NationMaster - Encyclopedia: Curved space

General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. It unifies special relativity and Isaac Newton's law of universal gravitation with the insight that gravitation is not viewed as being due to a force (in the traditional sense) but rather a manifestation of curved space and time, this curvature being produced by the mass-energy content of the spacetime. Image File history File links Please see the file description page for further information. ... It has been suggested that this article or section be merged into General relativity. ... Table of Geometry, from the 1728 Cyclopaedia. ... Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies. ... In physics, gravitation or gravity is the universal force of attraction between objects with mass. ... Albert Einstein photographed by Oren J. Turner in 1947. ... 1915 (MCMXV) was a common year starting on Friday (see link for calendar). ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... Sir Isaac Newton, PRS, (4 January [O.S. 25 December 1642] 1643 â€“ 31 March [O.S. 20 March] 1727) was an English physicist, mathematician, astronomer, alchemist, inventor, and natural philosopher who is generally regarded as one of the most influential scientists in history. ... It has been suggested that this article or section be merged into Gravity. ... In physics, a force is defined as a rate of change of momentum (Newtonian definition). ... Curvature is the amount by which a geometric object deviates from being flat. ... Mass is a property of a physical object that quantifies the amount of matter it contains. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ...

General relativity

Overview of GR
- History
- Mathematics
- Resources
- Tests
Black hole
Einstein equation
Equivalence principle
Event horizon
Exact solutions
FLRW metric
Gravitational lens
Gravitational radiation
Kerr metric
Quantum gravity
Schwarzschild metric
Singularity

Related topics

edit Image File history File links Download high resolution version (1024x768, 7 KB) Description: Gravitational light deflection at a neutron star Source: Gallery of Tempolimit Lichtgeschwindigkeit Date: 09. ... // Development Early investigations The development of general relativity began in 1907 with the publication of an article by Einstein on acceleration under special relativity. ... Notational point: General relativity articles using tensors will use the abstract index notation . ... // Books Popular Geroch, Robert (1981). ... Einsteins general theory of relativity was introduced in 1915. ... A black hole is a concentration of mass great enough that the force of gravity prevents anything past its event horizon from escaping it except through quantum tunnelling behaviour (known as Hawking Radiation). ... In physics, the Einstein field equation or Einstein equation is a differential equation in Einsteins theory of general relativity. ... THERE IS NO SUCH THING< MWAHAHAHAHAHAHAHA> In relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... Event Horizon is a 1997 science fiction and horror film. ... // Introduction In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field. ... The Friedmann-LemaÃ®tre-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. ... This article is in need of attention from an expert on the subject. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In general relativity, the Kerr metric describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... Quantum gravity is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... In Einsteins theory of general relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the gravitational field outside a spherical, non-rotating mass such as a (non-rotating) star, planet, or black hole. ... This article is in need of attention from an expert on the subject. ... Albert Einstein photographed by Oren J. Turner in 1947. ... Spiral Galaxy ESO 269-57 Astrophysics is the tree of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature and chemical composition) of astronomical objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... Gravity is a force of attraction that acts between bodies that have mass. ... Cosmology, as a branch of astrophysics, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ...

1 Overview
2 Predictions of General Relativity
3 Relationship to other physical theories
4 History
5 Status
6 Quotes
7 Notes
8 See also
9 References

Overview

Treatment of gravitation

In this theory, spacetime is treated as a 4-dimensional Lorentzian manifold which is curved by the presence of mass, energy, and momentum (or stress-energy) within it. The relationship between stress-energy and the curvature of spacetime is governed by the Einstein field equations. The motion of objects being influenced solely by the geometry of spacetime (inertial motion) occurs along special paths called timelike and null geodesics of spacetime. It has been suggested that this article or section be merged into General relativity. ... Illustration of spacetime curvature. ... Illustration of spacetime curvature. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... Mass is a property of a physical object that quantifies the amount of matter it contains. ... In physics, momentum is the product of the mass and velocity of an object. ... This article is in need of attention from an expert on the subject. ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... The principle of inertia is one of the fundamental laws of classical physics which are used to describe the motion of matter and how it is affected by applied forces. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... KK Null, a Japanese musician Null, a special value in computer programming. ... In physics, and specifically general relativity, geodesics are the world lines of a particle free from all external force. ...

