132. ^ For an account of the evolution of this concept, see Israel 1987. A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and apparent horizons cf. Hawking & Ellis 1973, pp. 312–320 or Wald 1984, sec. 12.2; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. Ashtekar & Krishnan 2004.
133. ^ For first steps, cf. Israel 1971; see Hawking & Ellis 1973, sec. 9.3 or Heusler 1996, ch. 9 and 10 for a derivation, and Heusler 1998 as well as Beig & Chruściel 2006 as overviews of more recent results.
134. ^ The laws of black hole mechanics were first described in Bardeen, Carter & Hawking 1973; a more pedagogical presentation can be found in Carter 1979; for a more recent review, see chapter 2 of Wald 2001. A thorough, book-length introduction including an introduction to the necessary mathematics Poisson 2004. For the Penrose process, see Penrose 1969.
135. ^ See Bekenstein 1973, Bekenstein 1974.
136. ^ The fact that black holes radiate, quantum mechanically, was first derived in Hawking 1975; a more thorough derivation can be found in Wald 1975. A review is given in chapter 3 of Wald 2001.
137. ^ Cf. Narlikar 1993, sec. 4.4.4 and 4.4.5.
138. ^ Horizons: cf. Rindler 2001, sec. 12.4. Unruh effect: Unruh 1976, cf. Wald 2001, chapter 3.
139. ^ See Hawking & Ellis 1973, section 8.1, Wald 1984, section 9.1.
140. ^ See Townsend 1997, chapter 2; a more extensive treatment of this solution can be found in Chandrasekhar 1983, chapter 3.
141. ^ See Townsend 1997, chapter 4; for a more extensive treatment, cf. Chandrasekhar 1983, chapter 6.
142. ^ See Ellis & van Elst 1999; a closer look at the singularity itself is taken in Börner 1993, sec. 1.2
143. ^ Namely when there are trapped null surfaces, cf. Penrose 1965.
144. ^ See Hawking 1966.
145. ^ The conjecture was made in Belinskii, Khalatnikov & Lifschitz 1971; for a more recent review, see Berger 2002. An accessible exposition is given by Garfinkle 2007.
146. ^ The restriction to future singularities naturally excludes initial singularities such as the big bang singularity, which in principle be visible to observers at later cosmic time. The cosmic censorship conjecture was first presented in Penrose 1969; a text-book level account is given in Wald 1984, pp. 302–305. For numerical results, see the review Berger 2002, sec. 2.1.
147. ^ Cf. Hawking & Ellis 1973, sec. 7.1.
148. ^ Arnowitt, Deser & Misner 1962; for a pedagogical introduction, see Misner, Thorne & Wheeler 1973, §21.4–§21.7.
149. ^ Fourès-Bruhat 1952 and Bruhat 1962; for a pedagogical introduction, see Wald 1984, ch. 10; an online review can be found in Reula 1998.
150. ^ See Gourgoulhon 2007; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see Lehner 2001.
151. ^ Cf. Misner, Thorne & Wheeler 1973, §20.4.
152. ^ Arnowitt, Deser & Misner 1962.
153. ^ Cf. Komar 1959; for a pedagogical introduction, see Wald 1984, sec. 11.2; although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes, cf. Ashtekar & Magnon-Ashtekar 1979.
154. ^ For a pedagogical introduction, see Wald 1984, sec. 11.2.
155. ^ See the various references given on p. 295 of Wald 1984; this is important for questions of stability—if there were negative mass states, then flat, empty Minkowski space, which has mass zero, could evolve into these states.
156. ^ E.g. Townsend 1997, ch. 5.
157. ^ Such quasi-local mass-energy definitions are the Hawking energy, Geroch energy, or Penrose's quasi-local energy-momentum based on twistor methods; cf. the review article Szabados 2004.
158. ^ An overview of quantum theory can be found in standard textbooks such as Messiah 1999; a more elementary account is given in Hey & Walters 2003.
159. ^ Cf. textbooks such as Ramond 1990, Weinberg 1995, or Peskin & Schroeder 1995; a more accessible overview can be found in Auyang 1995.
160. ^ Cf. Wald 1994 and Birrell & Davies 1984.
161. ^ For Hawking radiation Hawking 1975, Wald 1975; an accessible introduction to black hole evaporation can be found in Traschen 2000.
162. ^ Cf. chapter 3 in Wald 2001.
163. ^ Put simply, matter is the source of spacetime curvature, and once matter has quantum properties, we can expect spacetime to have them as well. Cf. section 2 in Carlip 2001.
164. ^ E.g. p. 407ff. in Schutz 2003.
165. ^ A timeline and overview can be found in Rovelli 2000.
166. ^ See Donoghue 1995.
167. ^ In particular, a technique known as renormalization, an integral part of deriving predictions which take into account higher-energy contributions, cf. chapters 17 and 18 of Weinberg 1996, fails in this case; cf. Goroff & Sagnotti 1985.
168. ^ An accessible introduction at the undergraduate level can be found in Zwiebach 2004; more complete overviews can be found in Polchinski 1998a and Polchinski 1998b.
169. ^ At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different modes of oscillation of one and the same type of fundamental string appear as particles with different (electric and other) charges, e.g. Ibanez 2000. The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity, e.g. Green, Schwarz & Witten 1987, sec. 2.3 and 5.3.
170. ^ E. g. Green, Schwarz & Witten 1987, sec. 4.2.
171. ^ E.g. Weinberg 2000, ch. 31.
172. ^ E.g. Townsend 1996, Duff 1996.
173. ^ Cf. section 3 in Kuchař 1973.
174. ^ These variables represent geometric gravity using mathematical analogues of electric and magnetic fields; cf. Ashtekar 1986, Ashtekar 1987.
175. ^ For a review, see Thiemann 2006; more extensive accounts can be found in Rovelli 1998, Ashtekar & Lewandowski 2004 as well as in the lecture notes Thiemann 2003.
176. ^ See e.g. the systematic expositions in Isham 1994 and Sorkin 1997.
177. ^ See Loll 1998.
178. ^ See Sorkin 2005.
179. ^ See ch. 33 in Penrose 2004 and references therein.
180. ^ Cf. Hawking 1987.
181. ^ E.g. Ashtekar 2007, Schwarz 2007.
182. ^ Cf. Maddox 1998, pp. 52–59 and 98–122; Penrose 2004, section 34.1 and chapter 30.
183. ^ Cf. the section Quantum gravity, above.
184. ^ Cf. the section Cosmology, above.
185. ^ See Nieto 2006.
186. ^ See Friedrich 2005.
187. ^ A review of the various problems and the techniques being developed to overcome them, see Lehner 2002.
188. ^ See Bartusiak 2000 for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as GEO 600 and LIGO.
189. ^ For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see Blanchet et al. 2008, and Arun et al. 2007; for a review of work on compact binaries, see Blanchet 2006 and Futamase & Itoh 2006; for a general review of experimental tests of general relativity, see Will 2006.
190. ^ A good starting point for a snapshot of present-day research in relativity is the electronic review journal Living Reviews in Relativity.

