
General relativity is a
theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
of
gravitation
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
developed by
Albert Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping of
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
.
By the beginning of the 20th century,
Newton's law of universal gravitation
Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is Proportionality (mathematics)#Direct proportionality, proportional to the product ...
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
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
s of
Mercury and other
planet
A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
s. General relativity also predicts novel effects of gravity, such as
gravitational wave
Gravitational waves are oscillations of the gravitational field that Wave propagation, travel through space at the speed of light; they are generated by the relative motion of gravity, gravitating masses. They were proposed by Oliver Heaviside i ...
s,
gravitational lens
A gravitational lens is matter, such as a galaxy cluster, cluster of galaxies or a point particle, that bends light from a distant source as it travels toward an observer. The amount of gravitational lensing is described by Albert Einstein's Ge ...
ing and an effect of gravity on time known as
gravitational time dilation
Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events, as measured by observers situated at varying distances from a gravitating mass. The lower the gravitational potential (the closer ...
. Many of these predictions have been confirmed by experiment or observation,
most recently gravitational waves.
General relativity has developed into an essential tool in modern
astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the ...
. It provides the foundation for the current understanding of
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
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
radiation
In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or a material medium. This includes:
* ''electromagnetic radiation'' consisting of photons, such as radio waves, microwaves, infr ...
emitted by certain types of astronomical objects (such as
active galactic nuclei
An active galactic nucleus (AGN) is a compact region at the center of a galaxy that emits a significant amount of energy across the electromagnetic spectrum, with characteristics indicating that this luminosity is not produced by the stars. Such e ...
or
microquasar
A microquasar, a smaller version of a quasar, is a compact region surrounding a stellar black hole with a mass several times that of its companion star, observable in sufficient details, in our own or nearby galaxy. The matter being pulled from ...
s). General relativity is also part of the framework of the standard
Big Bang
The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models based on the Big Bang concept explain a broad range of phenomena, including th ...
model of
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...
.
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
quantum physics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
to produce a complete and self-consistent theory of
quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
.
From special to general relativity
In September 1905,
Albert Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
published his theory of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity,
"On the Ele ...
, which reconciles
Newton's laws of motion
Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows:
# A body re ...
with
electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
(the interaction between objects with
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
). Special relativity introduced a new framework for all of physics by proposing new concepts of
space
Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
and time. Some then-accepted physical theories were inconsistent with that framework; a key example was Newton's theory of
gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
, 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
A thought experiment is an imaginary scenario that is meant to elucidate or test an argument or theory. It is often an experiment that would be hard, impossible, or unethical to actually perform. It can also be an abstract hypothetical that is ...
involving an observer in free fall to his fully geometric theory of gravity.
Equivalence principle
A person in a
free-falling elevator experiences
weightlessness
Weightlessness is the complete or near-complete absence of the sensation of weight, i.e., zero apparent weight. It is also termed zero g-force, or zero-g (named after the g-force) or, incorrectly, zero gravity.
Weight is a measurement of the fo ...
; 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
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity,
"On the Ele ...
: observers for whom
light
Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
travels along straight lines at constant speed.
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
equivalence principle
The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...
. 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.
Gravity and acceleration

Most effects of gravity vanish in
free fall
In classical mechanics, free fall is any motion of a physical object, body where gravity is the only force acting upon it.
A freely falling object may not necessarily be falling down in the vertical direction. If the common definition of the word ...
, but effects that seem the same as those of gravity can be ''produced'' by an
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
2, the
standard gravity
The standard acceleration of gravity or standard acceleration of free fall, often called simply standard gravity and denoted by or , is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is a constant ...
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 .
An observer in an accelerated reference frame must introduce what physicists call
fictitious force
A fictitious force, also known as an inertial force or pseudo-force, is a force that appears to act on an object when its motion is described or experienced from a non-inertial reference frame, non-inertial frame of reference. Unlike real forc ...
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''. The apparent magnitude of the fictitious forces always appears to be proportional to the mass of any object on which they actfor 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
Newton's law of gravitation
Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the s ...
.
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.

The first new effect is the
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
red-shifted, as may be calculated from
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity,
"On the Ele ...