It has been suggested that this article or section be merged with Newtonian mechanics. ... Gravity is a force of attraction that acts between bodies that have mass. ... Free Fall opens with one of the most stunning first paragraphs I have ever, or am ever likely to, read. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... An Ariane 5 expendable launch vehicle lifts off with the Rosetta spacecraft on March 2, 2004. ... In ordinary language, a trajectory is the path followed by a body moving through space, for instance, the path taken by a falling body or the orbit of a planet. ...

Justification

The justification for creating general relativity comes from the equivalence principle, which dictates that freefalling observers are the ones in inertial motion. A consequence of this insight is that inertial observers can accelerate with respect to each other. (Think of two balls falling on opposite sides of the Earth, for example.) This redefinition is incompatible with Newton's first law of motion, and cannot be accounted for in the Euclidean geometry of special relativity. To quote Einstein himself: THERE IS NO SUCH THING< MWAHAHAHAHAHAHAHA> In relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... Freefall or free fall in the strict sense is the condition of acceleration which is due only to gravity. ... The principle of inertia is one of the fundamental laws of classical physics which are used to describe the motion of matter and how it is affected by applied forces. ... Newtons laws of motion are three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... Euclid Euclidean geometry is a mathematical system due to the Hellenistic mathematician Euclid of Egypt. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ...

Thus the equivalence principle led Einstein to search for a gravitational theory which involves curved spacetimes.

Another motivating factor was the realization that relativity calls for gravitation to be expressed as a rank-two tensor, and not just a vector as was the case in Newtonian physics [2]. (An analogy is the electromagnetic field tensor of special relativity). Thus Einstein sought a rank-two tensor means of describing curved spacetimes surrounding massive objects. This effort came to fruition with the discovery of the Einstein field equations in 1915. In mathematics, a tensor is a generalized quantity or a certain kind of geometrical entity that includes all the ideas of scalars, vectors, matrices and linear operators. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In electromagnetism, the electromagnetic tensor, or electromagnetic field tensor, F, is defined as: where Ai is the vector potential. ...

Fundamental principles

General relativity is based on a set of fundamental principles which guided its development. These are:

(The equivalence principle, which was the starting point for the development of general relativity, ended up being a consequence of the general principle of relativity and the principle that inertial motion is geodesic motion.) The general principle of relativity as used in Einsteins general theory of relativity is that the laws of physics must take the same form in all reference frames. ... The principle of general covariance states that the laws of physics should take the same form in all coordinate systems. ... The principle of inertia is one of the fundamental laws of classical physics which are used to describe the motion of matter and how it is affected by applied forces. ... In physics, and specifically general relativity, geodesics are the world lines of a particle free from all external force. ... â€¹The template below has been proposed for deletion. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... THERE IS NO SUCH THING< MWAHAHAHAHAHAHAHA> In relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... // Development Early investigations The development of general relativity began in 1907 with the publication of an article by Einstein on acceleration under special relativity. ...

Spacetime as a curved Lorentzian manifold

In general relativity, the concept of spacetime (which was introduced by Hermann Minkowski for special relativity) is modified. In general relativity spacetime is World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... Hermann Minkowski. ...

The curvature of spacetime (caused by the presence of stress-energy) can be viewed intuitively in the following way. Placing a heavy object such as a bowling ball on a trampoline will produce a 'dent' in the trampoline. This is analogous to a large mass such as the Earth causing the local spacetime geometry to curve. This is represented by the image at the top of this article. The larger the mass, the bigger the amount of curvature. A relatively light object placed in the vicinity of the 'dent', such as a ping-pong ball, will accelerate towards the bowling ball in a manner governed by the 'dent'. Firing the ping-pong ball at just the right speed towards the 'dent' will result in the ping-pong ball 'orbiting' the bowling ball. This is analogous to the Moon orbiting the Earth, for example. Curvature is the amount by which a geometric object deviates from being flat. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ... 2-dimensional renderings (ie. ...

Similarly, in general relativity massive objects do not directly impart a force on other massive objects as hypothesized in Newton's action at a distance idea. Instead (in a manner analogous to the ping-pong ball's response to the bowling ball's dent rather than the bowling ball itself), other massive objects respond to how the first massive object curves spacetime. In Physics, action at a distance is the instantaneous interaction of two objects which are separated in space. ...

