View source for Physics:Introduction to general relativity

{{short description|Theory of gravity by Albert Einstein}}
{{introductory article|General relativity}}
[[File:Cassini-science-br.jpg|thumb|right|236px|High-precision test of general relativity by the Cassini space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) are delayed by the warping of spacetime (blue lines) due to the Sun's mass.]]
{{General relativity sidebar}}

'''General relativity''' is a [[Scientific theory|theory]] of gravitation developed by [[Biography:Albert Einstein|Albert Einstein]] between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping of [[Physics:Spacetime|spacetime]].

By the beginning of the 20th century, [[Physics:Newton's law of universal gravitation|Newton's law of universal gravitation]] had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was extremely successful at describing motion.

Experiments and observations show that Einstein's description of gravitation accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the [[Physics:Orbit|orbit]]s of [[Astronomy:Mercury (planet)|Mercury]] and other [[Astronomy:Planet|planet]]s. General relativity also predicts novel effects of gravity, such as [[Astronomy:Gravitational wave|gravitational wave]]s, [[Astronomy:Gravitational lens|gravitational lens]]ing and an effect of gravity on time known as [[Physics:Gravitational time dilation|gravitational time dilation]]. Many of these predictions have been confirmed by experiment or observation, [[Physics:Gravitational wave observation|most recently gravitational waves]].

General relativity has developed into an essential tool in modern [[Physics:Astrophysics|astrophysics]]. It provides the foundation for the current understanding of [[Astronomy:Black hole|black hole]]s, regions of space where the gravitational effect is strong enough that even light cannot escape. Their strong gravity is thought to be responsible for the intense [[Physics:Radiation|radiation]] emitted by certain types of astronomical objects (such as [[Active galactic nucleus|active galactic nuclei]] or [[Astronomy:Microquasar|microquasar]]s). General relativity is also part of the framework of the standard [[Astronomy:Big Bang|Big Bang]] model of [[Astronomy:Physical cosmology|cosmology]].

Although general relativity is not the only relativistic theory of gravity, it is the simplest one that is consistent with the experimental data.  Nevertheless, a number of open questions remain, the most fundamental of which is how general relativity can be reconciled with the laws of [[Physics:Introduction to quantum mechanics|quantum physics]] to produce a complete and self-consistent theory of [[Physics:Quantum gravity|quantum gravity]].

==From special to general relativity==

In September 1905, [[Biography:Albert Einstein|Albert Einstein]] published his theory of [[Physics:Special relativity|special relativity]], which reconciles [[Physics:Newton's laws of motion|Newton's laws of motion]] with [[Physics:Classical electromagnetism|electrodynamics]] (the interaction between objects with [[Physics:Electric charge|electric charge]]).  Special relativity introduced a new framework for all of physics by proposing new concepts of [[Space|space]] and time.  Some then-accepted physical theories were inconsistent with that framework; a key example was Newton's theory of gravity, which describes the mutual attraction experienced by bodies due to their mass.

Several physicists, including Einstein, searched for a theory that would reconcile Newton's law of gravity and special relativity.  Only Einstein's theory proved to be consistent with experiments and observations. To understand the theory's basic ideas, it is instructive to follow Einstein's thinking between 1907 and 1915, from his simple [[Thought experiment|thought experiment]] involving an observer in free fall to his fully geometric theory of gravity.<ref>This development is traced e.g. in {{harvnb|Renn|2005|loc=p. 110ff.}}, in chapters 9 through 15 of {{harvnb|Pais|1982}}, and in {{harvnb|Janssen|2005}}. A precis of Newtonian gravity can be found in {{harvnb|Schutz|2003|loc=chapters 2–4}}.  It is impossible to say whether the problem of Newtonian gravity crossed Einstein's mind before 1907, but, by his own admission, his first serious attempts to reconcile that theory with special relativity date to that year, cf. {{harvnb|Pais|1982|loc=p. 178}}.</ref>

===Equivalence principle===
{{main|Astronomy:Equivalence principle}}
A person in a [[Physics:Free fall|free-falling]] elevator experiences [[Physics:Weightlessness|weightlessness]]; objects either float motionless or drift at constant speed.  Since everything in the elevator is falling together, no gravitational effect can be observed.  In this way, the experiences of an observer in free fall are indistinguishable from those of an observer in deep space, far from any significant source of gravity.  Such observers are the privileged ("inertial") observers Einstein described in his theory of [[Physics:Special relativity|special relativity]]: observers for whom [[Physics:Light|light]] travels along straight lines at constant speed.<ref>This is described in detail in chapter 2 of {{harvnb|Wheeler|1990}}.</ref>

Einstein hypothesized that the similar experiences of weightless observers and inertial observers in special relativity represented a fundamental property of gravity, and he made this the cornerstone of his theory of general relativity, formalized in his [[Astronomy:Equivalence principle|equivalence principle]].  Roughly speaking, the principle states that a person in a free-falling elevator cannot tell that they are in free fall. Every experiment in such a free-falling environment has the same results as it would for an observer at rest or moving uniformly in deep space, far from all sources of gravity.<ref>While the equivalence principle is still part of modern expositions of general relativity, there are some differences between the modern version and Einstein's original concept, cf. {{harvnb|Norton|1985}}.</ref>

===Gravity and acceleration===
[[File:Elevator gravity.svg|thumb|right|236px|Ball falling to the floor in an accelerating rocket (left) and on Earth (right). The effect is identical.]]
Most effects of gravity vanish in free fall, but effects that seem the same as those of gravity can be ''produced'' by an [[Physics:Acceleration|accelerated]] frame of reference. An observer in a closed room cannot tell which of the following two scenarios is true:
* Objects are falling to the floor because the room is resting on the surface of the Earth and the objects are being pulled down by gravity.
* Objects are falling to the floor because the room is aboard a rocket in space, which is accelerating at 9.81 m/s<sup>2</sup>, the [[Physics:Standard gravity|standard gravity]] on Earth, and is far from any source of gravity. The objects are being pulled towards the floor by the same "inertial force" that presses the driver of an accelerating car into the back of their seat.

Conversely, any effect observed in an accelerated reference frame should also be observed in a gravitational field of corresponding strength.  This principle allowed Einstein to predict several novel effects of gravity in 1907, as explained in the next section.