; the second observer will measure a lower
frequency
Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
for the light than the first sent out. Conversely, light sent from the higher observer to the lower is
blue-shifted, that is, shifted towards higher frequencies. 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
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
* Ernst von Below (1863–1955), German World War I general
* Fred Belo ...
.
This gravitational frequency shift corresponds to a
gravitational time dilation
Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events, as measured by observers situated at varying distances from a gravitating mass. The lower the gravitational potential (the closer ...
: 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. This effect is minute, but it too has been confirmed experimentally in multiple experiments, as described
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
* Ernst von Below (1863–1955), German World War I general
* Fred Belo ...
.
In a similar way, Einstein predicted the
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.
Tidal effects

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
fictitious force
A fictitious force, also known as an inertial force or pseudo-force, is a force that appears to act on an object when its motion is described or experienced from a non-inertial reference frame, non-inertial frame of reference. Unlike real forc ...
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 – 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
center of gravity
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weighted relative position of the distributed mass sums to zero. For ...
. 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
skydiver
Parachuting and skydiving are methods of descending from a high point in an atmosphere to the ground or ocean surface with the aid of gravity, involving the control of speed during the descent using a parachute or multiple parachutes.
For hu ...
s on opposite sides of the Earth, the effect is large. Such differences in force are also responsible for the
tide
Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another.
Tide tables ...
s in the Earth's oceans, so the term "
tidal effect
The tidal force or tide-generating force is the difference in gravitational attraction between different points in a gravitational field, causing bodies to be pulled unevenly and as a result are being stretched towards the attraction. It is the ...
" is used for this phenomenon.
The equivalence between inertia and gravity cannot explain tidal effects – it cannot explain variations in the gravitational field. 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 (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
of
surfaces
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space.
Surface or surfaces may also refer to:
Mathematics
*Surface (mathematics), a generalization of a plane which needs not be flat
* Sur ...
. 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
fictitious force
A fictitious force, also known as an inertial force or pseudo-force, is a force that appears to act on an object when its motion is described or experienced from a non-inertial reference frame, non-inertial frame of reference. Unlike real forc ...
s have to be introduced in order to explain particle motion): this is analogous to the transition from a
Cartesian coordinate system
In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
(in which the coordinate lines are straight lines) to a
curved coordinate system (where coordinate lines need not be straight).
A deeper analogy relates tidal forces with a property of surfaces called ''
curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
''. 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
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*'' Equiva ...
to a
plane. In the summer of 1912, inspired by these analogies, Einstein searched for a geometric formulation of gravity.
The elementary objects of
geometry –
points
A point is a small dot or the sharp tip of something. Point or points may refer to:
Mathematics
* Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...
,
lines,
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s – are traditionally defined in three-dimensional
space
Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
or on two-dimensional
surfaces
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space.
Surface or surfaces may also refer to:
Mathematics
*Surface (mathematics), a generalization of a plane which needs not be flat
* Sur ...
. In 1907,
Hermann Minkowski
Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, o ...
, Einstein's former mathematics professor at the Swiss Federal Polytechnic, introduced
Minkowski space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
, a geometric formulation of Einstein's
special theory of relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 paper,
"On the Electrodynamics of Moving Bodies", the theory is presen ...
where the geometry included not only
space
Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
but also time. The basic entity of this new geometry is four-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
al
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. The orbits of moving bodies are
curves in spacetime; the orbits of bodies moving at constant speed without changing direction correspond to straight lines.
The geometry of general curved surfaces was developed in the early 19th century by
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
. This geometry had in turn been generalized to higher-dimensional spaces in
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
introduced by
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
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.
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
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...
influences spacetime's curvature. Having formulated what are now known as
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
The Royal Prussian Academy of Sciences () was an academy established in Berlin, Germany on 11 July 1700, four years after the Prussian Academy of Arts, or "Arts Academy," to which "Berlin Academy" may also refer. In the 18th century, when Frenc ...
in late 1915, culminating in his final presentation on November 25, 1915.