The mathematics of general relativity

Due to the expectation that spacetime is curved, Riemannian geometry (a type of non-Euclidean geometry) must be used. In essence, spacetime does not adhere to the "common sense" rules of Euclidean geometry, but instead objects that were initially traveling in parallel paths through spacetime (meaning that their velocities do not differ to first order in their separation) come to travel in a non-parallel fashion. This effect is called geodesic deviation, and it is used in general relativity as an alternative to gravity. For example, two people on the Earth heading due north from different positions on the equator are initially traveling on parallel paths, yet at the north pole those paths will cross. Similarly, two balls initially at rest with respect to and above the surface of the Earth (which are parallel paths by virtue of being at rest with respect to each other) come to have a converging component of relative velocity as both accelerate towards the center of the Earth due to their subsequent freefall. (Another way of looking at this is how a single ball moving in a purely timelike fashion parallel to the center of the Earth comes through geodesic motion to be moving towards the center of the Earth.) Notational point: General relativity articles using tensors will use the abstract index notation . ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... In differential geometry, the geodesic deviation equation is an equation involving the Riemann curvature tensor, which measures the change in separation of neighbouring geodesics. ... The equator is an imaginary circle drawn around a planet (or other astronomical object) at a distance halfway between the poles. ... The North Pole is the northernmost point on the Earth. ...

The requirements of the mathematics of general relativity are further modified by the other principles. Local Lorentz Invariance requires that the manifolds described in GR be 4-dimensional and Lorentzian instead of Riemannian. In addition, the principle of general covariance forces that math to be expressed using tensor calculus. Tensor calculus permits a manifold as mapped with a coordinate system to be equipped with a metric tensor of spacetime which describes the incremental (spacetime) intervals between coordinates from which both the geodesic equations of motion and the curvature tensor of the spacetime can be ascertained. In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... For more technical Wiki articles on tensors, see the section later in this article. ... A map is a simplified depiction of a space, a navigational aid which highlights relations between objects within that space. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

The Einstein field equations

The Einstein field equations (EFE) describe how stress-energy causes curvature of spacetime and are usually written in tensor form (using abstract index notation) as For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... In mathematics, a tensor is a generalized quantity or a certain kind of geometrical entity that includes all the ideas of scalars, vectors, matrices and linear operators. ... Abstract index notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...

where $G_{ab}$ is the Einstein tensor, $T_{ab}$ is the stress-energy tensor and $kappa$ is a constant. The tensors $G_{ab}$ and $T_{ab}$ are both rank 2 symmetric tensors, i.e. they can each be thought of as 4×4 matrices each of which contains 10 independent terms. Definition In differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds. ... This article is in need of attention from an expert on the subject. ...

The solutions of the EFE are metrics of spacetime. These metrics describe the structure of spacetime given the stress-energy and coordinate mapping used to obtain that solution. Being non-linear differential equations, the EFE often defy attempts to obtain an exact solution; however, many such solutions are known. In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... // Introduction In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field. ...

The EFE reduce to Newton's law of gravity in the limiting cases of a weak gravitational field and slow speed relative to the speed of light. In fact, the value of $kappa$ in the EFE is determined to be $kappa = 8 pi G / c^4$ by making these two approximations. Gravitation is the tendency of masses to move toward each other. ... The weak-field approximation in general relativity is used to describe the gravitational field very far from the source of gravity. ...

The EFE are the identifying feature of general relativity. Other theories built of the same premises include additional rules and/or constraints. The result almost invariably is a theory with different field equations (such as Brans-Dicke theory, teleparallelism, Rosen bimetric theory, and Einstein-Cartan theory). In mathematical physics, the Brans-Dicke theory of gravitation (sometimes called the Jordan/Brans/Dicke theory) is a well-known competitor of Einsteins theory of general relativity. ... // History Teleparallelism, also called distant parallelism, was an attempt by Einstein to unify electromagnetism and gravity. ... This article is in need of attention from an expert on the subject. ...

Coordinate vs. physical acceleration

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

In classical mechanics, space is preferentially mapped with a Cartesian coordinate system. Inertial motion then occurs as one moves through this space at a consistent coordinate rate with respect to time. Any change in this rate of progression must be due to a force, and therefore a physical and coordinate acceleration were in classical mechanics one and the same. It is important to note that in special relativity that same kind of Cartesian coordinate system was used, with time being added as a fourth dimension and defined for an observer using the Einstein synchronization procedure. As a result, physical and coordinate acceleration correspond in special relativity too, although their magnitudes may vary. It has been suggested that this article or section be merged with Newtonian mechanics. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ...

