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General covariance
 
 
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In theoretical physics
Theoretical physics

Theoretical physics employs mathematical models and abstractions of physics in an attempt to explain experimental data taken of the natural world....
, general covariance (also known as diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
 covariance or general invariance) is the invariance
Invariance

Invariance is a French journal edited by Jacques Camatte, published since 1968.It emerged from the Italian left-communist tradition associated with Amadeo Bordiga and it originally bore the subtitle "Invariance of the theory of the proletariat", indicating Bordiga's notion of the unchanging nature of communist theory....
 of the form of physical law
Physical law

A physical law or scientific law is a scientific generalization based on empiricism observations of physical behavior . Laws of nature are observable....
s under arbitrary differentiable
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
 coordinate transformation
Coordinate transformation

See:*Coordinate system#Transformations*List of canonical coordinate transformations*Coordinate rotation*Covariance and contravariance*Covariant transformation...
s. The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.

A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems, and is usually expressed in terms of tensor field
Tensor field

In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in general relativity, in the analysis of stress and strain tensor in materials, and in numerous applications in the physical sciences and en...
s.






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In theoretical physics
Theoretical physics

Theoretical physics employs mathematical models and abstractions of physics in an attempt to explain experimental data taken of the natural world....
, general covariance (also known as diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
 covariance or general invariance) is the invariance
Invariance

Invariance is a French journal edited by Jacques Camatte, published since 1968.It emerged from the Italian left-communist tradition associated with Amadeo Bordiga and it originally bore the subtitle "Invariance of the theory of the proletariat", indicating Bordiga's notion of the unchanging nature of communist theory....
 of the form of physical law
Physical law

A physical law or scientific law is a scientific generalization based on empiricism observations of physical behavior . Laws of nature are observable....
s under arbitrary differentiable
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
 coordinate transformation
Coordinate transformation

See:*Coordinate system#Transformations*List of canonical coordinate transformations*Coordinate rotation*Covariance and contravariance*Covariant transformation...
s. The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.

A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems, and is usually expressed in terms of tensor field
Tensor field

In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in general relativity, in the analysis of stress and strain tensor in materials, and in numerous applications in the physical sciences and en...
s. The classical (non-quantum
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
) theory of electrodynamics is one theory that has such a formulation.

Albert Einstein
Albert Einstein

Albert Einstein was a Germany-born theoretical physics. He is best known for his theory of relativity and specifically mass?energy equivalence, expressed by the equation E = mc2....
 proposed this principle for his special theory of relativity
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
; however, that theory was limited to space-time
Spacetime

In physics, spacetime is any mathematical model that combines space and Time in physics into a single continuum . Spacetime is usually interpreted with space being Three-dimensional space and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions....
 coordinate systems related to each other by uniform relative motions only, the so-called "inertial frames
Inertial frame of reference

In physics, an inertial frame of reference is a frame of reference, tied to the state of motion of an Observer , with the property that each physical law portrays itself in the same form in every inertial frame....
." Einstein recognized that the general principle of relativity
Principle of relativity

In physics, the principle of relativity is the requirement that the equations, describing the laws of physics, have the same form in all admissible frames of reference....
 should also apply to accelerated relative motions, and he used the newly developed tool of tensor calculus
Tensor field

In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in general relativity, in the analysis of stress and strain tensor in materials, and in numerous applications in the physical sciences and en...
 to extend the special theory's global Lorentz covariance
Global Lorentz covariance

A global Lorentz covariance or global Lorentz symmetry is a global symmetry and a Lorentz symmetry of space-time. Any theory which exhibits a global Lorentz symmetry is said to be globally Lorentzian....
 (applying only to inertial frames) to the more general local Lorentz covariance
Local Lorentz covariance

A local Lorentz covariance or local Lorentz symmetry is a local symmetry of space-time which reduces Principle of locality to a Lorentz symmetry....
 (which applies to all frames), eventually producing his general theory of relativity
General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
. The local reduction of the general metric tensor
Metric tensor

In the mathematics field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of Vector in Euclidean space....
 to the Minkowski metric
Minkowski space

In physics and mathematics, Minkowski space is the mathematical setting in which Albert Einstein theory of special relativity is most conveniently formulated....
 corresponds to free-falling (geodesic
Geodesic

In mathematics, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "Line " to "manifolds".In presence of a Metric , geodesics are defined to be the shortest path between points on the space....
) motion, in this theory, thus encompassing the phenomenon of gravitation
Gravitation

Gravitation is a natural phenomenon that gives weight to objects. In everyday life, attraction due to gravity is the result of the presence of relatively large bodies, such as the Earth and the Moon....
.

Much of the work on classical unified field theories
Classical unified field theories

Since the 1800s, some physicists have attempted to develop a single theoretical framework that can account for the fundamental forces of nature ? a unified field theory....
 consisted of attempts to further extend the general theory of relativity to interpret additional physical phenomena, particularly electromagnetism, within the framework of general covariance, and more specifically as purely geometric objects in the space-time continuum.

Remarks


The relationship between general covariance and general relativity may be summarized by quoting a standard textbook:

A more modern interpretation of the physical content of the original principle of general covariance is that the Lie group
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
 GL4(R) is a fundamental "external" symmetry
Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
 of the world. Other symmetries, including "internal" symmetries based on discrete groups
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
, now play a major role in fundamental physical theories.

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See also

  • Galilean invariance
    Galilean invariance

    Galilean invariance or Galilean relativity is a principle of relativity which states that the fundamental physical law are the same in all inertial frames....
  • Lorentz covariance
    Lorentz covariance

    In standard physics, Lorentz covariance is a key property of spacetime that follows from the special theory of relativity, where it applies globally....
  • Coordinate conditions
    Coordinate conditions

    In general relativity, the laws of physics can be expressed in a general covariance form. In other words, the real world does not care about our coordinate systems....
  • Covariance and contravariance
    Covariance and contravariance

    DefinitionIn mathematics and theoretical physics, covariance and contravariance refer to how coordinates change under a change of basis ....
  • Prior geometry
  • List of mathematical topics in relativity
    List of mathematical topics in relativity

    This is a list of mathematical topics in relativity , by Wikipedia page....
  • Harmonic coordinate condition
    Harmonic coordinate condition

    In general relativity, a harmonic coordinate xα is one which satisfies the Wave equation when regarded as a scalar field. Solutions of Laplace's equation are called harmonic functions....
  • Diffeomorphism
    Diffeomorphism

    In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
  • Coordinate-free approach
  • Inertial frame of reference
    Inertial frame of reference

    In physics, an inertial frame of reference is a frame of reference, tied to the state of motion of an Observer , with the property that each physical law portrays itself in the same form in every inertial frame....
  • Fictitious force
    Fictitious force

    A fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses in a non-inertial reference frame, such as a rotating reference frame....
  • Special relativity
    Special relativity

    Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
  • Symmetry in physics
    Symmetry in physics

    Symmetry in physics includes all features of a physical system that exhibit the property of symmetry?that is, under certain transformation , aspects of these systems are "unchanged", according to a particular observation....