General relativity

For the graduate textbook by Robert Wald, seeGeneral Relativity (book).

For a more accessible and less technical introduction to this topic, seeIntroduction to general relativity.

Slow motion computer simulation of the black holebinary systemGW150914 as seen by a nearby observer, during 0.33 s of its finalinspiral,merge,andringdown.The star field behind the black holes is being heavily distorted and appears to rotate and move, due to extremegravitational lensing,asspacetimeitself is distorted and dragged around by the rotatingblack holes.

General relativity,also known as thegeneral theory of relativity,and asEinstein's theory of gravity,is thegeometrictheoryofgravitationpublished byAlbert Einsteinin 1915 and is the current description of gravitation inmodern physics.Generalrelativitygeneralizesspecial relativityand refinesNewton's law of universal gravitation,providing a unified description of gravity as a geometric property ofspaceandtime,orfour-dimensionalspacetime.In particular, thecurvatureof spacetimeis directly related to theenergyandmomentumof whatever presentmatterandradiation.The relation is specified by theEinstein field equations,a system of second-orderpartial differential equations.

Newton's law of universal gravitation,which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyondNewton's law of universal gravitationinclassical physics.These predictions concern the passage of time, thegeometryof space, the motion of bodies infree fall,and the propagation of light, and includegravitational time dilation,gravitational lensing,thegravitational redshiftof light, theShapiro time delayandsingularities/black holes.So far, alltests of general relativityhave been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework forcosmology,thus leading to the discovery of theBig Bangandcosmic microwave backgroundradiation. Despite the introduction of a number ofalternative theories,general relativity continues to be the simplest theory consistent withexperimental data.

Reconciliation of general relativity with the laws ofquantum physicsremains a problem, however, as there is a lack of a self-consistent theory ofquantum gravity.It is not yet known how gravity can beunifiedwith the three non-gravitational forces:strong,weakandelectromagnetic.

Einstein's theory hasastrophysicalimplications, including the prediction ofblack holes—regions of space in which space and time are distorted in such a way that nothing, not evenlight,can escape from them. Black holes are the end-state formassive stars.Microquasarsandactive galactic nucleiare believed to bestellar black holesandsupermassive black holes.It also predictsgravitational lensing,where the bending of light results in multiple images of the same distant astronomical phenomenon. Other predictions include the existence ofgravitational waves,which have beenobserved directlyby the physics collaborationLIGOand other observatories. In addition, general relativity has provided the base ofcosmologicalmodels of anexpanding universe.

Widely acknowledged as a theory of extraordinarybeauty,general relativity has often been described as the most beautiful of all existing physical theories.

History

Main articles:History of general relativityandClassical theories of gravitation

Henri Poincaré's 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity. While others thought that gravity was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed thatgravitational wavespropagate at the speed of light.Soon afterwards, Einstein started thinking about how to incorporategravityinto his relativistic framework. In 1907, beginning with a simplethought experimentinvolving an observer in free fall (FFO), 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 presentation to thePrussian Academy of Sciencein November 1915 of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity.These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present.A version ofnon-Euclidean geometry,calledRiemannian geometry,enabled Einstein to develop general relativity by providing the key mathematical framework on which he fit his physical ideas of gravity.This idea was pointed out by mathematicianMarcel Grossmannand published by Grossmann and Einstein in 1913.

The Einstein field equations arenonlinearand considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicistKarl Schwarzschildfound the first non-trivial exact solution to the Einstein field equations, theSchwarzschild 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 toelectrically chargedobjects were taken, eventually resulting in theReissner–Nordström solution,which is now associated withelectrically charged black holes.In 1917, Einstein applied his theory to theuniverseas 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—thecosmological constant—to match that observational presumption.By 1929, however, the work ofHubbleand others had shown that our universe is expanding. This is readily described by the expanding cosmological solutions found byFriedmannin 1922, which do not require a cosmological constant.Lemaîtreused these solutions to formulate the earliest version of theBig Bangmodels, in which our universe has evolved from an extremely hot and dense earlier state.Einstein later declared the cosmological constant the biggest blunder of his life.

During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior toNewtonian gravity,being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein showed in 1915 how his theory explained theanomalous perihelion advanceof the planetMercurywithout any arbitrary parameters ( "fudge factors"),and in 1919 an expedition led byEddingtonconfirmed general relativity's prediction for the deflection of starlight by the Sun during the totalsolar eclipse of 29 May 1919,instantly making Einstein famous.Yet the theory remained outside the mainstream oftheoretical physicsand astrophysics until developments between approximately 1960 and 1975, now known as thegolden age of general relativity.Physicists began to understand the concept of a black hole, and to identifyquasarsas one of these objects' astrophysical manifestations.Ever more precise solar system tests confirmed the theory's predictive power,and relativistic cosmology also became amenable to direct observational tests.