In general relativity, the elegance of a flat spacetime and the ability to use a preferred coordinate system are lost (due to stress-energy curving spacetime and the principle of general covariance). Consequently, coordinate and physical accelerations become sundered. For example: Try using a radial coordinate system in classical mechanics. In this system, an inertially moving object which passes by (instead of through) the origin point is found to first be moving mostly inwards, then to be moving tangentially with respect to the origin, and finally to be moving outwards, yet is moving in a straight line. This is an example of an inertially moving object undergoing a coordinate acceleration, and the way this coordinate acceleration changes as the object travels are given by the geodesic equations for the manifold and coordinate system in use. To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ...

Another more direct example is the case of someone standing on the Earth, where they are at rest with respect to the surface coordinates for the Earth (latitude, longitude, and elevation) but are undergoing a continuous physical acceleration because the mechanical resistance of the Earth's surface keeps them from free falling.

Predictions of General Relativity

Gravitational effects

Acceleration effects

These effects occur in any accelerated frame of reference, and are therefore independent of the curvature of spacetime. (Note however that spacetime curvature usually is the source the causative acceleration when these effects are being observed.)

Redshift of spectral lines in the optical spectrum of a supercluster of distant galaxies (right), as compared to that of the Sun (left). ... Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye (visible light) or, in a technical or scientific context, electromagnetic radiation of any wavelength. ... The Pound-Rebka experiment is a well known experiment in general relativity. ... This article is in need of attention from an expert on the subject. ... The Hafele-Keating experiment was a test of the theory of relativity. ... Over fifty GPS satellites such as this NAVSTAR have been launched since 1978. ... In General relativity, the Shapiro effect, or gravitational time delay, is one of the four classic solar system tests of general relativity. ... Image File history File links Download high resolution version (1579x1224, 150 KB)A photograph of a solar eclipse, from a 1919 test of general relativity. ... Image File history File links Download high resolution version (1579x1224, 150 KB)A photograph of a solar eclipse, from a 1919 test of general relativity. ... Photo taken during the French 1999 eclipse An eclipse (Greek verb: ecleipo, to cease existing or calypse, to cover ) is an astronomical event that occurs when one celestial object moves into the shadow of another. ...

Bending of light

This bending also occurs in any accelerated frame of reference. However, the details of the bending and therefore the gravitational lensing effects are governed by spacetime curvature.

Photo taken during the French 1999 eclipse An eclipse (Greek verb: ecleipo, to cease existing or calypse, to cover ) is an astronomical event that occurs when one celestial object moves into the shadow of another. ... A gravitational lens is formed when the light from a very distant, bright source (such as a quasar) is bent around a massive object (such as a massive galaxy) between the source object and the observer. ... A gravitational lens is formed when the light from a very distant, bright source (such as a quasar) is bent around a massive object (such as a massive galaxy) between the source object and the observer. ... A beautiful example of an Einstein ring is the radio source B1938+666 discovered with the UK radiotelescope. ...

Orbital effects

These are ways in which the celestial mechanics of general relativity differs from that of classical mechanics. Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ...

Precession refers to a change in the direction of the axis of a rotating object. ... Another view of Keplerian orbital elements. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... Precession (also called gyroscopic precession) is the phenomenon by which the axis of a spinning object (e. ... The newton (symbol: N) is the SI unit of force. ... Gravity is a force of attraction that acts between bodies that have mass. ... Einsteins general theory of relativity was introduced in 1915. ... Composite Optical/X-ray image of the Crab Nebula pulsar, showing surrounding nebular gases stirred by the pulsars magnetic field and radiation. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... This article does not cite its references or sources. ... Gravity Probe B (GP-B) is a satellite-based mission to measure the stress-energy tensor (the distribution, and especially the motion, of matter) in and near Earth, and thus to test related models; in application of Einsteins general theory of relativity. ...

Rotational effects

According to Albert Einsteins theory of general relativity, space and time get pulled out of shape near a rotating body in a phenomenon referred to as frame-dragging. ... Gravity Probe B (GP-B) is a satellite-based mission to measure the stress-energy tensor (the distribution, and especially the motion, of matter) in and near Earth, and thus to test related models; in application of Einsteins general theory of relativity. ...

Black holes

Black holes are objects which have gravitationally collapsed behind an event horizon. In a "classical" black hole, nothing that enters can ever escape. However, Stephen Hawking has shown that black holes can "leak" energy, a phenomenon called Hawking radiation. A black hole is a concentration of mass great enough that the force of gravity prevents anything past its event horizon from escaping it except through quantum tunnelling behaviour (known as Hawking Radiation). ... Event Horizon is a 1997 science fiction and horror film. ... Stephen Hawking in 2005 Professor Stephen William Hawking, CH, CBE, FRS, (born January 8, 1942) is considered one of the worlds leading theoretical physicists. ... In physics, Hawking radiation is thermal radiation thought to be emitted by black holes due to quantum effects. ...