Geometry and gravitation
Paraphrasing
John Wheeler, Einstein's geometric theory of gravity can be summarized as: ''spacetime tells matter how to move; matter tells spacetime how to curve''. What this means is addressed in the following three sections, which explore the motion of so-called
test particle In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...
s, 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

In order to map a body's gravitational influence, it is useful to think about what physicists call probe or
test particle In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...
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
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
, 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
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
, or
curved, and in curved spacetime straight world lines may not exist. Instead, test particles move along lines called
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
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 or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional spac ...
, 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
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Discussion
Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...
, such as a
line of longitude or the
equator
The equator is the circle of latitude that divides Earth into the Northern Hemisphere, Northern and Southern Hemisphere, Southern Hemispheres of Earth, hemispheres. It is an imaginary line located at 0 degrees latitude, about in circumferen ...
. 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.
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
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. For objects massive enough that their 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.
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
mass
Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
. 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
energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
, as expressed by the formula
''E'' = ''mc'', 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
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
or the
binding energy
In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly use ...
of systems such as
nuclei or
molecule
A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s, contribute to that body's mass, and hence act as sources of gravity.
In special relativity, energy is closely connected to
momentum
In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
. 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
four-momentum
In special relativity, four-momentum (also called momentum–energy or momenergy) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum i ...
. 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
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
and
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
energy–momentum tensor Energy–momentum may refer to:
* Four-momentum
* Stress–energy tensor
* Energy–momentum relation
{{dab ...
.
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
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
, in which the geometric properties of a space (or a spacetime) are described by a quantity called a
metric
Metric or metrical may refer to:
Measuring
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
...
. The metric encodes the information needed to compute the fundamental geometric notions of distance and angle in a curved space (or spacetime).

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
latitude
In geography, latitude is a geographic coordinate system, geographic coordinate that specifies the north-south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at t ...
and
longitude
Longitude (, ) is a geographic coordinate that specifies the east- west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lett ...
. 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 , while for someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely . 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.
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
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mos ...
, which describes exactly how the
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
, 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
energy–momentum tensor Energy–momentum may refer to:
* Four-momentum
* Stress–energy tensor
* Energy–momentum relation
{{dab ...
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
Einstein tensor
In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field e ...
, which describes some aspects of the way spacetime is curved. ''Einstein's equation'' then states that
:
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
gravitational constant
The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
of Newtonian gravity, and ''c'' is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
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.
A solution of these equations describes a particular geometry of
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
; for example, the
Schwarzschild solution describes the geometry around a spherical, non-rotating mass such as a
star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
or a
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
, whereas the
Kerr solution
The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric tensor, metric is an Exact solutions in general relativity, e ...
describes a rotating black hole. Still other solutions can describe a
gravitational wave
Gravitational waves are oscillations of the gravitational field that Wave propagation, travel through space at the speed of light; they are generated by the relative motion of gravity, gravitating masses. They were proposed by Oliver Heaviside i ...
or, in the case of the
Friedmann–Lemaître–Robertson–Walker solution, an expanding universe. The simplest solution is the uncurved
Minkowski spacetime
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a s ...
, the spacetime described by special relativity.
Experiments
No scientific theory is self-evidently true; each is a model that must be checked by experiment.
Newton's law of gravity
Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the sq ...
was accepted because it accounted for the motion of planets and moons in the
Solar System
The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...
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:

* Newtonian gravity predicts that the
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
which a single
planet
A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
traces around a perfectly spherical
star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
should be an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
. 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
anomalous perihelion shift. The first measurement of this effect, for the 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
radio telescope
A radio telescope is a specialized antenna (radio), antenna and radio receiver used to detect radio waves from astronomical radio sources in the sky. Radio telescopes are the main observing instrument used in radio astronomy, which studies the r ...
s. General relativity predicts the correct anomalous perihelion shift for all planets where this can be measured accurately (
Mercury,
Venus
Venus is the second planet from the Sun. It is often called Earth's "twin" or "sister" planet for having almost the same size and mass, and the closest orbit to Earth's. While both are rocky planets, Venus has an atmosphere much thicker ...
and the Earth).