General relativity has acquired a reputation as a theory of extraordinary beauty.Subrahmanyan Chandrasekharhas noted that at multiple levels, general relativity exhibits whatFrancis Baconhas termed a "strangeness in the proportion" (i.e.elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and timeversusmatter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were the principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory.Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency.

In the preface toRelativity: The Special and the General Theory,Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated."

From classical mechanics to general relativity

General relativity can be 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 a geometric description. The combination of this description with the laws of special relativity results in aheuristicderivation of general relativity.

Geometry of Newtonian gravity

According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket (left) as it does on Earth (right), provided that the acceleration of the rocket is equal to 9.8 m/s²(the acceleration due to gravity at the surface of the Earth).

At the base ofclassical mechanicsis the notion that abody's motion can be described as a combination of free (orinertial) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's secondlaw of motion,which states that the netforceacting on a body is equal to that body's (inertial)massmultiplied by itsacceleration.The preferred inertial motions are related to the geometry of space and time: in the standardreference framesof classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths aregeodesics,straightworld linesincurved spacetime.

Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such aselectromagnetismorfriction), can be used to define the geometry of space, as well as a timecoordinate.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 ofEötvösand its successors (seeEötvös experiment), there is a universality of free fall (also known as the weakequivalence principle,or the universal equality of inertial and passive-gravitational mass): the trajectory of atest bodyin free fall depends only on its position and initial speed, but not on any of its material properties.A simplified version of this is embodied inEinstein's elevator experiment,illustrated in the figure on the right: for an observer in an enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration.

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 specificconnectionwhich depends on thegradientof thegravitational potential.Space, in this construction, still has the ordinaryEuclidean geometry.However, spacetimeas 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 notintegrable.From this, one can deduce that spacetime is curved. The resultingNewton–Cartan theoryis a geometric formulation of Newtonian gravity using onlycovariantconcepts, i.e. a description which is valid in any desired coordinate system.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.

Relativistic generalization

Light cone

As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely alimiting caseof (special) relativistic mechanics.In the language ofsymmetry:where gravity can be neglected, physics isLorentz invariantas in special relativity rather thanGalilei invariantas in classical mechanics. (The defining symmetry of special relativity is thePoincaré group,which includes translations, rotations, boosts and reflections.) The differences between the two become significant when dealing with speeds approaching thespeed of light,and with high-energy phenomena.

With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for eacheventA,there is a set of events that can, in principle, either influence or be influenced byAvia signals or interactions that do not need to travel faster than light (such as eventBin the image), and a set of events for which such an influence is impossible (such as eventCin the image). These sets areobserver-independent.In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines aconformal structureor conformal geometry.

Special relativity is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no globalinertial frames.Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straighttime-likelines 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.

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 ofpreferred 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.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 theEinstein equivalence principle,a crucial guiding principle for generalizing special-relativistic physics to include gravity.

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 theMinkowski 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. Themetric tensorthat 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- orpseudo-Riemannianmetric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, theLevi-Civita connection,and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitablelocally inertial coordinates,the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).

Einstein's equations

Main articles:Einstein field equationsandMathematics of general relativity

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 theenergy–momentum tensor,which includes bothenergyand momentumdensitiesas well asstress:pressureand shear.Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that thefield equationfor gravity relates this tensor and theRicci 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 isdivergence-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-manifoldcounterparts,covariant derivativesstudied 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 nontrivial set of equations are what are called Einstein's (field) equations:

Einstein's field equations

$G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,$

On the left-hand side is theEinstein tensor,$G_{\mu \nu }$,which is symmetric and a specific divergence-free combination of the Ricci tensor$R_{\mu \nu }$and the metric. In particular,

$R=g^{\mu \nu }R_{\mu \nu }$

is the curvature scalar. The Ricci tensor itself is related to the more generalRiemann curvature tensoras

$R_{\mu \nu }={R^{\alpha }}_{\mu \alpha \nu }.$

On the right-hand side,$\kappa$is a constant and$T_{\mu \nu }$is the energy–momentum tensor. All tensors are written inabstract index notation.Matching the theory's prediction to observational results forplanetaryorbitsor, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant$\kappa$is found to be${\textstyle \kappa ={\frac {8\pi G}{c^{4}}}}$,where$G$is theNewtonian constant of gravitationand$c$the speed of light in vacuum.When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations,

$R_{\mu \nu }=0.$

In general relativity, theworld lineof a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic.