Cosmological effects

Hubbles law is the statement in physical cosmology that the redshift in light coming from distant galaxies is proportional to their distance. ... The Friedmann-LemaÃ®tre-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. ... Edwin Hubble Edwin Powell Hubble (November 20, 1889 â€“ September 28, 1953) was an American astronomer, noted for his discovery of galaxies beyond the Milky Way and the cosmological Redshift. ... 1929 (MCMXXIX) was a common year starting on Tuesday (link will take you to calendar). ... Hubbles law is the statement in astronomy that the redshift in light coming from distant galaxies is proportional to their distance. ... According to the Big Bang theory, the universe emerged from an extremely dense and hot state (bottom). ... This article is in need of attention from an expert on the subject. ... When any patch of the sky is observed where no individual sources can be discerned, and the effects of interplanetary dust, and interstellar matter are taken into account, there is still radiation. ... Arno Allan Penzias (born April 26, American physicist. ... Robert Woodrow Wilson (born January 10, 1936) is an American physicist. ... 1965 (MCMLXV) was a common year starting on Friday (the link is to a full 1965 calendar). ... In cosmology, dark energy is a hypothetical form of energy which permeates all of space and has strong negative pressure. ...

Other predictions

In relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Composite Optical/X-ray image of the Crab Nebula pulsar, showing surrounding nebular gases stirred by the pulsars magnetic field and radiation. ... In physics, the graviton is a hypothetical elementary particle that transmits the force of gravity in most quantum gravity systems. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... A subatomic particle is a particle smaller than an atom: it may be elementary or composite. ... Image:Quadrupole magnet. ... Multipole moments in mathematics and mathematical physics are an orthogonal basis for the decomposition of a function, based on the response of a field to point sources that are brought infinitely close to each other. ... The Earths magnetic field, which is approximately a dipole. ...

Relationship to other physical theories

Classical mechanics and special relativity

Classical mechanics and special relativity are lumped together here because special relativity is in many ways intermediate between general relativity and classical mechanics, and shares many attributes with classical mechanics.

Note that in the discussion which follows, the mathematics of general relativity is used heavily. Also note that under the principle of minimal coupling, the physical equations of special relativity can be turned into their general relativity equivalent by replacing the Minkowski metric (η_ab) with the relevant metric of spacetime (g_ab) and by replacing any regular derivatives with covariant derivatives. In the discussions that follow, the change of metrics is implied. Notational point: General relativity articles using tensors will use the abstract index notation . ...

Inertia

In both classical mechanics and special relativity, space and then spacetime were assumed to be flat. In the language of tensor calculus, this meant that R^a_bcd = 0, where R^a_bcd is the Riemann curvature tensor. In addition, the coordinate system itself was also assumed to be Cartesian. These restrictions permit inertial motion to be described mathematically as In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

Note that in classical mechanics, x^a is three-dimensional and τ ≡ t, where t is coordinate time. Proper time is time as measured by the clock for an observer who is traveling through spacetime. ...

In general relativity, these restrictions on the shape of spacetime and on the coordinate system to be used are lost. Therefore a different definition of inertial motion is required. In relativity, inertial motion occurs along timelike or null geodesics as parameterized by proper time. This is expressed mathematically by the geodesic equation: In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ...

Since x is a rank one tensor, these equations are four in number, with each one describing the second derivative of a coordinate with respect to proper time. (Note that under the Minkowski metric of special relativity, the values of the connections are all zeros. This is what turns the general relativity geodesic equations into $ddot{x}^a = 0$ for special relativity.) In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ... In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

Gravitation

For gravitation, the relationship between Newton's theory of gravity and general relativity is governed by the correspondence principle: General relativity must produce the same results as gravity does for the cases where Newtonian physics has been shown to be accurate. Gravity is a force of attraction that acts between bodies that have mass. ... In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ...