## References

Popular books
• Geroch, Robert (1981). General Relativity from A to B. Chicago: University of Chicago Press. ISBN 0-226-28864-1.
• Lieber, Lillian (2008). The Einstein Theory of Relativity: A Trip to the Fourth Dimension. Philadelphia: Paul Dry Books, Inc.. ISBN 978-1-58988-044-3.
• Wald, Robert M. (1992). Space, Time, and Gravity: the Theory of the Big Bang and Black Holes. Chicago: University of Chicago Press. ISBN 0-226-87029-4.
• Callahan, James J. (2000). The Geometry of Spacetime: an Introduction to Special and General Relativity. New York: Springer. ISBN 0-387-98641-3.
• Taylor, Edwin F.; Wheeler, John Archibald (2000). Exploring Black Holes: Introduction to General Relativity. Addison Wesley. ISBN 0-201-38423-X.
• Cheng, Ta-Pei (2005). Relativity, Gravitation and Cosmology: a Basic Introduction. Oxford and New York: Oxford University Press. ISBN 0198529570.
• Gron, O.; Hervik, S. (2007), Einstein's General theory of Relativity, Springer, ISBN 978-0-387-69199-2
• Hartle, James B. (2003). Gravity: an Introduction to Einstein's General Relativity. San Francisco: Addison-Wesley. ISBN 0-8053-8662-9.
• Hughston, L. P. & Tod, K. P. (1991). Introduction to General Relativity. Cambridge: Cambridge University Press. ISBN 0-521-33943-X.
• d'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Oxford University Press. ISBN 0-19-859686-3.

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General Relativity, also known as the General Theory of Relativity, is an extension of special relativity, dealing with curved coordinate systems, accelerating frames of reference, curvilinear motion, and curvature of spacetime itself. It could be said that general relativity is to special relativity as vector calculus is to vector algebra. General relativity is best known for its formulation of gravity as a fictitious force arising from the curvature of spacetime. In fact, "general relativity" and "Einstein's formulation of gravity" are nearly synonymous in many people's minds.

The general theory of relativity was first published by Albert Einstein in 1916.