An observer in an accelerated reference frame must introduce what physicists call [[Physics:Fictitious force|fictitious force]]s to account for the acceleration experienced by the observer and objects around them. In the example of the driver being pressed into their seat, the force felt by the driver is one example; another is the force one can feel while pulling the arms up and out if attempting to spin around like a top.  Einstein's master insight was that the constant, familiar pull of the Earth's gravitational field is fundamentally the same as these fictitious forces.<ref>E. g. {{harvnb|Janssen|2005|loc=p. 64f}}. Einstein himself also explains this in section XX of his non-technical book Einstein 1961. Following earlier ideas by [[Biography:Ernst Mach|Ernst Mach]], Einstein also explored centrifugal forces and their gravitational analogue, cf. {{harvnb|Stachel|1989}}.</ref>  The apparent magnitude of the fictitious forces always appears to be proportional to the mass of any object on which they act – for instance, the driver's seat exerts just enough force to accelerate the driver at the same rate as the car.  By analogy, Einstein proposed that an object in a gravitational field should feel a gravitational force proportional to its mass, as embodied in [[Physics:Newton's law of universal gravitation|Newton's law of gravitation]].<ref>Einstein explained this in section XX of Einstein 1961.  He considered an object "suspended" by a rope from the ceiling of a room aboard an accelerating rocket: from inside the room it looks as if gravitation is pulling the object down with a force proportional to its mass, but from outside the rocket it looks as if the rope is simply transferring the acceleration of the rocket to the object, and must therefore exert just the "force" to do so.</ref>

===Physical consequences===
In 1907, Einstein was still eight years away from completing the general theory of relativity. Nonetheless, he was able to make a number of novel, testable predictions that were based on his starting point for developing his new theory: the equivalence principle.<ref>More specifically, Einstein's calculations, which are described in chapter 11b of {{harvnb|Pais|1982}}, use the equivalence principle, the equivalence of gravity and inertial forces, and the results of special relativity for the propagation of light and for accelerated observers (the latter by considering, at each moment, the instantaneous [[Physics:Inertial frame of reference|inertial frame of reference]] associated with such an accelerated observer).</ref>

[[File:Gravitational red-shifting2.png|thumb|left|200px|The gravitational redshift of a light wave as it moves upwards against a gravitational field (caused by the yellow star below)]]

The first new effect is the [[Physics:Gravitational redshift|gravitational frequency shift]] of light. Consider two observers aboard an accelerating rocket-ship.  Aboard such a ship, there is a natural concept of "up" and "down": the direction in which the ship accelerates is "up", and free-floating objects accelerate in the opposite direction, falling "downward". Assume that one of the observers is "higher up" than the other. When the lower observer sends a light signal to the higher observer, the acceleration of the ship causes the light to be [[Astronomy:Redshift|red-shifted]], as may be calculated from [[Physics:Special relativity|special relativity]]; the second observer will measure a lower [[Physics:Frequency|frequency]] for the light than the first sent out. Conversely, light sent from the higher observer to the lower is [[Astronomy:Blueshift|blue-shifted]], that is, shifted towards higher frequencies.<ref>This effect can be derived directly within special relativity, either by looking at the equivalent situation of two observers in an accelerated rocket-ship or by looking at a falling elevator; in both situations, the frequency shift has an equivalent description as a Doppler shift between certain inertial frames. For simple derivations of this, see {{harvnb|Harrison|2002}}.</ref> Einstein argued that such frequency shifts must also be observed in a gravitational field. This is illustrated in the figure at left, which shows a light wave that is gradually red-shifted as it works its way upwards against the gravitational acceleration. This effect has been confirmed experimentally, as described below.

This gravitational frequency shift corresponds to a [[Physics:Gravitational time dilation|gravitational time dilation]]: Since the "higher" observer measures the same light wave to have a lower frequency than the "lower" observer, time must be passing faster for the higher observer.  Thus, time runs more slowly for observers the lower they are in a gravitational field.

It is important to stress that, for each observer, there are no observable changes of the flow of time for events or processes that are at rest in his or her reference frame.  Five-minute-eggs as timed by each observer's clock have the same consistency; as one year passes on each clock, each observer ages by that amount; each clock, in short, is in perfect agreement with all processes happening in its immediate vicinity.  It is only when the clocks are compared between separate observers that one can notice that time runs more slowly for the lower observer than for the higher.<ref>See chapter 12 of {{harvnb|Mermin|2005}}.</ref>  This effect is minute, but it too has been confirmed experimentally in multiple experiments, as described below.

In a similar way, Einstein predicted the [[Astronomy:Tests of general relativity#Deflection of light by the Sun|gravitational deflection of light]]: in a gravitational field, light is deflected downward, to the center of the gravitational field.  Quantitatively, his results were off by a factor of two; the correct derivation requires a more complete formulation of the theory of general relativity, not just the equivalence principle.<ref>Cf. {{harvnb|Ehlers|Rindler|1997}}; for a non-technical presentation, see {{harvnb|Pössel|2007}}.</ref>

===Tidal effects===
[[File:Tide fall.png|thumb|150px|Two bodies falling towards the center of the Earth accelerate towards each other as they fall.]]
The equivalence between gravitational and inertial effects does not constitute a complete theory of gravity. When it comes to explaining gravity near our own location on the Earth's surface, noting that our reference frame is not in free fall, so that [[Physics:Fictitious force|fictitious force]]s are to be expected, provides a suitable explanation. But a freely falling reference frame on one side of the Earth cannot explain why the people on the opposite side of the Earth experience a gravitational pull in the opposite direction.

A more basic manifestation of the same effect involves two bodies that are falling side by side towards the Earth, with a similar position and velocity. In a reference frame that is in free fall alongside these bodies, they appear to hover weightlessly&nbsp;– but not exactly so.  These bodies are not falling in precisely the same direction, but towards a single point in space: namely, the Earth's [[Physics:Center of mass|center of gravity]]. Consequently, there is a component of each body's motion towards the other (see the figure). In a small environment such as a freely falling lift, this relative acceleration is minuscule, while for skydivers on opposite sides of the Earth, the effect is large.  Such differences in force are also responsible for the [[Astronomy:Tide|tide]]s in the Earth's oceans, so the term "tidal effect" is used for this phenomenon.

The equivalence between inertia and gravity cannot explain tidal effects&nbsp;– it cannot explain variations in the gravitational field.<ref>These and other tidal effects are described in {{harvnb|Wheeler|1990|pp=83–91}}.</ref> For that, a theory is needed which describes the way that matter (such as the large mass of the Earth) affects the inertial environment around it.