* According to general relativity, light does not travel along straight lines when it propagates in a gravitational field. Instead, it is
deflected in the presence of massive bodies. In particular, starlight is deflected as it passes near the Sun, leading to apparent shifts of up to 1.75
arc seconds
A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...
in the stars' positions in the sky (an arc second is equal to 1/3600 of a
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
solar eclipse
A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby obscuring the view of the Sun from a small part of Earth, totally or partially. Such an alignment occurs approximately every six months, during the eclipse season i ...
. In this way, a British expedition to West Africa in 1919, directed by
Arthur Eddington
Sir Arthur Stanley Eddington, (28 December 1882 – 22 November 1944) was an English astronomer, physicist, and mathematician. He was also a philosopher of science and a populariser of science. The Eddington limit, the natural limit to the lu ...
, confirmed that Einstein's prediction was correct, and the Newtonian predictions wrong, via observation of the
May 1919 eclipse. Eddington's results were not very accurate; subsequent observations of the deflection of the light of distant
quasar
A quasar ( ) is an extremely Luminosity, luminous active galactic nucleus (AGN). It is sometimes known as a quasi-stellar object, abbreviated QSO. The emission from an AGN is powered by accretion onto a supermassive black hole with a mass rangi ...
s by the Sun, which utilize highly accurate techniques of
radio astronomy
Radio astronomy is a subfield of astronomy that studies Astronomical object, celestial objects using radio waves. It started in 1933, when Karl Jansky at Bell Telephone Laboratories reported radiation coming from the Milky Way. Subsequent observat ...
, have confirmed Eddington's results with significantly better precision (the first such measurements date from 1967, the most recent comprehensive analysis from 2004).
*
Gravitational redshift
In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well lose energy. This loss of energy correspo ...
was first measured in a laboratory setting in 1959 by
Pound and Rebka. It is also seen in astrophysical measurements, notably for light escaping the
white dwarf
A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
Sirius B
Sirius is the brightest star in the night sky. Its name is derived from the Greek word (Latin script: ), meaning 'glowing' or 'scorching'. The star is designated Canis Majoris, Latinized to Alpha Canis Majoris, and abbre ...
. The related
gravitational time dilation
Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events, as measured by observers situated at varying distances from a gravitating mass. The lower the gravitational potential (the closer ...
effect has been measured by transporting
atomic clock
An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwee ...
s to altitudes of between tens and tens of thousands of kilometers (first by
Hafele and Keating in 1971; most accurately to date by
Gravity Probe A
Gravity Probe A (GP-A) was a space-based experiment to test the equivalence principle, a feature of Einstein's theory of relativity. It was performed jointly by the Smithsonian Astrophysical Observatory and the National Aeronautics and Space Adm ...
launched in 1976).
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. These three experiments justified adopting general relativity over Newton's theory and, incidentally, over a number of
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
A gyroscope (from Ancient Greek γῦρος ''gŷros'', "round" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining Orientation (geometry), orientation and angular velocity. It is a spinning wheel or disc in ...
travelling through space. One of these effects,
geodetic precession, has been tested with the
Lunar Laser Ranging Experiment
Lunar Laser Ranging (LLR) is the practice of measuring Lunar distance (astronomy), the distance between the surfaces of the Earth and the Moon using Lidar, laser ranging. The distance can be calculated from the Round-trip delay, round-trip time ...
(high-precision measurements of the orbit of the
Moon
The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
). Another, which is related to rotating masses, is called
frame-dragging
Frame-dragging is an effect on spacetime, predicted by Albert Einstein's General relativity, general theory of relativity, that is due to non-static stationary distributions of mass–energy. A stationary Field (physics), field is one that is ...
. The geodetic and frame-dragging effects were both tested by the
Gravity Probe B
Gravity Probe B (GP-B) was a satellite-based experiment whose objective was to test two previously-unverified predictions of general relativity: the geodetic effect and frame-dragging. This was to be accomplished by measuring, very precisely, t ...
satellite experiment launched in 2004, with results confirming relativity to within 0.5% and 15%, respectively, as of December 2008.
