In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the

invariance Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iterat ...

of the ''form'' of physical laws 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, which describes the dynamics of spacetime, 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 (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...

) theory of electrodynamics is one theory that has such a formulation. Albert Einstein proposed this principle for his special theory of relativity; however, that theory was limited to spacetime coordinate systems related to each other by uniform '' inertial'' motion.Extract of page 367
/ref> Einstein recognized that the

general principle of relativity In physics, the principle of relativity is the requirement that the equations describing the physical law, laws of physics have the same form in all admissible frames of reference. For example, in the framework of special relativity the Maxwell ...

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 tensor corresponds to free-falling (

geodesic In geometry, a geodesic () is a curve representing in some sense the 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 connection. ...

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

gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...

. 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 In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the ''form'' of physical laws under arbitrary differentiable coordinate transformations. The essential idea is ...

is that the

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

GL₄(R) is a fundamental "external"

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

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= http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf , archive-date= 2002-10-18 , url-status=live ("archive" version is re-typset, 460 kbytes) General relativity Differential geometry Diffeomorphisms

Overview

Remarks

See also

Notes

References

External links