===From acceleration to geometry===
While Einstein was exploring the equivalence of gravity and acceleration as well as the role of tidal forces, he discovered several analogies with the [[Geometry|geometry]] of [[Surface (mathematics)|surfaces]].  An example is the transition from an inertial reference frame (in which free particles coast along straight paths at constant speeds) to a rotating reference frame (in which [[Physics:Fictitious force|fictitious force]]s have to be introduced in order to explain particle motion): this is analogous to the transition from a [[Cartesian coordinate system]] (in which the coordinate lines are straight lines) to a [[Curvilinear coordinates|curved coordinate system]] (where coordinate lines need not be straight).

A deeper analogy relates tidal forces with a property of surfaces called ''[[Curvature|curvature]]''. For gravitational fields, the absence or presence of tidal forces determines whether or not the influence of gravity can be eliminated by choosing a freely falling reference frame. Similarly, the absence or presence of curvature determines whether or not a surface is [[Isometry|equivalent]] to a [[Plane (mathematics)|plane]]. In the summer of 1912, inspired by these analogies, Einstein searched for a geometric formulation of gravity.<ref>Tides and their geometric interpretation are explained in chapter 5 of {{harvnb|Wheeler|1990}}.  This part of the historical development is traced in {{harvnb|Pais|1982|loc=section 12b}}.</ref>

The elementary objects of [[Geometry|geometry&nbsp;]]– [[Point (geometry)|points]], lines, [[Triangle|triangle]]s&nbsp;– are traditionally defined in three-dimensional [[Space|space]] or on two-dimensional [[Surface (mathematics)|surfaces]]. In 1907, [[Biography:Hermann Minkowski|Hermann Minkowski]], Einstein's former mathematics professor at the Swiss Federal Polytechnic, introduced [[Physics:Minkowski space|Minkowski space]], a geometric formulation of Einstein's [[Physics:Special theory of relativity|special theory of relativity]] where the geometry included not only [[Space|space]] but also time. The basic entity of this new geometry is four-[[Physics:Dimension|dimension]]al [[Physics:Spacetime|spacetime]]. The orbits of moving bodies are [[World line|curves in spacetime]]; the orbits of bodies moving at constant speed without changing direction correspond to straight lines.<ref>For elementary presentations of the concept of spacetime, see the first section in chapter 2 of {{harvnb|Thorne|1994}}, and {{harvnb|Greene|2004|loc=p. 47–61}}. More complete treatments on a fairly elementary level can be found e.g. in {{harvnb|Mermin|2005}} and in {{harvnb|Wheeler|1990|loc=chapters 8 and 9}}.</ref>

The geometry of general curved surfaces was developed in the early 19th century by [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]]. This geometry had in turn been generalized to higher-dimensional spaces in [[Riemannian geometry]] introduced by [[Biography:Bernhard Riemann|Bernhard Riemann]] in the 1850s. With the help of Riemannian geometry, Einstein formulated a geometric description of gravity in which Minkowski's spacetime is replaced by distorted, curved spacetime, just as curved surfaces are a generalization of ordinary plane surfaces. '''Embedding Diagrams''' are used to illustrate curved spacetime in educational contexts.<ref>{{cite journal | last1 = Marolf | first1 = Donald | year = 1999 | title = Spacetime Embedding Diagrams for Black Holes | journal = General Relativity and Gravitation | volume = 31 | issue = 6| pages = 919–944 |arxiv=gr-qc/9806123|bibcode = 1999GReGr..31..919M |doi = 10.1023/A:1026646507201 | s2cid = 12502462 }}</ref><ref>See {{harvnb|Wheeler|1990|loc=chapters 8 and 9}} for vivid illustrations of curved spacetime.</ref>

After he had realized the validity of this geometric analogy, it took Einstein a further three years to find the missing cornerstone of his theory: the equations describing how [[Physics:Matter|matter]] influences spacetime's curvature. Having formulated what are now known as [[Physics:Einstein's equations|Einstein's equations]] (or, more precisely, his field equations of gravity), he presented his new theory of gravity at several sessions of the Prussian Academy of Sciences in late 1915, culminating in his final presentation on November 25, 1915.<ref>Einstein's struggle to find the correct field equations is traced in chapters 13–15 of {{harvnb|Pais|1982}}.</ref>

==Geometry and gravitation==

Paraphrasing [[Biography:John Archibald Wheeler|John Wheeler]], Einstein's geometric theory of gravity can be summarized as: ''spacetime tells matter how to move; matter tells spacetime how to curve''.<ref>E.g. p. xi in {{harvnb|Wheeler|1990}}.</ref> What this means is addressed in the following three sections, which explore the motion of so-called [[Physics:Test particle|test particles]], examine which properties of matter serve as a source for gravity, and, finally, introduce Einstein's equations, which relate these matter properties to the curvature of spacetime.

===Probing the gravitational field===
[[File:Earth geo.png|thumb|right|236px|Converging geodesics: two lines of longitude (green) that start out in parallel at the equator (red) but converge to meet at the pole]]

In order to map a body's gravitational influence, it is useful to think about what physicists call probe or [[Physics:Test particle|test particle]]s: particles that are influenced by gravity, but are so small and light that we can neglect their own gravitational effect. In the absence of gravity and other external forces, a test particle moves along a straight line at a constant speed. In the language of [[Physics:Spacetime|spacetime]], this is equivalent to saying that such test particles move along straight world lines in spacetime. In the presence of gravity, spacetime is non-Euclidean, or [[Curvature|curved]], and in curved spacetime straight world lines may not exist. Instead, test particles move along lines called [[Geodesic|geodesic]]s, which are "as straight as possible", that is, they follow the shortest path between starting and ending points, taking the curvature into consideration.

A simple analogy is the following: In [[Geodesy|geodesy]], the science of measuring Earth's size and shape, a geodesic is the shortest route between two points on the Earth's surface. Approximately, such a route is a segment of a [[Great circle|great circle]], such as a line of longitude or the [[Astronomy:Equator|equator]]. These paths are certainly not straight, simply because they must follow the curvature of the Earth's surface. But they are as straight as is possible subject to this constraint.