General relativity, like quantum mechanics (relativity and quantum mechanics are the two theories comprising "modern physics") both have reputations for being notoriously complicated and difficult to understand. In fact, in the early decades of the 20th century, general relativity had a sort of cult status in this regard. General relativity and quantum mechanics are both advanced college-level and postgraduate level topics. We won't attempt to give a comprehensive explanation of general relativity at the expert level. But we will attempt to give a rough outline, for reasonably advanced students, of the general relativistic formulation of gravity, below. A somewhat simpler, but, we hope, still reasonably literate, introduction for students may be found here.

Modern science does not say that Newtonian (classical) gravity is wrong. It is obviously very very nearly correct. In the weak field approximation, such as one finds in our solar system, the differences between general relativity and Newtonian gravity are minuscule. It takes very sensitive tests to show the difference. The history of those tests is a fascinating subject, and will be covered near the end of this article. But in all tests conducted so far, where there are discrepancies between the predictions of general relativity and Newtonian gravity (or other competing theories for that matter), experimental results have shown general relativity to be a better description.

Outside of the solar system, one can find stronger gravitational fields, and other phenomena, such as quasars and neutron stars, that permit even more definitive tests. General relativity appears to pass those tests as well.

This is not to say, by any means, that general relativity is the ultimate, perfect theory. It has never been unified with modern formulations of quantum mechanics, and it is therefore known to be incorrect at extremely small scales. Just as Newtonian gravity is very nearly correct, and completely correct for its time, general relativity is believed to be very nearly correct, but not completely so. Contemporary speculation on the next step involves extremely esoteric notions such as string theory, gravitons, and "quantum loop gravity".

The theory was inspired by a thought experiment developed by Einstein involving two elevators. The first elevator is stationary on the Earth, while the other is being pulled through space at a constant acceleration of g. Einstein realized that any physical experiment carried out in the elevators would give the same result. This realization is known as the equivalence principle and it states that accelerating frames of reference and gravitational fields are indistinguishable. General relativity is the theory of gravity that incorporates special relativity and the equivalence principle.

Click here for video lectures by Kip Thorne of Caltech on the mathematics of General Relativity.

## Curved space-time

Curvature of the four-dimensional space-time is the basis of general relativity. A curved space is difficult to conceive particularly the fourth dimension is peculiar. Einstein calls it t=x4. It seems simpler to consider it as an imaginary number ict where i is the quadratic root of -1 and c the speed of light. Then the space-time has the following four dimensions: (x,y,z,w=ict).

## Riemann coordinates

Understanding of general relativity, like restricted relativity, will be easier by using two dimensions (x, y=ict) instead of four. With this representation, we will have a riemannian instead of pseudo-riemannian space. Cartesian coordinates are the most common reference system. The Earth, being spherical, is not a flat space and the Pythagorean theorem is valid only locally. The cartesian frame changes its orientation from place to place but the law of gravity is the same in Paris or in Valparaiso. The Riemann coordinates are local cartesian coordinates. They are such that the Pythagorean theorem is valid even on a curved surface. It is not necessary to know the transformation from curved coordinates to use them. They are not always suitable, for example, it is necessary to compute the Riemann tensor in Gauss (e.g. spherical) coordinates in order to obtain the Schwarzschild metric.

## The metric

The metric of a euclidean space represents, in the plane, the Pythagorean theorem.

 $\left. ds^2= dx^2 + dy^2\right.$

The metric of a curved surface is, according to Gauss:

 $\left. ds^2= g_{xx} dx^2 + 2 g_{xy} dx dy + g_{yy} dy^2\right.$

where the gij are the coefficients of the metric. Every curved surface may be approximated, locally, by the osculating paraboloid, becoming the tangent plane z=0 when the principal curvatures kx et ky cancel:

 $z= \frac{1}{2}\left(k_{x} x^2 + k_{y} y^2\right)$

Indeed, in the frame used, the axes Ox and Oy are in the tangent plane z=0, the origin of the coordinates, x=0, y=0 being at the contact point. The Gauss curvature is, by definition, the product of the principal curvatures:

 $K=k_{x}k_{y}= \frac{\partial ^2 z}{\partial x^2} \frac{\partial ^2 z}{\partial y^2}$

In order to be in Riemann coordinates, it remains to orientate the axes Ox and Oy in such a manner that the metric be diagonal (the computation is given in [1]):

 $ds^2= dx^2 + \left[1 - K\left( x^2 + y^2\right)\right] dy^2$

where K= kxky is the Gaussian curvature. In this expression, we have gxx=1, gxy=0 and

 $g_{yy} =1 - K\left( x^2 + y^2\right)$

It is not necessary to determine the principal directions to work with the Riemann coordinates since the laws of physics are invariant under a frame change. It is also not necessary to change the scales of the coordinate axes to get a metric with coefficients equal to one. It only assumed that it is always possible to change the coordinates in such a way that the Pythagorean theorem is verified locally, at the contact point, taken as the origin of the coordinates. In Riemann coordinates, all the paraboloids, including the sphere, locally, have the same metric, provided thet have the same Gaussian curvature.