Around a spherically symmetric object, the theory of gravity predicts that objects will be physically accelerated towards the center on the object by the rule where

In the weak-field approximation of general relativity, an identical coordinate acceleration must exist. For the Schwarzschild solution (which is the simplest possible spacetime surrounding a massive object), the same acceleration as that which (in Newtonian physics) is created by gravity is obtained when a constant of integration is set equal to 2m (where m=MG/c^2). For more information, see Deriving the Schwarzschild solution. The weak-field approximation in general relativity is used to describe the gravitational field very far from the source of gravity. ... The Schwarzschild solution is one of the simplest and useful solutions of the Einstein field equations (see general relativity). ...

Transition from Newtonian mechanics to general relativity

Some of the basic concepts of general relativity can be outlined outside the relativistic domain. In particular, the idea that mass/energy generates curvature in space and that curvature affects the motion of masses can be illustrated in a Newtonian setting. Some of the basic concepts of General Relativity can be outlined outside the relativistic domain. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... Curvature is the amount by which a geometric object deviates from being flat. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... The original version of the physical discipline of mechanics, due to Sir Isaac Newton, who developed the theory over a period from about 1664, until the publication of his great work, known as the Principia, in 1687. ...

General relativity generalizes the geodesic equation and the field equation to the relativistic realm in which trajectories in space are replaced with Fermi-Walker transport along world lines in spacetime. The equations are also generalized to more complicated curvatures. In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... Fermi Walker transport is a process in General relativity to define a coordinate system or reference frame such that all curvature in the frame is due to the presence of mass/energy density and not to arbitrary spin or rotation of the frame. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ...

Transition from special relativity to general relativity

The basic structure of general relativity, including the geodesic equation and Einstein field equation, can be obtained from special relativity by examining the kinetics and dynamics of a particle in a circular orbit about the earth. The basic structure of general relativity, including the geodesic equation and Einstein field equation, can be obtained from special relativity by examining the dynamics of a particle in a circular orbit about the earth. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... Kinetics refers to two different areas of science: Chemical kinetics studies reaction rates. ... The word dynamics can refer to: a branch of mechanics; see dynamics (mechanics) the volume of music; see dynamics (music) DYNAMIC+ This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. ...

Conservation of energy-momentum

In classical mechanics, conservation of energy and momentum are handled separately. Conservation of energy is possibly the most important, and certainly the most practically useful of several conservation laws in physics. ... In physics, momentum is the product of the mass and velocity of an object. ...

In special relativity, energy and momentum are joined in the four-momentum and the stress-energy tensors. For any self-contained system or for any physical interaction, the total energy-momentum is conserved in the sense that: In special relativity, four-momentum is a four-vector that replaces classical momentum; the four-momentum of a particle is defined as the particles mass times the particles four-velocity. ... This article is in need of attention from an expert on the subject. ...

For general relativity, this relationship is modified to account for curvature, becoming In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. ...

$nabla_b , {T_a}^b = partial_b , {T_a}^b + {Gamma^b}_{cb} , {T_a}^c + {Gamma^c}_{ab} , {T_c}^b = 0$ , where

Unlike classical mechanics and special relativity, it is not usually possible to unambiguously define the total energy and momentum in general relativity, so the conservation laws are local statements only (see ADM energy, though). This often causes confusion in time-dependent spacetimes which apparently do not conserve energy, although the local law is always satisfied. In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In theoretical physics, the ADM energy (short for Arnowitt, Deser and Misner) is a special way to define the energy in general relativity which is only applicable to some special geometries of spacetime that asymptotically approach a well-defined metric tensor at infinity - for example the asymptotically Minkowski space. ...

Electromagnetism

Electromagnetism sounded the death knell for classical mechanics, since Maxwell's Equations are not Galilean invariant. This created a dilemma that was resolved by the advent of special relativity. Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... Galilean invariance is a principle which states that the fundamental laws of physics are the same in all inertial (uniform-velocity) frames of reference. ...

$partial_a,F^{,ab} = (4pi/c),J^{,b}$ and
$partial^{a},F^{,bc} + partial^{b} , F^{,ca} + partial^{c} , F^{,ab} = 0$ , where

The effect of an electromagnetic field on a charged object of mass m is then In electromagnetism, the electromagnetic tensor, or electromagnetic field tensor, F, is defined as: where Ai is the vector potential. ... In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density where c is the speed of light, ρ the charge density, and j the conventional current density. ...

In general relativity, Maxwell's equations become In special relativity, four-momentum is a four-vector that replaces classical momentum; the four-momentum of a particle is defined as the particles mass times the particles four-velocity. ...

$nabla_a,F^{,ab} = (4pi/c),J^{,b}$ and
$nabla^a,F^{,bc} + nabla^b , F^{,ca} + nabla^c , F^{,ab} = 0$ .