The properties of geodesics differ from those of straight lines. For example, on a plane, parallel lines never meet, but this is not so for geodesics on the surface of the Earth: for example, lines of longitude are parallel at the equator, but intersect at the poles. Analogously, the world lines of test particles in free fall are spacetime geodesics, the straightest possible lines in spacetime.  But still there are crucial differences between them and the truly straight lines that can be traced out in the gravity-free spacetime of special relativity.  In special relativity, parallel geodesics remain parallel. In a gravitational field with tidal effects, this will not, in general, be the case. If, for example, two bodies are initially at rest relative to each other, but are then dropped in the Earth's gravitational field, they will move towards each other as they fall towards the Earth's center.<ref>A thorough, yet accessible account of basic differential geometry and its application in general relativity can be found in {{harvnb|Geroch|1978}}.</ref>

Compared with planets and other astronomical bodies, the objects of everyday life (people, cars, houses, even mountains) have little mass. Where such objects are concerned, the laws governing the behavior of test particles are sufficient to describe what happens. Notably, in order to deflect a test particle from its geodesic path, an external force must be applied. A chair someone is sitting on applies an external upwards force preventing the person from falling freely towards [[Travel to the Earth's center|the center of the Earth]] and thus following a geodesic, which they would otherwise be doing without the chair there, or any other matter in between them and the center point of the Earth. In this way, general relativity explains the daily experience of gravity on the surface of the Earth ''not'' as the downwards pull of a gravitational force, but as the upwards push of external forces. These forces deflect all bodies resting on the Earth's surface from the geodesics they would otherwise follow.<ref>See chapter 10 of {{harvnb|Wheeler|1990}}.</ref> For matter objects whose own gravitational influence cannot be neglected, the laws of motion are somewhat more complicated than for test particles, although it remains true that spacetime tells matter how to move.<ref>In fact, when starting from the complete theory, Einstein's equation can be used to derive these more complicated laws of motion for matter as a consequence of geometry, but deriving from this the motion of idealized test particles is a highly non-trivial task, cf. {{harvnb|Poisson|2004}}.</ref>

===Sources of gravity===
In Newton's description of gravity, the gravitational force is caused by matter. More precisely, it is caused by a specific property of material objects: their [[Physics:Mass|mass]]. In Einstein's theory and related theories of gravitation, curvature at every point in spacetime is also caused by whatever matter is present. Here, too, mass is a key property in determining the gravitational influence of matter. But in a relativistic theory of gravity, mass cannot be the only source of gravity. Relativity links mass with energy, and energy with momentum.

The equivalence between mass and [[Physics:Energy|energy]], as expressed by the formula [[Mass–energy equivalence|''E''&nbsp;=&nbsp;''mc''{{smallsup|2}}]], is the most famous consequence of special relativity. In relativity, mass and energy are two different ways of describing one physical quantity. If a physical system has energy, it also has the corresponding mass, and vice versa. In particular, all properties of a body that are associated with energy, such as its [[Physics:Temperature|temperature]] or the [[Physics:Binding energy|binding energy]] of systems such as [[Physics:Atomic nucleus|nuclei]] or [[Physics:Molecule|molecule]]s, contribute to that body's mass, and hence act as sources of gravity.<ref>A simple explanation of mass–energy equivalence can be found in sections 3.8 and 3.9 of {{harvnb|Giulini|2005}}.</ref>

In special relativity, energy is closely connected to [[Physics:Momentum|momentum]]. In special relativity, just as space and time are different aspects of a more comprehensive entity called spacetime, energy and momentum are merely different aspects of a unified, four-dimensional quantity that physicists call [[Physics:Four-momentum|four-momentum]]. In consequence, if energy is a source of gravity, momentum must be a source as well. The same is true for quantities that are directly related to energy and momentum, namely internal [[Physics:Pressure|pressure]] and [[Physics:Tension|tension]]. Taken together, in general relativity it is mass, energy, momentum, pressure and tension that serve as sources of gravity: they are how matter tells spacetime how to curve. In the theory's mathematical formulation, all these quantities are but aspects of a more general physical quantity called the [[Stress–energy tensor|energy–momentum tensor]].<ref>See chapter 6 of {{harvnb|Wheeler|1990}}.</ref>

===Einstein's equations===
[[Physics:Einstein's equations|Einstein's equations]] are the centerpiece of general relativity. They provide a precise formulation of the relationship between spacetime geometry and the properties of matter, using the language of mathematics. More concretely, they are formulated using the concepts of [[Riemannian geometry]], in which the geometric properties of a space (or a spacetime) are described by a quantity called a [[Physics:Metric tensor|metric]]. The metric encodes the information needed to compute the fundamental geometric notions of distance and angle in a curved space (or spacetime).

[[File:Metric globe.png|thumb|right|236px|Distances, at different latitudes, corresponding to 30 degrees difference in longitude]]

A spherical surface like that of the Earth provides a simple example. The location of any point on the surface can be described by two coordinates: the geographic [[Earth:Latitude|latitude]] and [[Earth:Longitude|longitude]]. Unlike the Cartesian coordinates of the plane, coordinate differences are not the same as distances on the surface, as shown in the diagram on the right: for someone at the equator, moving 30 degrees of longitude westward (magenta line) corresponds to a distance of roughly {{convert|3300|km|mi|sp=us}}, while for someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely {{convert|1900|km|mi|sp=us}}. Coordinates therefore do not provide enough information to describe the geometry of a spherical surface, or indeed the geometry of any more complicated space or spacetime. That information is precisely what is encoded in the metric, which is a function defined at each point of the surface (or space, or spacetime) and relates coordinate differences to differences in distance. All other quantities that are of interest in geometry, such as the length of any given curve, or the angle at which two curves meet, can be computed from this metric function.<ref>For a more detailed definition of the metric, but one that is more informal than a textbook presentation, see chapter 14.4 of {{harvnb|Penrose|2004}}.</ref>

The metric function and its rate of change from point to point can be used to define a geometrical quantity called the [[Riemann curvature tensor]], which describes exactly how the [[Riemannian manifold]], the spacetime in the theory of relativity, is curved at each point. As has already been mentioned, the matter content of the spacetime defines another quantity, the [[Stress–energy tensor|energy–momentum tensor]] '''T''', and the principle that "spacetime tells matter how to move, and matter tells spacetime how to curve" means that these quantities must be related to each other. Einstein formulated this relation by using the Riemann curvature tensor and the metric to define another geometrical quantity '''G''', now called the [[Physics:Einstein tensor|Einstein tensor]], which describes some aspects of the way spacetime is curved. ''Einstein's equation'' then states that

: <math>\mathbf{G}=\frac{8\pi G}{c^4}\mathbf{T},</math>

i.e., up to a constant multiple, the quantity '''G''' (which measures curvature) is equated with the quantity '''T''' (which measures matter content). Here, ''G'' is the [[Physics:Gravitational constant|gravitational constant]] of Newtonian gravity, and ''c'' is the [[Physics:Speed of light|speed of light]] from special relativity.