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
binary pulsars. In such a star system, two highly compact
neutron star
A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
s orbit each other. At least one of them is a
pulsar
A pulsar (''pulsating star, on the model of quasar'') is a highly magnetized rotating neutron star that emits beams of electromagnetic radiation out of its Poles of astronomical bodies#Magnetic poles, magnetic poles. This radiation can be obse ...
– 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.
One particular set of observations is related to eminently useful practical applications, namely to
satellite navigation system
A satellite or an artificial satellite is an object, typically a spacecraft, placed into orbit around a celestial body. They have a variety of uses, including communication relay, weather forecasting, navigation ( GPS), broadcasting, scientif ...
s such as the
Global Positioning System
The Global Positioning System (GPS) is a satellite-based hyperbolic navigation system owned by the United States Space Force and operated by Mission Delta 31. It is one of the global navigation satellite systems (GNSS) that provide ge ...
(GPS) that are used for both precise
positioning Positioning may refer to:
* Positioning (marketing), creating an identity in the minds of a target market
* Positioning theory, a theory in social psychology
* Positioning (critical literacy), reader context
* Positioning (telecommunications), a t ...
and
timekeeping
Time is the continuous progression of existence that occurs in an apparently irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequence events, to compa ...
. Such systems rely on two sets of
atomic clocks
An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwee ...
: 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
universal coordinated time
Coordinated Universal Time (UTC) is the primary time standard globally used to regulate clocks and time. It establishes a reference for the current time, forming the basis for civil time and time zones. UTC facilitates international communica ...
) are testament to the validity of the relativistic predictions.
A number of other tests have probed the validity of various versions of the
equivalence principle
The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...
; 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.
Astrophysical applications
Models based on general relativity play an important role in
astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the ...
; the success of these models is further testament to the theory's validity.
Gravitational lensing

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
quasar
A quasar ( ) is an extremely Luminosity, luminous active galactic nucleus (AGN). It is sometimes known as a quasi-stellar object, abbreviated QSO. The emission from an AGN is powered by accretion onto a supermassive black hole with a mass rangi ...
can pass along one side of a massive
galaxy
A galaxy is a Physical system, system of stars, stellar remnants, interstellar medium, interstellar gas, cosmic dust, dust, and dark matter bound together by gravity. The word is derived from the Ancient Greek, Greek ' (), literally 'milky', ...
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 lens
A lens is a transmissive optics, optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a #Compound lenses, compound lens consists of several simple ...
es, and hence the corresponding gravitational effect is called
gravitational lensing
A gravitational lens is matter, such as a galaxy cluster, cluster of galaxies or a point particle, that bends light from a distant source as it travels toward an observer. The amount of gravitational lensing is described by Albert Einstein's Ge ...
.
Observational astronomy
Observational astronomy is a division of astronomy that is concerned with recording data about the observable universe, in contrast with theoretical astronomy, which is mainly concerned with calculating the measurable implications of physical ...
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
mass
Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
distribution responsible for the light deflection. In particular, gravitational lensing provides one way to measure the distribution of
dark matter
In astronomy, dark matter is an invisible and hypothetical form of matter that does not interact with light or other electromagnetic radiation. Dark matter is implied by gravity, gravitational effects that cannot be explained by general relat ...
, 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.
Gravitational waves
Gravitational wave
Gravitational waves are oscillations of the gravitational field that Wave propagation, travel through space at the speed of light; they are generated by the relative motion of gravity, gravitating masses. They were proposed by Oliver Heaviside i ...
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
In fluid dynamics, gravity waves are waves in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere and the oc ...
s of
fluid dynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
, which are a different concept.
In February 2016, the Advanced
LIGO
The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a large-scale physics experiment and observatory designed to detect cosmic gravitational waves and to develop gravitational-wave observations as an astronomical tool. Prior to LIG ...
team announced that they had directly
observed gravitational waves from a
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
merger.
Indirectly, the effect of gravitational waves had been detected in observations of specific binary stars. Such pairs of stars
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
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
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
neutron star
A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
, for which gravitational wave emission is much stronger than for ordinary stars. Also, a pulsar emits a narrow beam of
electromagnetic radiation
In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength ...