Thegeodesic equationis:

${d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0,$

where$s$is a scalar parameter of motion (e.g. theproper time), and$\Gamma ^{\mu }{}_{\alpha \beta }$areChristoffel symbols(sometimes called theaffine connectioncoefficients orLevi-Civita connectioncoefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and thesummation conventionis used for repeated indices$\alpha$and$\beta$.The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous toNewton's laws of motionwhich likewise provide formulae for the acceleration of a particle. This equation of motion employs theEinstein notation,meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of atest particlewhose motion is described by the geodesic equation.

Total force in general relativity

See also:Two-body problem in general relativity

In general relativity, the effectivegravitational potential energyof an object of massmrevolving around a massive central bodyMis given by

$U_{f}(r)=-{\frac {GMm}{r}}+{\frac {L^{2}}{2mr^{2}}}-{\frac {GML^{2}}{mc^{2}r^{3}}}$

A conservative totalforcecan then be obtained as itsnegative gradient

$F_{f}(r)=-{\frac {GMm}{r^{2}}}+{\frac {L^{2}}{mr^{3}}}-{\frac {3GML^{2}}{mc^{2}r^{4}}}$

whereLis theangular momentum.The first term represents theforce of Newtonian gravity,which is described by the inverse-square law. The second term represents thecentrifugal forcein the circular motion. The third term represents the relativistic effect.

Alternatives to general relativity

Main article:Alternatives to general relativity

There arealternatives to general relativitybuilt upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples areWhitehead's theory,Brans–Dicke theory,teleparallelism,f(R) gravityandEinstein–Cartan theory.

Definition and basic applications

See also:Mathematics of general relativityandPhysical theories modified by general relativity

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 ametrictheory of gravitation. At its core areEinstein's equations,which describe the relation between the geometry of a four-dimensionalpseudo-Riemannian manifoldrepresenting spacetime, and theenergy–momentumcontained in that spacetime.Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such asfree-fall,orbital motion, andspacecrafttrajectories), 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.The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativistJohn Archibald Wheeler,spacetime tells matter how to move; matter tells spacetime how to curve.

While general relativity replaces thescalargravitational potential of classical physics by a symmetricrank-twotensor,the latter reduces to the former in certainlimiting cases.Forweak gravitational fieldsandslow speedrelative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.

As it is constructed using tensors, general relativity exhibitsgeneral covariance:its laws—and further laws formulated within the general relativistic framework—take on the same form in allcoordinate systems.Furthermore, the theory does not contain any invariant geometric background structures, i.e. it isbackground independent.It thus satisfies a more stringentgeneral principle of relativity,namely that thelaws of physicsare the same for all observers.Locally,as expressed in the equivalence principle, spacetime isMinkowskian,and the laws of physics exhibitlocal Lorentz invariance.

Model-building

The core concept of general-relativistic model-building is that of asolution 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.

Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.Nevertheless, a number ofexact solutionsare known, although only a few have direct physical applications.The best-known exact solutions, and also those most interesting from a physics point of view, are theSchwarzschild solution,theReissner–Nordström solutionand theKerr metric,each corresponding to a certain type of black hole in an otherwise empty universe,and theFriedmann–Lemaître–Robertson–Walkerandde Sitter universes,each describing an expanding cosmos.Exact solutions of great theoretical interest include theGödel universe(which opens up the intriguing possibility oftime travelin curved spacetimes), theTaub–NUT solution(a model universe that ishomogeneous,butanisotropic), andanti-de Sitter space(which has recently come to prominence in the context of what is called theMaldacena conjecture).

Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently bynumerical integrationon a computer, or by considering small perturbations of exact solutions. In the field ofnumerical 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.In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such asnaked singularities.Approximate solutions may also be found byperturbation theoriessuch aslinearized gravityand its generalization, thepost-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.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.

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 many years of research that followed Einstein's initial publication.

Gravitational time dilation and frequency shift

Main article:Gravitational time dilation

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

Assuming that the equivalence principle holds,gravity influences the passage of time. Light sent down into agravity wellisblueshifted,whereas light sent in the opposite direction (i.e., climbing out of the gravity well) isredshifted;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.

Gravitational redshift has been measured in the laboratoryand using astronomical observations.Gravitational time dilation in the Earth's gravitational field has been measured numerous times usingatomic clocks,while ongoing validation is provided as a side effect of the operation of theGlobal Positioning System(GPS).Tests in stronger gravitational fields are provided by the observation ofbinary pulsars.All results are in agreement with general relativity.However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.

Light deflection and gravitational time delay

Main articles:Schwarzschild geodesics,Kepler problem in general relativity,Gravitational lens,andShapiro delay

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

General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a star. This effect was initially confirmed by observing the light of stars or distant quasars being deflected as it passes theSun.