This equation is often referred to in the plural as ''Einstein's equations'', since the quantities '''G''' and '''T''' are each determined by several functions of the coordinates of spacetime, and the equations equate each of these component functions.<ref>The geometrical meaning of Einstein's equations is explored in chapters 7 and 8 of {{harvnb|Wheeler|1990}}; cf. box 2.6 in {{harvnb|Thorne|1994}}. An introduction using only very simple mathematics is given in chapter 19 of {{harvnb|Schutz|2003}}.</ref> A solution of these equations describes a particular geometry of [[Physics:Spacetime|spacetime]]; for example, the [[Astronomy:Schwarzschild metric|Schwarzschild solution]] describes the geometry around a spherical, non-rotating mass such as a [[Astronomy:Star|star]] or a [[Astronomy:Black hole|black hole]], whereas the [[Astronomy:Kerr metric|Kerr solution]] describes a rotating black hole. Still other solutions can describe a [[Astronomy:Gravitational wave|gravitational wave]] or, in the case of the [[Astronomy:Friedmann–Lemaître–Robertson–Walker metric|Friedmann–Lemaître–Robertson–Walker solution]], an expanding universe.  The simplest solution is the uncurved [[Physics:Minkowski spacetime|Minkowski spacetime]], the spacetime described by special relativity.<ref>The most important solutions are listed in every [[Physics:General relativity#Further reading|textbook on general relativity]]; for a (technical) summary of our current understanding, see {{harvnb|Friedrich|2005}}.</ref>

==Experiments==

No scientific theory is self-evidently true; each is a model that must be checked by experiment. Newton's law of gravity was accepted because it accounted for the motion of planets and moons in the [[Astronomy:Solar System|Solar System]] with considerable accuracy. As the precision of experimental measurements gradually improved, some discrepancies with Newton's predictions were observed, and these were accounted for in the general theory of relativity. Similarly, the predictions of general relativity must also be checked with experiment, and Einstein himself devised three tests now known as the classical tests of the theory:
[[File:Newtonianvseinsteinianorbits.gif|thumb|right|200px|Newtonian (red) vs. Einsteinian orbit (blue) of a single planet orbiting a spherical star]]
* Newtonian gravity predicts that the [[Physics:Orbit|orbit]] which a single [[Astronomy:Planet|planet]] traces around a perfectly spherical [[Astronomy:Star|star]] should be an [[Ellipse|ellipse]]. Einstein's theory predicts a more complicated curve: the planet behaves as if it were travelling around an ellipse, but at the same time, the ellipse as a whole is rotating slowly around the star. In the diagram on the right, the ellipse predicted by Newtonian gravity is shown in red, and part of the orbit predicted by Einstein in blue. For a planet orbiting the Sun, this deviation from Newton's orbits is known as the [[Astronomy:Tests of general relativity#Perihelion precession of Mercury|anomalous perihelion shift]].  The first measurement of this effect, for the planet [[Astronomy:Mercury (planet)|Mercury]], dates back to 1859. The most accurate results for Mercury and for other planets to date are based on measurements which were undertaken between 1966 and 1990, using [[Astronomy:Radio telescope|radio telescope]]s.<ref>More precisely, these are VLBI measurements of planetary positions; see chapter 5 of {{harvnb|Will|1993}} and section 3.5 of {{harvnb|Will|2006}}.</ref>  General relativity predicts the correct anomalous perihelion shift for all planets where this can be measured accurately ([[Astronomy:Mercury (planet)|Mercury]], [[Astronomy:Venus|Venus]] and the Earth).
* According to general relativity, light does not travel along straight lines when it propagates in a gravitational field. Instead, it is [[Astronomy:Tests of general relativity#Deflection of light by the Sun|deflected]] in the presence of massive bodies. In particular, starlight is deflected as it passes near the Sun, leading to apparent shifts of up 1.75 arc seconds in the stars' positions in the sky (an arc second is equal to 1/3600 of a [[Degree (angle)|degree]]). In the framework of Newtonian gravity, a heuristic argument can be made that leads to light deflection by half that amount. The different predictions can be tested by observing stars that are close to the Sun during a [[Earth:Solar eclipse|solar eclipse]]. In this way, a British expedition to West Africa in 1919, directed by [[Biography:Arthur Eddington|Arthur Eddington]], confirmed that Einstein's prediction was correct, and the Newtonian predictions wrong, via observation of the [[Astronomy:Solar eclipse of May 29, 1919|May 1919 eclipse]]. Eddington's results were not very accurate; subsequent observations of the deflection of the light of distant [[Astronomy:Quasar|quasar]]s by the Sun, which utilize highly accurate techniques of [[Astronomy:Radio astronomy|radio astronomy]], have confirmed Eddington's results with significantly better precision (the first such measurements date from 1967, the most recent comprehensive analysis from 2004).<ref>For the historical measurements, see {{harvnb|Hartl|2005}}, {{harvnb|Kennefick|2005}}, and {{harvnb|Kennefick|2007}}; Soldner's original derivation in the framework of Newton's theory is {{harvnb|von Soldner|1804}}. For the most precise measurements to date, see {{harvnb|Bertotti|2005}}.</ref>
* [[Physics:Gravitational redshift|Gravitational redshift]] was first measured in a laboratory setting in 1959 by [[Physics:Pound–Rebka experiment|Pound and Rebka]]. It is also seen in astrophysical measurements, notably for light escaping the [[Astronomy:White dwarf|white dwarf]] Sirius B. The related [[Physics:Gravitational time dilation|gravitational time dilation]] effect has been measured by transporting [[Engineering:Atomic clock|atomic clock]]s to altitudes of between tens and tens of thousands of kilometers (first by [[Physics:Hafele–Keating experiment|Hafele and Keating]] in 1971; most accurately to date by [[Physics:Gravity Probe A|Gravity Probe A]] launched in 1976).<ref>See {{harvnb|Kennefick|2005}} and chapter 3 of {{harvnb|Will|1993}}. For the Sirius B measurements, see {{harvnb|Trimble|Barstow|2007}}.</ref>
Of these tests, only the perihelion advance of Mercury was known prior to Einstein's final publication of general relativity in 1916.  The subsequent experimental confirmation of his other predictions, especially the first measurements of the deflection of light by the sun in 1919, catapulted Einstein to international stardom.<ref>{{harvnb|Pais|1982}}, Mercury on pp. 253–254, Einstein's rise to fame in sections 16b and 16c.</ref>  These three experiments justified adopting general relativity over Newton's theory and, incidentally, over a number of [[Physics:Alternatives to general relativity|alternatives to general relativity]] that had been proposed.