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,
Russell Hulse and
Joseph Taylor, were awarded the
Nobel Prize in Physics
The Nobel Prize in Physics () is an annual award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions to mankind in the field of physics. It is one of the five Nobel Prizes established by the ...
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.
Currently, a number of land-based
gravitational wave detectors are in operation, and a mission to launch a space-based detector,
LISA
Lisa or LISA may refer to:
People
People with the mononym
* Lisa (Japanese musician, born 1974), stylized "LISA"
* Lisa, stagename of Japanese singer Lisa Komine (born 1978)
* Lisa (South Korean singer) (born 1980)
* Lisa (Japanese musician, b ...
, is currently under development, with a precursor mission (
LISA Pathfinder
LISA Pathfinder, formerly Small Missions for Advanced Research in Technology-2 (SMART-2), was an ESA spacecraft that was launched on 3 December 2015 on board Vega flight VV06. The mission tested technologies needed for the Laser Interferometer ...
) 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
universe
The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
fractions of a second after the
Big Bang
The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models based on the Big Bang concept explain a broad range of phenomena, including th ...
.
Black holes

When mass is concentrated into a sufficiently
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
region of space, general relativity predicts the formation of a
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
– 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
evolution
Evolution is the change in the heritable Phenotypic trait, characteristics of biological populations over successive generations. It occurs when evolutionary processes such as natural selection and genetic drift act on genetic variation, re ...
of massive
star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s. On the other hand,
supermassive black hole
A supermassive black hole (SMBH or sometimes SBH) is the largest type of black hole, with its mass being on the order of hundreds of thousands, or millions to billions, of times the mass of the Sun (). Black holes are a class of astronomical ...
s with the mass of
million
1,000,000 (one million), or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian ''millione'' (''milione'' in modern Italian), from ''mille'', "thousand", plus the ...
s or
billions of
Sun
The Sun is the star at the centre of the Solar System. It is a massive, nearly perfect sphere of hot plasma, heated to incandescence by nuclear fusion reactions in its core, radiating the energy from its surface mainly as visible light a ...
s are assumed to reside in the cores of most
galaxies
A galaxy is a system of stars, stellar remnants, interstellar gas, dust, and dark matter bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar Sys ...
, and they play a key role in current models of how galaxies have formed over the past billions of years.
Matter falling onto a compact object is one of the most efficient mechanisms for releasing
energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
in the form of
radiation
In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or a material medium. This includes:
* ''electromagnetic radiation'' consisting of photons, such as radio waves, microwaves, infr ...
, 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
A quasar ( ) is an extremely Luminosity, luminous active galactic nucleus (AGN). It is sometimes known as a quasi-stellar object, abbreviated QSO. The emission from an AGN is powered by accretion onto a supermassive black hole with a mass rangi ...
and other types of
active galactic nuclei
An active galactic nucleus (AGN) is a compact region at the center of a galaxy that emits a significant amount of energy across the electromagnetic spectrum, with characteristics indicating that this luminosity is not produced by the stars. Such e ...
. 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.
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
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.
Cosmology

One of the most important aspects of general relativity is that it can be applied to the
universe
The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
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
homogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
and
isotropic
In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
. Such comparatively simple universes can be described by simple solutions of Einstein's equations. The current
cosmological models of the universe are obtained by combining these simple solutions to general relativity with theories describing the properties of the universe's
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...
content, namely
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
,
nuclear- and
particle physics
Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
. According to these models, our present universe emerged from an extremely dense high-temperature state – the
Big Bang
The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models based on the Big Bang concept explain a broad range of phenomena, including th ...
– roughly 14
billion
Billion is a word for a large number, and it has two distinct definitions:
* 1,000,000,000, i.e. one thousand million, or (ten to the ninth power), as defined on the short scale. This is now the most common sense of the word in all varieties of ...
year
A year is a unit of time based on how long it takes the Earth to orbit the Sun. In scientific use, the tropical year (approximately 365 Synodic day, solar days, 5 hours, 48 minutes, 45 seconds) and the sidereal year (about 20 minutes longer) ...
s ago and has been
expanding ever since.