This and related predictions follow from the fact that light follows what is called a light-like ornull geodesic—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of theinvarianceof lightspeed in special relativity.As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion),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,the angle of deflection resulting from such calculations is only half the value given by general relativity.

Closely related to light deflection is the Shapiro Time Delay, 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.In theparameterized 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.

Gravitational waves

Main article:Gravitational wave

Ring of test particles deformed by a passing (linearized, amplified for better visibility) gravitational wave

Predicted in 1916by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous toelectromagnetic waves.On 11 February 2016, the Advanced LIGO team announced that they haddirectly detected gravitational wavesfrom apairof black holesmerging.

The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right).Since Einstein's equations arenon-linear,arbitrarily strong gravitational waves do not obeylinear superposition,making their description difficult. However, linear approximations of 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 beFourier decomposed.

Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty spaceorGowdy universes,varieties of an expanding cosmos filled with gravitational waves.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.

Orbital effects and the relativity of direction

Main article:Two-body problem in general relativity

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. The influence of other planets is ignored.

Main article:Apsidal precession

In general relativity, theapsidesof any orbit (the point of the orbiting body's closest approach to the system'scenter of mass) willprecess;the orbit is not anellipse,but akin to an ellipse that rotates on its focus, resulting in arose 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 atest particle.For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier byUrbain Le Verrierin 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.

The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)or the much more generalpost-Newtonian formalism.It is due to the influence of gravity on the geometry of space and to the contribution ofself-energyto a body's gravity (encoded in thenonlinearityof Einstein's equations).Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus, and Earth),as well as in binary pulsar systems, where it is larger by fiveorders of magnitude.

In general relativity the perihelion shift$\sigma$,expressed in radians per revolution, is approximately given by

$\sigma ={\frac {24\pi ^{3}L^{2}}{T^{2}c^{2}(1-e^{2})}}\,$

where:

• $L$is thesemi-major axis
• $T$is theorbital period
• $c$is the speed of light in vacuum
• $e$is theorbital eccentricity

Orbital decay

Orbital decay for PSR J0737−3039: time shift (ins), tracked over 16 years (2021).

According to general relativity, abinary systemwill 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 theSolar Systemor for ordinarydouble stars,the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbitingneutron stars,one of which is apulsar: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. Because neutron stars are immensely compact, significant amounts of energy are emitted in the form of gravitational radiation.

The first observation of a decrease in orbital period due to the emission of gravitational waves was made byHulseandTaylor,using the binary pulsarPSR1913+16they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993Nobel Prizein physics.Since then, several other binary pulsars have been found, in particular the double pulsarPSR J0737−3039,where both stars are pulsarsand which was last reported to also be in agreement with general relativity in 2021 after 16 years of observations.

Geodetic precession and frame-dragging

Main articles:Geodetic precessionandFrame dragging

Several relativistic effects are directly related to the relativity of direction.One isgeodetic precession:the axis direction of agyroscopein 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").For the Moon–Earth system, this effect has been measured with the help oflunar laser ranging.More recently, it has been measured for test masses aboard the satelliteGravity Probe Bto a precision of better than 0.3%.

Near a rotating mass, there are gravitomagnetic orframe-draggingeffects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme forrotating black holeswhere, for any object entering a zone known as theergosphere,rotation is inevitable.Such effects can again be tested through their influence on the orientation of gyroscopes in free fall.Somewhat controversial tests have been performed using theLAGEOSsatellites, confirming the relativistic prediction.Also theMars Global Surveyorprobe around Mars has been used.

Astrophysical applications

Gravitational lensing

Main article: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.Depending on the configuration, scale, and mass distribution, there can be two or more images, a bright ring known as anEinstein ring,or partial rings called arcs. Theearliest examplewas discovered in 1979;since then, more than a hundred gravitational lenses have been observed.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 "microlensingevents "have been observed.

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

Gravitational-wave astronomy

Main articles:Gravitational waveandGravitational-wave astronomy

Artist's impression of the space-borne gravitational wave detectorLISA

Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (seeOrbital decay,above). Detection of these waves is a major goal of current relativity-related research.Several land-basedgravitational wave detectorsare currently in operation, most notably theinterferometric detectorsGEO 600,LIGO(two detectors),TAMA 300andVIRGO.Variouspulsar timing arraysare usingmillisecond pulsarsto detect gravitational waves in the 10⁻⁹to 10⁻⁶hertzfrequency range, which originate from binary supermassive blackholes.A European space-based detector,eLISA / NGO,is currently under development,with a precursor mission (LISA Pathfinder) having launched in December 2015.