Further tests of general relativity include precision measurements of the Shapiro effect or gravitational time delay for light, measured in 2002 by the Cassini space probe. One set of tests focuses on effects predicted by general relativity for the behavior of gyroscopes travelling through space. One of these effects, [[Physics:Geodetic effect|geodetic precession]], has been tested with the Lunar Laser Ranging Experiment (high-precision measurements of the orbit of the [[Astronomy:Moon|Moon]]). Another, which is related to rotating masses, is called [[Physics:Frame-dragging|frame-dragging]]. The geodetic and frame-dragging effects were both tested by the [[Physics:Gravity Probe B|Gravity Probe B]] satellite experiment launched in 2004, with results confirming relativity to within 0.5% and 15%, respectively, as of December 2008.<ref>{{citation |author1=Everitt, C.W.F. |author2=Parkinson, B.W. |url=http://einstein.stanford.edu/content/final_report/GPB_Final_NASA_Report-020509-web.pdf |title=Gravity Probe B Science Results—NASA Final Report |year=2009 |access-date=2009-05-02 }}</ref>

By cosmic standards, gravity throughout the solar system is weak. Since the differences between the predictions of Einstein's and Newton's theories are most pronounced when gravity is strong, physicists have long been interested in testing various relativistic effects in a setting with comparatively strong gravitational fields. This has become possible thanks to precision observations of [[Astronomy:Binary pulsar|binary pulsar]]s. In such a star system, two highly compact [[Astronomy:Neutron star|neutron star]]s orbit each other. At least one of them is a [[Astronomy:Pulsar|pulsar]]&nbsp;– an astronomical object that emits a tight beam of radiowaves. These beams strike the Earth at very regular intervals, similarly to the way that the rotating beam of a lighthouse means that an observer sees the lighthouse blink,  and can be observed as a highly regular series of pulses. General relativity predicts specific deviations from the regularity of these radio pulses. For instance, at times when the radio waves pass close to the other neutron star, they should be deflected by the star's gravitational field. The observed pulse patterns are impressively close to those predicted by general relativity.<ref>{{harvnb|Kramer|2004}}.</ref>

One particular set of observations is related to eminently useful practical applications, namely to satellite navigation systems such as the [[Engineering:Global Positioning System|Global Positioning System]] that are used for both precise [[Earth:Navigation|positioning]] and timekeeping. Such systems rely on two sets of atomic clocks: clocks aboard satellites orbiting the Earth, and reference clocks stationed on the Earth's surface. General relativity predicts that these two sets of clocks should tick at slightly different rates, due to their different motions (an effect already predicted by special relativity) and their different positions within the Earth's gravitational field. In order to ensure the system's accuracy, either the satellite clocks are slowed down by a relativistic factor, or that same factor is made part of the evaluation algorithm. In turn, tests of the system's accuracy (especially the very thorough measurements that are part of the definition of [[UTC|universal coordinated time]]) are testament to the validity of the relativistic predictions.<ref>An accessible account of relativistic effects in the global positioning system can be found in {{harvnb|Ashby|2002}}; details are given in {{harvnb|Ashby|2003}}.</ref>

A number of other tests have probed the validity of various versions of the [[Astronomy:Equivalence principle|equivalence principle]]; strictly speaking, all measurements of gravitational time dilation are tests of the weak version of that principle, not of general relativity itself. So far, general relativity has passed all observational tests.<ref>An accessible introduction to tests of general relativity is {{harvnb|Will|1993}}; a more technical, up-to-date account is {{harvnb|Will|2006}}.</ref>

==Astrophysical applications==

Models based on general relativity play an important role in [[Physics:Astrophysics|astrophysics]]; the success of these models is further testament to the theory's validity.

===Gravitational lensing===
[[File:Einstein cross.jpg|thumb|220px|The [[Astronomy:Einstein Cross|Einstein Cross]]: four images of the same distant quasar, produced by a gravitational lens (the much closer foreground galaxy Huchra's lens)]]

Since light is deflected in a gravitational field, it is possible for the light of a distant object to reach an observer along two or more paths. For instance, light of a very distant object such as a [[Astronomy:Quasar|quasar]] can pass along one side of a massive [[Astronomy:Galaxy|galaxy]] and be deflected slightly so as to reach an observer on Earth, while light passing along the opposite side of that same galaxy is deflected as well, reaching the same observer from a slightly different direction. As a result, that particular observer will see one astronomical object in two different places in the night sky.  This kind of focussing is well known when it comes to optical lenses, and hence the corresponding gravitational effect is called [[Physics:Gravitational lensing|gravitational lensing]].<ref>The geometry of such situations is explored in chapter 23 of {{harvnb|Schutz|2003}}.</ref>

[[Astronomy:Observational astronomy|Observational astronomy]] uses lensing effects as an important tool to infer properties of the lensing object. Even in cases where that object is not directly visible, the shape of a lensed image provides information about the [[Physics:Mass|mass]] distribution responsible for the light deflection. In particular, gravitational lensing provides one way to measure the distribution of [[Astronomy:Dark matter|dark matter]], which does not give off light and can be observed only by its gravitational effects. One particularly interesting application are large-scale observations, where the lensing masses are spread out over a significant fraction of the observable universe, and can be used to obtain information about the large-scale properties and evolution of our cosmos.<ref>Introductions to gravitational lensing and its applications can be found on the webpages {{harvnb|Newbury|1997}} and {{harvnb|Lochner|2007}}.</ref>

===Gravitational waves===
[[Astronomy:Gravitational wave|Gravitational wave]]s, a direct consequence of Einstein's theory, are distortions of geometry that propagate at the speed of light, and can be thought of as ripples in spacetime. They should not be confused with the [[Gravity wave|gravity wave]]s of [[Physics:Fluid dynamics|fluid dynamics]], which are a different concept.