Einstein's equations can be generalized by adding a term called the
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is a coefficient that Albert Einstein initially added to his field equations of general rel ...
. When this term is present,
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
accelerating
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are vector quantities (in that they have magnit ...
expansion consistent with a cosmological constant – or, equivalently, with a particular and ubiquitous kind of
dark energy
In physical cosmology and astronomy, dark energy is a proposed form of energy that affects the universe on the largest scales. Its primary effect is to drive the accelerating expansion of the universe. It also slows the rate of structure format ...
– has steadily been accumulating.
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.
In contrast to all other modern theories of
fundamental interaction
In physics, the fundamental interactions or fundamental forces are interactions in nature that appear not to be reducible to more basic interactions. There are four fundamental interactions known to exist:
* gravity
* electromagnetism
* weak int ...
s, general relativity is a
classical theory: it does not include the effects of
quantum physics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. The quest for a quantum version of general relativity addresses one of the most fundamental
open questions in physics. While there are promising candidates for such a theory of
quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
, notably
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
and
loop quantum gravity
Loop quantum gravity (LQG) is a theory of quantum gravity that incorporates matter of the Standard Model into the framework established for the intrinsic quantum gravity case. It is an attempt to develop a quantum theory of gravity based direc ...
, 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
Singularity or singular point may refer to:
Science, technology, and mathematics Mathematics
* Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
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
beginning of the universe.
Other attempts to modify general relativity have been made in the context of
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...
. In the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely
dark energy
In physical cosmology and astronomy, dark energy is a proposed form of energy that affects the universe on the largest scales. Its primary effect is to drive the accelerating expansion of the universe. It also slows the rate of structure format ...
and
dark matter
In astronomy, dark matter is an invisible and hypothetical form of matter that does not interact with light or other electromagnetic radiation. Dark matter is implied by gravity, gravitational effects that cannot be explained by general relat ...
. 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
modified Newtonian dynamics
Modified Newtonian dynamics (MOND) is a theory that proposes a modification of Newton's laws to account for observed properties of galaxies. Modifying Newton's law of gravity results in modified gravity, while modifying Newton's second law resul ...
.
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, and ever more comprehensive computer simulations of specific spacetimes (such as those describing merging black holes) are run.
More than one hundred years after the theory was first published, research is more active than ever.
[A good starting point for a snapshot of present-day research in relativity is the electronic review journa]
Living Reviews in Relativity
See also
*
General relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
*
Introduction to the mathematics of general relativity
The mathematics of general relativity is complicated. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be s ...
*
Special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity,
"On the Ele ...
*
History of general relativity
History is the systematic study of the past, focusing primarily on the human past. As an academic discipline, it analyses and interprets evidence to construct narratives about what happened and explain why it happened. Some theorists categ ...
*
Tests of general relativity
Tests of general relativity serve to establish observational evidence for the theory of general relativity. The first three tests, proposed by Albert Einstein in 1915, concerned the "anomalous" precession of the perihelion of Mercury (planet), Me ...
*
Numerical relativity
Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars a ...
*
Derivations of the Lorentz transformations
There are many ways to derive the Lorentz transformations using a variety of physical principles, ranging from Maxwell's equations to Einstein's postulates of special relativity, and mathematical tools, spanning from elementary algebra and hyperb ...
*
List of books on general relativity
References
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External links
''Additional resources, including more advanced material, can be found in
General relativity resources.''
Einstein Online Website featuring articles on a variety of aspects of relativistic physics for a general audience, hosted by the
Max Planck Institute for Gravitational Physics
NCSA Spacetime Wrinkles Website produced by the
numerical relativity
Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars a ...
group at the
National Center for Supercomputing Applications
The National Center for Supercomputing Applications (NCSA) is a unit of the University of Illinois Urbana-Champaign, and provides high-performance computing resources to researchers in the United States. NCSA is currently led by Professor Bill ...
, featuring an elementary introduction to general relativity, black holes and
gravitational wave
Gravitational waves are oscillations of the gravitational field that Wave propagation, travel through space at the speed of light; they are generated by the relative motion of gravity, gravitating masses. They were proposed by Oliver Heaviside i ...
s
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Gravity