The equation for the effect of the electromagnetic field remains the same, although the change of metrics will modify its results.

Quantum mechanics

Unsolved problems in physics: How can the theory of quantum mechanics be merged with the theory of general relativity to produce a so-called "theory of everything"?

General relativity is incompatible with quantum mechanics; it is generally held that one of the most important unsolved problems in modern physics is the problem of obtaining a true quantum theory of gravitation. Unsuccessful attempts at obtaining such theories include supergravity, a field theory which attempted to unify general relativity with supersymmetry. At present, leading contenders which may turn out to solve this problem include M-theory and loop quantum gravity. Of these two, M-theory is significantly more ambitious in that it attempts to unify gravitation with the other known fundamental forces of Nature, whereas loop quantum gravity "merely" attempts to provide a viable quantum theory of gravitation with a well-defined classical limit which agrees with general relativity. Image File history File links Question_dropshade. ... This is an incomplete list of some of the unsolved problems in physics. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... A theory of everything (TOE) is a theory of theoretical physics and mathematics that fully explains and links together all known physical phenomena. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... In theoretical physics, a supergravity theory is a field theory combining supersymmetry and general relativity. ... Field theory (mathematics), the theory of the algebraic concept of field. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... M-theory is a solution proposed for the unknown theory of everything which would combine all five superstring theories and 11-dimensional supergravity together. ... Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. ...

Alternative theories

Well known classical theories of gravitation other than general relativity include: In theoretical physics, the current Gold Standard Theory of Gravitation is the general theory of relativity. ...

Even for "weak field" observations confined to our Solar system, various alternative theories of gravity predict quantitatively distinct deviations from Newtonian gravity. In the weak-field, slow-motion limit, it is possible to define 10 experimentally measurable parameters which completely characterize predictions of any such theory. This system of these parameters, which can be roughly thought of as describing a kind of ten dimensional "superspace" made from a certain class of classical gravitation theories, is known as PPN formalism (Parametric Post-Newtonian formalism). [3] Current bounds on the PPN parameters [4] are compatible with GR. In theoretical physics, NordstrÃ¶ms theory of gravitation was an early competitor of general relativity. ... Alfred North Whitehead, OM (February 15, 1861 â€“ December 30, 1947) was a British mathematician who evolved into a philosopher. ... Clifford Martin Will (b. ... George David Birkhoff (21 March 1884, Overisel, Michigan - 12 November 1944) was a leading American mathematician of his day. ... In mathematical physics, the Brans-Dicke theory of gravitation (sometimes called the Jordan/Brans/Dicke theory) is a well-known competitor of Einsteins theory of general relativity. ... Hermann Weyl Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ... In physics, Kaluza-Klein theory (or KK theory, for short) is a model which sought to unify the two fundamental forces of gravitation and electromagnetism. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Pascual Jordan (October 18, 1902 in Hanover - July 31, 1980 in Hamburg) was a German physicist. ... Richard Phillips Feynman (May 11, 1918 â€“ February 15, 1988) (surname pronounced FINE-man; in IPA) was one of the most influential American physicists of the 20th century, expanding greatly the theory of quantum electrodynamics, quark theory, and the physics of the superfluidity of supercooled liquid helium. ... This article is in need of attention from an expert on the subject. ... // Mathmatics In mathematics, the term torsion has several meanings, mostly unrelated to each other. ... // History Teleparallelism, also called distant parallelism, was an attempt by Einstein to unify electromagnetism and gravity. ... Nonsymmetric Gravitational Theory is a modification of Einsteins theory of General Relativity that tries to explain the mystery of Dark Matter. ... Categories: Possible copyright violations ... Self creation cosmology (SCC) theories are theories in which the universes mass is created out of its self contained gravitational and scalar fields. ... In mathematical physics, the Brans-Dicke theory of gravitation (sometimes called the Jordan/Brans/Dicke theory) is a well-known competitor of Einsteins theory of general relativity. ... Gravity Probe B (GP-B) is a satellite-based mission to measure the stress-energy tensor (the distribution, and especially the motion, of matter) in and near Earth, and thus to test related models; in application of Einsteins general theory of relativity. ... This article is in need of attention from an expert on the subject. ...

See in particular confrontation between Theory and Experiment in Gravitational Physics, a review paper by Clifford Will.