In February 2016, the Advanced [[Astronomy:LIGO|LIGO]] team announced that they had directly [[Physics:Gravitational wave observation|observed gravitational waves]] from a [[Astronomy:Black hole|black hole]] merger.<ref name='Abbot'>{{cite journal |title=Observation of Gravitational Waves from a Binary Black Hole Merger| author=B. P. Abbott |collaboration=LIGO Scientific and Virgo Collaborations |journal=Physical Review Letters| year=2016| volume=116|issue=6| doi=10.1103/PhysRevLett.116.061102|arxiv = 1602.03837 |bibcode = 2016PhRvL.116f1102A | pmid=26918975 | page=061102| s2cid=124959784}}</ref>

Indirectly, the effect of gravitational waves had been detected in observations of specific binary stars. Such pairs of stars [[Physics:Orbit|orbit]] each other and, as they do so, gradually lose energy by emitting gravitational waves. For ordinary stars like the Sun, this energy loss would be too small to be detectable, but this energy loss was observed in 1974 in a [[Astronomy:Binary pulsar|binary pulsar]] called PSR1913+16. In such a system, one of the orbiting stars is a pulsar. This has two consequences: a pulsar is an extremely dense object known as a [[Astronomy:Neutron star|neutron star]], for which gravitational wave emission is much stronger than for ordinary stars. Also, a pulsar emits a narrow beam of [[Physics:Electromagnetic radiation|electromagnetic radiation]] from its magnetic poles. As the pulsar rotates, its beam sweeps over the Earth, where it is seen as a regular series of radio pulses, just as a ship at sea observes regular flashes of light from the rotating light in a lighthouse. This regular pattern of radio pulses functions as a highly accurate "clock". It can be used to time the double star's orbital period, and it reacts sensitively to distortions of spacetime in its immediate neighborhood.

The discoverers of PSR1913+16, [[Biography:Russell Alan Hulse|Russell Hulse]] and Joseph Taylor, were awarded the Nobel Prize in Physics in 1993. Since then, several other binary pulsars have been found. The most useful are those in which both stars are pulsars, since they provide accurate tests of general relativity.<ref>{{harvnb|Schutz|2003|loc=pp. 317–321}}; {{harvnb|Bartusiak|2000|loc=pp. 70–86}}.</ref>

Currently, a number of land-based [[Physics:Gravitational wave detector|gravitational wave detector]]s are in operation, and a mission to launch a space-based detector, [[Astronomy:Laser Interferometer Space Antenna|LISA]], is currently under development, with a precursor mission ([[Astronomy:LISA Pathfinder|LISA Pathfinder]]) which was launched in 2015. Gravitational wave observations can be used to obtain information about compact objects such as neutron stars and black holes, and also to probe the state of the early [[Astronomy:Universe|universe]] fractions of a second after the [[Astronomy:Big Bang|Big Bang]].<ref>The ongoing search for gravitational waves is described in {{harvnb|Bartusiak|2000}} and in {{harvnb|Blair|McNamara|1997}}.</ref>

===Black holes===
[[File:M87 jet.jpg|thumb|right|200px|Black hole-powered jet emanating from the central region of the galaxy [[Astronomy:Messier 87|M87]]]]
When mass is concentrated into a sufficiently [[Physics:Hoop conjecture|compact]] region of space, general relativity predicts the formation of a [[Astronomy:Black hole|black hole]]&nbsp;– a region of space with a gravitational effect so strong that not even light can escape. Certain types of black holes are thought to be the final state in the [[Astronomy:Stellar evolution|evolution]] of massive [[Astronomy:Star|star]]s. On the other hand, [[Astronomy:Supermassive black hole|supermassive black hole]]s with the mass of millions or billions of [[Astronomy:Sun|Sun]]s are assumed to reside in the cores of most [[Astronomy:Galaxy|galaxies]], and they play a key role in current models of how galaxies have formed over the past billions of years.<ref>For an overview of the history of black hole physics from its beginnings in the early 20th century to modern times, see the very readable account by {{harvnb|Thorne|1994}}.  For an up-to-date account of the role of black holes in structure formation, see {{harvnb|Springel|White|Jenkins|Frenk|2005}}; a brief summary can be found in the related article {{harvnb|Gnedin|2005}}.</ref>

Matter falling onto a compact object is one of the most efficient mechanisms for releasing [[Physics:Energy|energy]] in the form of [[Physics:Radiation|radiation]], and matter falling onto black holes is thought to be responsible for some of the brightest astronomical phenomena imaginable. Notable examples of great interest to astronomers are quasars and other types of [[Active galactic nucleus|active galactic nuclei]]. Under the right conditions, falling matter accumulating around a black hole can lead to the formation of jets, in which focused beams of matter are flung away into space at speeds near that of light.<ref>See chapter 8 of {{harvnb|Sparke|Gallagher|2007}} and {{harvnb|Disney|1998}}.  A treatment that is more thorough, yet involves only comparatively little mathematics can be found in {{harvnb|Robson|1996}}.</ref>

There are several properties that make black holes the most promising sources of gravitational waves. One reason is that black holes are the most compact objects that can orbit each other as part of a binary system; as a result, the gravitational waves emitted by such a system are especially strong. Another reason follows from what are called [[Astronomy:No-hair theorem|black-hole uniqueness theorems]]: over time, black holes retain only a minimal set of distinguishing features (these theorems have become known as "no-hair" theorems), regardless of the starting geometric shape. For instance, in the long term, the collapse of a hypothetical matter cube will not result in a cube-shaped black hole. Instead, the resulting black hole will be indistinguishable from a black hole formed by the collapse of a spherical mass. In its transition to a spherical shape, the black hole formed by the collapse of a more complicated shape will emit gravitational waves.<ref>An elementary introduction to the black hole uniqueness theorems can be found in {{harvnb|Chrusciel|2006}} and in {{harvnb|Thorne|1994|loc=pp. 272–286}}.</ref>

===Cosmology===
[[File:WMAP image of the CMB anisotropy.jpg|thumb|236px|An image, created using data from the WMAP satellite telescope, of the radiation emitted no more than a few hundred thousand years after the Big Bang]]
One of the most important aspects of general relativity is that it can be applied to the [[Astronomy:Universe|universe]] as a whole. A key point is that, on large scales, our universe appears to be constructed along very simple lines: all current observations suggest that, on average, the structure of the cosmos should be approximately the same, regardless of an observer's location or direction of observation: the universe is approximately [[Physics:Homogeneity|homogeneous]] and isotropic. Such comparatively simple universes can be described by simple solutions of Einstein's equations. The current [[Astronomy:Physical cosmology|cosmological models]] of the universe are obtained by combining these simple solutions to general relativity with theories describing the properties of the universe's [[Physics:Matter|matter]] content, namely [[Physics:Thermodynamics|thermodynamics]], [[Physics:Nuclear physics|nuclear-]] and [[Physics:Particle physics|particle physics]]. According to these models, our present universe emerged from an extremely dense high-temperature state&nbsp;– the [[Astronomy:Big Bang|Big Bang]]&nbsp;– roughly 14 billion [[Year|year]]s ago and has been expanding ever since.<ref>Detailed information can be found in Ned Wright's Cosmology Tutorial and FAQ, {{harvnb|Wright|2007}}; a very readable introduction is {{harvnb|Hogan|1999}}. Using undergraduate mathematics but avoiding the advanced mathematical tools of general relativity, {{harvnb|Berry|1989}} provides a more thorough presentation.</ref>