Observations of gravitational waves promise to complement observations in theelectromagnetic spectrum.They are expected to yield information about black holes and other dense objects such as neutron stars and white dwarfs, about certain kinds ofsupernovaimplosions, and about processes in the very early universe, including the signature of certain types of hypotheticalcosmic string.In February 2016, the Advanced LIGO team announced that they had detected gravitational waves from a black hole merger.

Black holes and other compact objects

Main article:Black hole

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

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 ofstellar evolution,neutron stars of around 1.4solar 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.Usually a galaxy has onesupermassive black holewith a few million to a fewbillionsolar masses in its center,and its presence is thought to have played an important role in the formation of the galaxy and larger cosmic structures.

Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.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.In particular, accretion can lead torelativistic jets,focused beams of highly energetic particles that are being flung into space at almost light speed. General relativity plays a central role in modelling all these phenomena,and observations provide strong evidence for the existence of black holes with the properties predicted by the theory.

Black holes are also sought-after targets in the search for gravitational waves (cf.Gravitational waves,above). Mergingblack hole binariesshould 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.The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about the supermassive black hole's geometry.

Cosmology

Main article:Physical cosmology

This blue horseshoe is a distant galaxy that has been magnified and warped into a nearly complete ring by the strong gravitational pull of the massive foreground luminous red galaxy.

The current models of cosmology are based onEinstein's field equations,which include the cosmological constant$\Lambda$since it has important influence on the large-scale dynamics of the cosmos,

$R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }+\Lambda \ g_{\mu \nu }={\frac {8\pi G}{c^{4}}}\,T_{\mu \nu }$

where$g_{\mu \nu }$is the spacetime metric.Isotropicand homogeneous solutions of these enhanced equations, theFriedmann–Lemaître–Robertson–Walker solutions,allow physicists to model a universe that has evolved over the past 14billionyears from a hot, early Big Bang phase.Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,further observational data can be used to put the models to the test.Predictions, all successful, include the initial abundance of chemical elements formed in a period ofprimordial nucleosynthesis,the large-scale structure of the universe,and the existence and properties of a "thermalecho "from the early cosmos, thecosmic background radiation.

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% of all matter appears to be dark matter, which has mass (or, equivalently, gravitational influence), but does not interact electromagnetically and, hence, cannot be observed directly.There is no generally accepted description of this new kind of matter, within the framework of knownparticle physicsor otherwise.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 unusualequation of state,known asdark energy,the nature of which remains unclear.

Aninflationary phase,an additional phase of strongly accelerated expansion at cosmic times of around 10⁻³³seconds, 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.Recent measurements of the cosmic background radiation have resulted in the first evidence for this scenario.However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations.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 bangsingularity.An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed(cf. the section onquantum gravity,below).

Exotic solutions: time travel, warp drives

Kurt Gödelshowedthat solutions to Einstein's equations exist that containclosed timelike curves(CTCs), which allow for loops in time. The solutions require extreme physical conditions unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then, other—similarly impractical—GR solutions containing CTCs have been found, such as theTipler cylinderandtraversable wormholes.Stephen Hawkingintroducedchronology protection conjecture,which is an assumption beyond those of standard general relativity to preventtime travel.

Someexact solutions in general relativitysuch asAlcubierre drivepresent examples ofwarp drivebut these solutions requires exotic matter distribution, and generally suffers from semiclassical instability.

Advanced concepts

Asymptotic symmetries

Main article:Bondi–Metzner–Sachs group

The spacetime symmetry group forspecial relativityis thePoincaré group,which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries if any might apply in General Relativity. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity,viz.,the Poincaré group.

In 1962Hermann Bondi,M. G. van der Burg, A. W. MetznerandRainer K. Sachsaddressed thisasymptotic symmetryproblem in order to investigate the flow of energy at infinity due to propagatinggravitational waves.Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at light-like infinity to characterize what it means to say a metric is asymptotically flat, making noa prioriassumptions about the nature of the asymptotic symmetry group—not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known assupertranslations.This implies the conclusion that General Relativity (GR) doesnotreduce to special relativity in the case of weak fields at long distances. It turns out that the BMS symmetry, suitably modified, could be seen as a restatement of the universalsoft graviton theoreminquantum field theory(QFT), which relates universal infrared (soft) QFT with GR asymptotic spacetime symmetries.

Causal structure and global geometry

Main article:Causal structure

Penrose–Carter diagram of an infiniteMinkowski universe

In general relativity, no material body can catch up with or overtake a light pulse. No influence from an eventAcan reach any other locationXbefore light sent out atAtoX.In consequence, an exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. This structure can be displayed usingPenrose–Carter diagramsin 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 standardspacetime diagrams.

Aware of the importance of causal structure,Roger Penroseand others developed what is known asglobal 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 theRaychaudhuri equation,and additional non-specific assumptions about the nature of matter (usually in the form ofenergy conditions) are used to derive general results.

