theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

, general covariance, also known as

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

covariance or general invariance, consists of the invariance of the ''form'' of

physical law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...

s under arbitrary differentiable coordinate transformations. 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. While this concept is exhibited by

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 ...

, which describes the dynamics 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 ...

, one should not expect it to hold in less fundamental theories. For matter fields taken to exist independently of the background, it is almost never the case that their equations of motion will take the same form in curved space that they do in flat space.

Overview

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 fields. The classical (non-

quantum In physics, a quantum (: quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a property can be "quantized" is referred to as "the hypothesis of quantization". This me ...

) theory of electrodynamics is one theory that has such a formulation.

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 ...

proposed this principle for his special theory of relativity; however, that theory was limited to

coordinate systems related to each other by uniform '' inertial'' motion, meaning relative motion in any straight line without acceleration.Extract of page 367
/ref> Einstein recognized that the general principle of relativity should also apply to accelerated relative motions, and he used the newly developed tool of tensor calculus to extend the special theory's global Lorentz covariance (applying only to inertial frames) to the more general local Lorentz covariance (which applies to all frames), eventually producing his general theory of relativity. The local reduction of the

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...

to the

Minkowski metric In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model ...

tensor corresponds to free-falling (

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 ...

) motion, in this theory, thus encompassing the phenomenon 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 ...

. Much of the work on classical unified field theories 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 spacetime 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 In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

GL₄(R) is a fundamental "external"

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

of the world. Other symmetries, including "internal" symmetries based on compact groups, now play a major role in fundamental physical theories.

Notes

References

* See ''section 7.1''.

External links

*{{cite journal , last = Norton , first = J.D. , title = General covariance and the foundations of general relativity: eight decades of dispute , journal = Reports on Progress in Physics , volume = 56 , pages = 791–858 , publisher = IOP Publishing , year = 1993 , issue = 7 , url = http://www.pitt.edu/~jdnorton/papers/decades.pdf , bibcode = 1993RPPh...56..791N , doi = 10.1088/0034-4885/56/7/001 , s2cid = 250902085 , access-date = 2018-10-17 , archive-url = https://web.archive.org/web/20171124074404/http://www.pitt.edu/~jdnorton/papers/decades.pdf , archive-date = 2017-11-24 , url-status = bot: unknown ("archive" version is re-typset, 460 kbytes) General relativity Differential geometry Diffeomorphisms

Overview

Remarks

See also

Notes

References

External links