## Riemann tensor

Gauss found a formula of the curvature K of a surface with a computation, complicated in Gaussian coordinates but much simpler in Riemannian coordinates where the curvature and the Riemann tensor are equal (in two dimensions) [1]:

 $R_{xyxy}= -\frac{1}{2}\left(\frac{\partial ^2 g_{xx}}{\partial y^2} + \frac{\partial ^2 g_{yy}}{\partial x^2}\right)$

Let us check that the Riemann tensor is equal to the total Gauss curvature:

 $R_{xyxy}= -\frac{1}{2}\left(\frac{\partial ^2 g_{xx}}{\partial y^2} + \frac{\partial ^2 g_{yy}}{\partial x^2}\right)= 0 -\frac{1}{2}(-2K)=K$

We have also, by partial derivation of the coefficients of the metric:

 $\frac{\partial ^2 g_{xx}}{\partial x^2} + \frac{\partial ^2 g_{xx}}{\partial y^2}= 0$

The same for gyy

 $\frac{\partial ^2 g_{yy}}{\partial x^2} + \frac{\partial ^2 g_{yy}}{\partial y^2}= -4K$

We have obtained a Laplace equation and a Poisson equation.

## Einstein equations in vacuum

Einstein's hypothesis is that the curvature of space-time is zero in the vacuum which is thus a flat space. This is true in two dimensions where the Gaussian curvature is zero. In higher dimensions, only the Ricci tensor is zero according to the Einstein equation. In matter, the Ricci tensor is different from zero. We shall not consider this case, here, but it should be considered to describe the universe which contains matter. The Einstein equations are, in the vacuum:

 $\left.R_{ik}= 0\right.$

Rik is a complicated function of the various componants of the Riemann tensorRijkl and of the metric gik. The Ricci tensor, like the Riemann tensor dépends only on the coefficients of the metric. The Christoffel symbols http://en.wikipedia.org/wiki/Christoffel_symbol are then unnecessary intermediaries. In two dimensions, the Ricci tensor has two components each proportional to the single component of the Riemann tensor. Therefore there is only one Einstein equation in two dimensions:

 $\left.R_{xyxy}= 0\right.$

In two dimensions and in Riemann coordinates, the Riemann tensor is equal to the Gaussian curvature K, which is zero in the vacuum. Then the coefficients of the metric have to satisfy the Laplace equation Δgxx=0 and Δgyy=0. But, in two dimensions, the Laplace equation diverges unless the coefficients of the metric are constants, corresponding to a pseudo-euclidean space. In three and four dimensions, the Ricci tensor has to be zero, the corresponding space is called Ricci flat. The calculation is too complicated to be given here.

## Gravitational waves

Replacing y by ict in the Laplace equation, one obtains the d'Alembert equation of the plane gravitational waves for the coefficients of the metric:

 $\frac{\partial ^2 g_{xx}}{\partial x^2} - \frac{1}{c^2}\frac{\partial ^2 g_{xx}}{\partial t^2} = 0$
 $\frac{\partial ^2 g_{tt}}{\partial x^2} - \frac{1}{c^2}\frac{\partial ^2 g_{tt}}{\partial t^2} = 0$

The gravitational waves have not yet been detected.

## Einstein and Newton

The two-dimensional Laplace equation may be extrapolated in higher spaces with small curvature. In three dimensions, spherical symmetry and time independent metric, the Einstein equations reduce to the radial laplacian:

 $\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dg_{rr}}{dr}\right)= 0$

and for gtt. Its solution is the Coulomb potential in 1/r:

 $ds^2= g_{rr} dr^2 - g_{tt} c^2 dt^2= \left(A+ \frac{B}{r}\right) dr^2 -c^2 \left(A'+ \frac{B'}{r}\right) dt^2$

The correspondence principle with special relativity will give us the integration constants A and A'. For r=∞, we have:

 $ds^2= \left.A dr^2 -c^2 A' dt^2\right.$

It should be the Minkowski metric:

 $ds^2=\left.dr^2-c^2dt^2\right.$

Identifying these two metrics, we get A=A'=1. To obtain B', we apply the correspondence principle with the newtonian gravitation of a light particle on a circular trajectory around a highly attracting star similar to a black hole. Then dr=0, the metric is simplified:

 $\left. ds^2= -c^2g_{tt} dt^2 = -c^2\left( 1+ \frac{B'}{R}\right) dt^2\right.$

The Minkowski metric may be written

 $ds^2=-\left(1 -\frac{v^2}{c^2}\right)dt^2$

where v=dr/dt is the velocity of the particle. For a photon, v=c, ds=0: the length of a light trajectory is zero. It is the shortest way possible. Assuming that this remains true in general relativity, we have the condition:

 $1+ \frac{B'}{R}=0$

which gives R=-B'. The trajectory being a circle and the curvature of space small, we may apply newtonian mechanics. The kinetic energy is equal to the newtonian gravitation potential:

 $\frac{1}{2}mv^2=\frac{GM}{R}$

where G is the gravitation constant, M the mass of the attracting star and c the speed of light. Replacing R with -B' and v with c, we get:

 $B'= -\frac{2GM}{c^2}$

According to Einstein, the determinant (or its trace for low gravitation) of the metric should be equal to one. This can be shown by solving the four-dimensional Einstein equations for a static and spherically symmetric gravitational field. Therefore we may write B=-B' and obtain an approximation of the Schwarzschid metric:

 $\left. ds^2= \left(1+ \frac{2GM}{c^2r}\right) dr^2 -c^2 \left(1- \frac{2GM}{c^2r}\right) dt^2\right.$

This metric gives a light deviation by the sun twice as predicted by the newtonian theory or by the first Einstein theory of 1911 where time is dilated by gravitation. In his 1916 theory, gravitation dilates time and contracts space.

## References

1. 1.0 1.1 Bernard Schaeffer, Relativités et quanta clarifiés, Publibook, 2007

# Wikibooks

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## Index

 Tensors and geometry Tensors and matter-energy The stress-energy tensor from the action Physics in curved space Applications Celestial mechanics Approximations Precession of Mercury Global positioning system General Relativity/Cosmology Friedman-Robertson-Walker cosmology Gravitational waves Black holes Birkhoff's theorem Schwarzschild metric Eddington-Finkelstein coordinates Kruskal extension Kerr-Neuman black hole Reissner-Nordström black hole What is a singularity? BKL singularity Singularity theorems More on Tensors The Tensor Product Contraction Tensor Symmetries How Tensors Transform The Metric Tensor Differential geometry Differentiable manifolds Vector bundles Tangent and cotangent bundles Diffeomorphisms Tensor bundles and jet bundles Connections over vector bundles Lie derivatives Differential forms Exterior derivatives and de Rham cohomology Chain homology Stoke's theorem Pseudo Riemannian manifolds Metric tensor Torsion Levi-Civita connection Conformal rescalings and the Weyl tensor Coordinates and component notation Einstein-Hilbert action Causal structure Penrose diagrams Moving frames Principal bundles Frame bundle External bundle Frame bundle connection Palatini action The equivalence between the Palatini and the Einstein-Hilbert action Spinor reps of the Lorentz group Spinor bundles Dirac and Majorana spinors in curved spacetime

## References

• D'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Clarendon Press. ISBN 0-19-859686-3.
• Wald, Robert M. (1984). General Relativity. University of Chicago. ISBN 0-226-87033-2.
• P. A. M. Dirac (1996). General Theory of Relativity. Princeton University Press. ISBN 0-691-01146-X.
• Misner, Thorne, Wheeler (1973). Gravitation. W H Freeman and Company. ISBN 0-7167-0344-0.

# Simple English

General relativity is a theory of space and time created by Albert Einstein and published in 1915. The central idea of general relativity is that space and time are two aspects of something called spacetime, which is curved in the presence of matter, energy, and momentum, in a way that Einstein wrote down in the Einstein Field Equations.

In general relativity, freefall is inertial motion instead of being at rest on a massive body such as the Earth, as described by the equivalence principle.

General relativity has made many successful predictions. These include:

• The bending of light as it passes the Sun by twice the Newtonian value for an object traveling at the speed of light. This was confirmed by Eddington in 1919, and the announcement forced scientists to take general relativity seriously.
• The perihelion of the planet Mercury advances more than is expected under Newtonian physics. General relativity accounts for the difference between what is seen and what is expected without it.
• The gravitational redshift effect, by which the wavelength of light increases as it moves away from a massive object. This was confirmed by the Pound-Rebka experiment.
• The Shapiro effect, under which light appears to slow down as it passes close to a massive object. This was first confirmed in the 1960s for space probes headed towards the planet Venus.