Horizons

Main articles:Horizon (general relativity),No hair theorem,andBlack hole mechanics

Using global geometry, some spacetimes can be shown to contain boundaries calledhorizons,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 thehoop conjecture,the relevant length scale is theSchwarzschild radius), 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'shorizon,is not a physical barrier.

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 astaticblack hole) and the axisymmetricKerr solution(used to describe a rotating,stationaryblack hole, and introducing interesting features such as the ergosphere). Using global geometry, later studies have revealed more general properties of black holes. With time they become rather simple objects characterized by eleven parameters specifying: electric charge, mass–energy,linear momentum,angular momentum,and location at a specified time. This is stated by theblack hole uniqueness theorem:"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.

Even more remarkably, there is a general set of laws known asblack hole mechanics,which is analogous to thelaws 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 theentropyof a thermodynamic system. This limits the energy that can be extracted by classical means from a rotating black hole (e.g. by thePenrose process).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.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 the black hole area to decrease as long as other processes ensure that entropy increases overall. As thermodynamical objects with nonzero temperature, black holes should emitthermal radiation.Semiclassical calculations indicate that indeed they do, with the surface gravity playing the role of temperature inPlanck's law.This radiation is known asHawking radiation(cf. thequantum theory section,below).

There are many 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).Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons associated with a semiclassical radiation known asUnruh radiation.

Singularities

Main article:Spacetime singularity

Another general 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 asspacetime 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 theRicci scalar,take on infinite values.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,or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole.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.

Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artifact of idealization.The famoussingularity 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 stageand also at the beginning of a wide class of expanding universes.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 theBKL conjecture).Thecosmic censorship hypothesisstates 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.

Evolution equations

Main article:Initial value formulation (general relativity)

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 thetime evolutionof the metric tensor. It must be combined with acoordinate condition,which is analogous togauge fixingin other field theories.

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 "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example is theADM formalism.These decompositions show that the spacetime evolution equations of general relativity are well-behaved: solutions alwaysexist,and are uniquely defined, once suitable initial conditions have been specified.Such formulations of Einstein's field equations are the basis of numerical relativity.

Global and quasi-local quantities

Main article:Mass in general relativity

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.

Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass)or suitable symmetries (Komar mass).If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is theBondi massat null infinity.Just as inclassical physics,it can be shown that these masses are positive.Corresponding global definitions exist for momentum and angular momentum.There have also been a number of attempts to definequasi-localquantities, 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 aboutisolated systems,such as a more precise formulation of the hoop conjecture.

Relationship with quantum theory

If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles tosolid-state physics,would be the other.However, how to reconcile quantum theory with general relativity is still an open question.

Quantum field theory in curved spacetime

Main article:Quantum field theory in curved spacetime

Ordinaryquantum 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.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 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.Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known asHawking radiationleading to the possibility that theyevaporateover time.As briefly mentionedabove,this radiation plays an important role for the thermodynamics of black holes.

Quantum gravity

Main article:Quantum gravity

See also:String theory,Canonical general relativity,Loop quantum gravity,Causal dynamical triangulation,andCausal sets

Projection of aCalabi–Yau manifold,one of the ways ofcompactifyingthe extra dimensions posited by string theory

The demand for consistency between a quantum description of matter and a geometric description of spacetime,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.Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.

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.Some have argued that at low energies, this approach proves successful, in that it results in an acceptableeffective (quantum) field theoryof gravity.At very high energies, however, the perturbative results are badly divergent and lead to models devoid of predictive power ( "perturbativenon-renormalizability").

Simplespin networkof the type used in loop quantum gravity

One attempt to overcome these limitations isstring theory,a quantum theory not ofpoint particles,but of minute one-dimensional extended objects.The theory promises to be aunified descriptionof all particles and interactions, including gravity;the price to pay is unusual features such as sixextra dimensionsof space in addition to the usual three.In what is called thesecond superstring revolution,it was conjectured that both string theory and a unification of general relativity andsupersymmetryknown assupergravityform part of a hypothesized eleven-dimensional model known asM-theory,which would constitute a uniquely defined and consistent theory of quantum gravity.

Another approach starts with thecanonical quantizationprocedures of quantum theory. Using the initial-value-formulation of general relativity (cf.evolution equationsabove), the result is theWheeler–deWitt equation(an analogue of theSchrödinger equation) which, regrettably, turns out to be ill-defined without a proper ultraviolet (lattice) cutoff.However, with the introduction of what are now known asAshtekar variables,this leads to a promising model known asloop quantum gravity.Space is represented by a web-like structure called aspin network,evolving over time in discrete steps.

Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being the lattice theory of gravity based on the FeynmanPath Integralapproach andRegge calculus,dynamical triangulations,causal sets,twistor modelsor the path integral based models ofquantum cosmology.

Observation of gravitational waves from binary black hole merger GW150914

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.

Current status

General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests. However, there are strong indications that the theory is incomplete.The problem of quantum gravity and the question of the reality of spacetime singularities remain open.Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.

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,while numerical relativists run increasingly powerful computer simulations (such as those describing merging black holes).In February 2016, it was announced that the existence of gravitational waves was directly detected by the Advanced LIGO team on 14 September 2015.A century after its introduction, general relativity remains a highly active area of research.

See also

• Alcubierre drive– Hypothetical FTL transportation by warping space (warp drive)
• Alternatives to general relativity– Proposed theories of gravity
• Contributors to general relativity
• Derivations of the Lorentz transformations
• Ehrenfest paradox– Paradox in special relativity
• Einstein–Hilbert action– Concept in general relativity
• Einstein's thought experiments– Albert Einstein's hypothetical situations to argue scientific points
• General relativity priority dispute– Debate about credit for general relativity
• Introduction to the mathematics of general relativity
• Nordström's theory of gravitation– Predecessor to the theory of relativity
• Problems with Einstein's general theory of relativity
• Ricci calculus– Tensor index notation for tensor-based calculations
• Timeline of gravitational physics and relativity

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Further reading

Popular books

• Einstein, A.(1916),Relativity: The Special and the General Theory,Berlin,ISBN978-3-528-06059-6{{citation}}:CS1 maint: location missing publisher (link)
• Geroch, R.(1981),General Relativity from A to B,Chicago: University of Chicago Press,ISBN978-0-226-28864-2
• Lieber, Lillian(2008),The Einstein Theory of Relativity: A Trip to the Fourth Dimension,Philadelphia: Paul Dry Books, Inc.,ISBN978-1-58988-044-3
• Schutz, Bernard F.(2001), "Gravitational radiation", in Murdin, Paul (ed.),Encyclopedia of Astronomy and Astrophysics,Institute of Physics Pub.,ISBN978-1-56159-268-5
• Thorne, Kip;Hawking, Stephen (1994).Black Holes and Time Warps: Einstein's Outrageous Legacy.New York: W. W. Norton.ISBN0-393-03505-0.
• Wald, Robert M.(1992),Space, Time, and Gravity: the Theory of the Big Bang and Black Holes,Chicago: University of Chicago Press,ISBN978-0-226-87029-8
• Wheeler, John;Ford, Kenneth (1998),Geons, Black Holes, & Quantum Foam: a life in physics,New York: W. W. Norton,ISBN978-0-393-31991-0

Beginning undergraduate textbooks

• Callahan, James J. (2000),The Geometry of Spacetime: an Introduction to Special and General Relativity,New York: Springer,ISBN978-0-387-98641-8
• Taylor, Edwin F.; Wheeler, John Archibald (2000),Exploring Black Holes: Introduction to General Relativity,Addison Wesley,ISBN978-0-201-38423-9

Advanced undergraduate textbooks

• Cheng, Ta-Pei (2005),Relativity, Gravitation and Cosmology: a Basic Introduction,Oxford and New York: Oxford University Press,ISBN978-0-19-852957-6
• Dirac, Paul(1996),General Theory of Relativity,Princeton University Press,ISBN978-0-691-01146-2
• Gron, O.; Hervik, S. (2007),Einstein's General theory of Relativity,Springer,ISBN978-0-387-69199-2
• Hartle, James B.(2003),Gravity: an Introduction to Einstein's General Relativity,San Francisco: Addison-Wesley,ISBN978-0-8053-8662-2
• Hughston, L. P.;Tod, K. P. (1991),Introduction to General Relativity,Cambridge: Cambridge University Press,ISBN978-0-521-33943-8
• d'Inverno, Ray (1992),Introducing Einstein's Relativity,Oxford: Oxford University Press,ISBN978-0-19-859686-8
• Ludyk, Günter (2013).Einstein in Matrix Form(1st ed.). Berlin: Springer.ISBN978-3-642-35797-8.
• Møller, Christian (1955) [1952],The Theory of Relativity,Oxford University Press,OCLC7644624
• Moore, Thomas A (2012),A General Relativity Workbook,University Science Books,ISBN978-1-891389-82-5
• Schutz, B. F.(2009),A First Course in General Relativity(Second ed.), Cambridge University Press,ISBN978-0-521-88705-2