Einstein's equations can be generalized by adding a term called the [[Astronomy:Cosmological constant|cosmological constant]].  When this term is present, [[Physics:Vacuum|empty space]] itself acts as a source of attractive (or, less commonly, repulsive) gravity. Einstein originally introduced this term in his pioneering 1917 paper on cosmology, with a very specific motivation: contemporary cosmological thought held the universe to be static, and the additional term was required for constructing static model universes within the framework of general relativity. When it became apparent that the universe is not static, but expanding, Einstein was quick to discard this additional term. Since the end of the 1990s, however, astronomical evidence indicating an [[Physics:Acceleration|accelerating]] expansion consistent with a cosmological constant&nbsp;– or, equivalently, with a particular and ubiquitous kind of [[Astronomy:Dark energy|dark energy]]&nbsp;– has steadily been accumulating.<ref>Einstein's original paper is {{harvnb|Einstein|1917}}; good descriptions of more modern developments can be found in {{harvnb|Cowen|2001}} and {{harvnb|Caldwell|2004}}.</ref>

==Modern research==

General relativity is very successful in providing a framework for accurate models which describe an impressive array of physical phenomena. On the other hand, there are many interesting open questions, and in particular, the theory as a whole is almost certainly incomplete.<ref>Cf. {{harvnb|Maddox|1998|loc=pp. 52–59 and 98–122}}; {{harvnb|Penrose|2004|loc=section 34.1 and chapter 30}}.</ref>

In contrast to all other modern theories of [[Physics:Fundamental interaction|fundamental interaction]]s, general relativity is a [[Physics:Classical physics|classical]] theory: it does not include the effects of [[Physics:Introduction to quantum mechanics|quantum physics]]. The quest for a quantum version of general relativity addresses one of the most fundamental [[Physics:List of unsolved problems in physics|open questions]] in physics. While there are promising candidates for such a theory of [[Physics:Quantum gravity|quantum gravity]], notably [[Physics:String theory|string theory]] and [[Physics:Loop quantum gravity|loop quantum gravity]], there is at present no consistent and complete theory. It has long been hoped that a theory of quantum gravity would also eliminate another problematic feature of general relativity: the presence of spacetime singularities. These singularities are boundaries ("sharp edges") of spacetime at which geometry becomes ill-defined, with the consequence that general relativity itself loses its predictive power. Furthermore, there are so-called singularity theorems which predict that such singularities ''must'' exist within the universe if the laws of general relativity were to hold without any quantum modifications. The best-known examples are the singularities associated with the model universes that describe black holes and the [[Astronomy:Big Bang|beginning of the universe]].<ref>With a focus on string theory, the search for quantum gravity is described in {{harvnb|Greene|1999}}; for an account from the point of view of loop quantum gravity, see {{harvnb|Smolin|2001}}.</ref>

Other attempts to modify general relativity have been made in the context of [[Astronomy:Cosmology|cosmology]]. In the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely [[Astronomy:Dark energy|dark energy]] and [[Astronomy:Dark matter|dark matter]]. There have been several controversial proposals to remove the need for these enigmatic forms of matter and energy, by modifying the laws governing gravity and the dynamics of cosmic expansion, for example [[Astronomy:Modified Newtonian dynamics|modified Newtonian dynamics]].<ref>For dark matter, see {{harvnb|Milgrom|2002}}; for dark energy, {{harvnb|Caldwell|2004}}</ref>

Beyond the challenges of quantum effects and cosmology, research on general relativity is rich with possibilities for further exploration:  mathematical relativists explore the nature of singularities and the fundamental properties of Einstein's equations,<ref>See {{harvnb|Friedrich|2005}}.</ref> and ever more comprehensive computer simulations of specific spacetimes (such as those describing merging black holes) are run.<ref>A review of the various problems and the techniques being developed to overcome them, see {{harvnb|Lehner|2002}}.</ref>
More than one hundred years after the theory was first published, research is more active than ever.<ref>A good starting point for a snapshot of present-day research in relativity is the electronic review journal [http://relativity.livingreviews.org Living Reviews in Relativity].</ref>

==See also==
{{cols}}
* [[Physics:General relativity|General relativity]]
* [[Introduction to the mathematics of general relativity]]
* [[Physics:Special relativity|Special relativity]]
* [[Physics:History of general relativity|History of general relativity]]
* [[Astronomy:Tests of general relativity|Tests of general relativity]]
* [[Physics:Numerical relativity|Numerical relativity]]
* [[Physics:Derivations of the Lorentz transformations|Derivations of the Lorentz transformations]]
* [[Physics:General relativity#Further reading|List of books on general relativity]]{{colend}}

==References==
{{reflist|colwidth=30em}}

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{{Refend}}

==External links==
''Additional resources, including more advanced material, can be found in [[Physics:General relativity resources|General relativity resources]].''
* [https://www.einstein-online.info/en/ Einstein Online]. Website featuring articles on a variety of aspects of relativistic physics for a general audience, hosted by the [[Organization:Max Planck Institute for Gravitational Physics|Max Planck Institute for Gravitational Physics]]
* [https://web.archive.org/web/20090603075208/http://archive.ncsa.uiuc.edu/Cyberia/NumRel/NumRelHome.html NCSA Spacetime Wrinkles]. Website produced by the [[Physics:Numerical relativity|numerical relativity]] group at the [[Organization:National Center for Supercomputing Applications|National Center for Supercomputing Applications]], featuring an elementary introduction to general relativity, black holes and [[Astronomy:Gravitational wave|gravitational wave]]s

{{Relativity}}
{{Introductory science articles}}

{{DEFAULTSORT:General relativity, Introduction to}}
[[Category:General relativity| ]]
[[Category:Gravity]]

{{Sourceattribution|Introduction to general relativity}}