History

General relativity was developed by Einstein in a process that began in 1907 with the publication of an article on the influence of gravity and acceleration on the behavior of light in special relativity. Most of this work was done in the years 1911–1915, beginning with the publication of a second article on the effect of gravitation on light. By 1912, Einstein was actively seeking a theory in which gravitation was explained as a geometric phenomenon. In 1915, these efforts culminated in the publication of the Einstein field equations, which are a set of differential equations. // Development Early investigations The development of general relativity began in 1907 with the publication of an article by Einstein on acceleration under special relativity. ... Einsteins general theory of relativity was introduced in 1915. ... 1907 (MCMVII) was a common year starting on Tuesday (see link for calendar) of the Gregorian calendar (or a common year starting on Wednesday of the 13-day-slower Julian calendar). ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... 1911 (MCMXI) was a common year starting on Sunday (click on link for calendar). ... 1915 (MCMXV) was a common year starting on Friday (see link for calendar). ... 1912 (MCMXII) was a leap year starting on Monday in the Gregorian calendar (or a leap year starting on Tuesday in the 13-day-slower Julian calendar). ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ...

Since 1915, the development of general relativity has focused on solving the field equations for various cases. This generally means finding metrics which correspond to realistic physical scenarios. The interpretation of the solutions and their possible experimental and observational testing also constitutes a large part of research in GR. In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ...

The expansion of the universe created an interesting episode for general relativity. Starting in 1922, researchers found that cosmological solutions of the Einstein field equations call for an expanding universe. Einstein did not believe in an expanding universe, and so he added a cosmological constant to the field equations to permit the creation of static universe solutions. In 1929, Edwin Hubble found evidence that the universe is expanding. This resulted in Einstein dropping the cosmological constant, referring to it as "the biggest blunder in my career". Wikipedia does not yet have an article with this exact name. ... 1922 (MCMXXII) was a common year starting on Sunday (see link for calendar). ... The cosmological constant (usually denoted by the Greek capital letter lambda: Î›) occurs in Einsteins theory of general relativity. ... 1929 (MCMXXIX) was a common year starting on Tuesday (link will take you to calendar). ... Edwin Hubble Edwin Powell Hubble (November 20, 1889 â€“ September 28, 1953) was an American astronomer, noted for his discovery of galaxies beyond the Milky Way and the cosmological Redshift. ...

Progress in solving the field equations and understanding the solutions has been ongoing. Notable solutions have included the Schwarzschild solution (1916), the Reissner-Nordström solution and the Kerr solution. Introduction In Einsteins theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. ... 1916 (MCMXVI) is a leap year starting on Saturday (link will take you to calendar) // Events January-February January 1 - The Royal Army Medical Corps first successful blood transfusion using blood that had been stored and cooled. ... In physics and astronomy, a Reissner-NordstrÃ¸m black hole is a black hole that carries electric charge , no angular momentum, and mass . ... In general relativity, the Kerr metric describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ...

Observationally, general relativity has a history too. The perihelion precession of Mercury was the first evidence that general relativity is correct. Eddington's 1919 expedition in which he confirmed Einstein's prediction for the deflection of light by the Sun helped to cement the status of general relativity as a likely true theory. Since then, many observations have confirmed the predictions of general relativity. These include studies of binary pulsars, observations of radio signals passing the limb of the Sun, and even the GPS system. For more information, see the Tests of general relativity article. Over fifty GPS satellites such as this NAVSTAR have been launched since 1978. ... Einsteins general theory of relativity was introduced in 1915. ...

Status

The status of general relativity is decidedly mixed. On the one hand, it is a highly successful model of gravitation and cosmology which has passed every unambiguous test that it has been subjected to so far, both observationally and experimentally. It is therefore almost universally accepted by the scientific community.

On the other hand, general relativity is inconsistent with quantum mechanics, and the singularities of black holes also raise some disconcerting issues. So at the same time as it is accepted, there is also a sense that there may well be something beyond Einstein's theory still yet to be found.

Currently, better tests of general relativity are needed. Even the most recent binary pulsar discoveries only test general relativity to the first order of deviation from Newtonian projections in the post-Newtonian parameterizations. Some way of testing second and higher order terms is needed, and may shed light on how reality differs from Einstein's theory (if it does).

Quotes

John Archibald Wheeler (born July 9, 1911) is an American theoretical physicist. ... Max Born Max Born (born December 11, 1882 in Breslau, died January 5, 1970 in GÃ¶ttingen) was a German mathematician and physicist of Jewish heritage. ...

Notes

Contents

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