Graduate textbooks

• Carroll, Sean M.(2004),Spacetime and Geometry: An Introduction to General Relativity,San Francisco: Addison-Wesley,ISBN978-0-8053-8732-2
• Grøn, Øyvind;Hervik, Sigbjørn (2007),Einstein's General Theory of Relativity,New York: Springer,ISBN978-0-387-69199-2
• Landau, Lev D.;Lifshitz, Evgeny F.(1980),The Classical Theory of Fields (4th ed.),London: Butterworth-Heinemann,ISBN978-0-7506-2768-9
• Stephani, Hans (1990),General Relativity: An Introduction to the Theory of the Gravitational Field,Cambridge: Cambridge University Press,ISBN978-0-521-37941-0
• Will, Clifford;Poisson, Eric (2014).Gravity: Newtonian, Post-Newtonian, Relativistic.Cambridge University Press.ISBN978-1-107-03286-6.
• Charles W. Misner;Kip S. Thorne;John Archibald Wheeler(1973),Gravitation,W. H. Freeman, Princeton University Press,ISBN0-7167-0344-0
• R.K. Sachs;H. Wu (1977),General Relativity for Mathematicians,Springer-Verlag,ISBN1461299055
• Wald, Robert M.(1984).General Relativity.Chicago: University of Chicago Press.ISBN0-226-87032-4.OCLC10018614.

Specialists' books

• Hawking, Stephen;Ellis, George(1975).The Large Scale Structure of Space-time.Cambridge University Press.ISBN978-0-521-09906-6.
• Poisson, Eric(2007).A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics.Cambridge University Press.ISBN978-0-521-53780-3.

Journal articles

• Einstein, Albert(1916),"Die Grundlage der allgemeinen Relativitätstheorie",Annalen der Physik,49(7): 769–822,Bibcode:1916AnP...354..769E,doi:10.1002/andp.19163540702See alsoEnglish translation at Einstein Papers Project
• Flanagan, Éanna É.; Hughes, Scott A. (2005), "The basics of gravitational wave theory",New J. Phys.,7(1): 204,arXiv:gr-qc/0501041,Bibcode:2005NJPh....7..204F,doi:10.1088/1367-2630/7/1/204
• Landgraf, M.; Hechler, M.; Kemble, S. (2005), "Mission design for LISA Pathfinder",Class. Quantum Grav.,22(10): S487–S492,arXiv:gr-qc/0411071,Bibcode:2005CQGra..22S.487L,doi:10.1088/0264-9381/22/10/048,S2CID119476595
• Nieto, Michael Martin (2006),"The quest to understand the Pioneer anomaly"(PDF),Europhysics News,37(6): 30–34,arXiv:gr-qc/0702017,Bibcode:2006ENews..37f..30N,doi:10.1051/epn:2006604,archived(PDF)from the original on 24 September 2015
• Shapiro, I. I.;Pettengill, Gordon; Ash, Michael; Stone, Melvin; Smith, William; Ingalls, Richard; Brockelman, Richard (1968), "Fourth test of general relativity: preliminary results",Phys. Rev. Lett.,20(22): 1265–1269,Bibcode:1968PhRvL..20.1265S,doi:10.1103/PhysRevLett.20.1265
• Valtonen, M. J.; Lehto, H. J.; Nilsson, K.; Heidt, J.; Takalo, L. O.; Sillanpää, A.; Villforth, C.; Kidger, M.; et al. (2008), "A massive binary black-hole system in OJ 287 and a test of general relativity",Nature,452(7189): 851–853,arXiv:0809.1280,Bibcode:2008Natur.452..851V,doi:10.1038/nature06896,PMID18421348,S2CID4412396

External links

• Einstein OnlineArchived1 June 2014 at theWayback Machine– Articles on a variety of aspects of relativistic physics for a general audience; hosted by theMax Planck Institute for Gravitational Physics
• GEO600 home page,the official website of the GEO600 project.
• LIGO Laboratory
• NCSA Spacetime Wrinkles– produced by the numerical relativity group at theNCSA,with an elementary introduction to general relativity

• Courses
• Lectures
• Tutorials

• Einstein's General Theory of RelativityonYouTube(lecture byLeonard Susskindrecorded 22 September 2008 atStanford University).
• Series of lectures on General Relativitygiven in 2006 at theInstitut Henri Poincaré(introductory/advanced).
• General Relativity TutorialsbyJohn Baez.
• Brown, Kevin."Reflections on relativity".Mathpages.com.Archived fromthe originalon 18 December 2015.Retrieved29 May2005.
• Carroll, Sean M. (1997). "Lecture Notes on General Relativity".arXiv:gr-qc/9712019.
• Moor, Rafi."Understanding General Relativity".Retrieved11 July2006.
• Waner, Stefan."Introduction to Differential Geometry and General Relativity".Retrieved5 April2015.
• The Feynman Lectures on Physics Vol. II Ch. 42: Curved Space