Editing General covariance

{{Short description|Principle stating that physical laws are the same in all coordinate systems}}
In [[theoretical physics]], '''general covariance''', also known as '''[[diffeomorphism]] covariance''' or '''general invariance''', consists of the [[Invariant (physics)|invariance]] of the ''form'' of [[physical law]]s under arbitrary [[Derivative|differentiable]] [[coordinate transformation]]s. The essential idea is that coordinates do not exist ''a&nbsp;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,<ref>More precisely, only coordinate systems related through sufficiently differentiable transformations are considered.</ref> and is usually expressed in terms of [[tensor field]]s. The classical (non-[[Quantum mechanics|quantum]]) theory of [[electrodynamics]] is one theory that has such a formulation.

[[Albert Einstein]] proposed this principle for his [[special relativity|special theory of relativity]]; however, that theory was limited to [[spacetime]] coordinate systems related to each other by uniform ''[[inertial frame of reference|inertial]]'' motion, meaning relative motion in any straight line without acceleration.<ref>{{cite book |title=The Formative Years of Relativity: The History and Meaning of Einstein's Princeton Lectures |edition=illustrated |first1=Hanoch |last1=Gutfreund |first2=Jürgen |last2=Renn |publisher=Princeton University Press |year=2017 |isbn=978-1-4008-8868-9 |page=376 |url=https://books.google.com/books?id=VYi9DgAAQBAJ}} [https://books.google.com/books?id=VYi9DgAAQBAJ&pg=PA367 Extract of page 367]</ref> Einstein recognized that the [[principle of relativity#General principle of relativity|general principle of relativity]] should also apply to accelerated relative motions, and he used the newly developed tool of [[Tensor field#Tensor calculus|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 relativity|general theory of relativity]]. The local reduction of the [[metric tensor]] to the [[Minkowski space|Minkowski metric]] tensor corresponds to free-falling ([[Geodesics in general relativity|geodesic]]) motion, in this theory, thus encompassing the phenomenon of [[gravitation]].

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:<ref name="GravitationP431">{{cite book | title=[[Gravitation (book)|Gravitation]] | author-link1=Charles W. Misner|author1=Charles W. Misner|author-link2=Kip S. Thorne|author2=Kip S. Thorne|author-link3=John Archibald Wheeler|author3=John Archibald Wheeler | year=1973 | publisher=Freeman | isbn=0-7167-0344-0 | page=431}}</ref>

{{Quote|Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics. Einstein described both demands by a single phrase, "general covariance". The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion.}}

A more modern interpretation of the physical content of the original [[principle of covariance|principle of general covariance]] is that the [[Lie group]] GL<sub>4</sub>('''R''') is a fundamental "external" [[symmetry]] of the world.  Other symmetries, including "internal" symmetries based on compact [[Group (mathematics)|groups]], now play a major role in fundamental physical theories.

== See also ==

{{div col|colwidth=30em}}
* [[Coordinate conditions]]
* [[Coordinate-free]]
* [[Background independence]]
* [[Differential geometry]]
* [[Diffeomorphism]]
* [[Covariance and contravariance of vectors|Covariance and contravariance]]
* [[Covariant derivative]]
* [[Fictitious force]]
* [[Galilean invariance]]
* [[Gauge covariant derivative]]
* [[General covariant transformations]]
* [[Harmonic coordinate condition]]
* [[Inertial frame of reference]]
* [[Lorentz covariance]]
* [[Principle of covariance]]
* [[Special relativity]]
* [[Symmetry in physics]]
{{div col end}}

== Notes ==
{{reflist}}

== References ==
* {{cite book |author1=Ohanian, Hans C. |author2=Ruffini, Remo | title=Gravitation and Spacetime | edition=2nd | location=New York | publisher=[[W. W. Norton]] | year=1994 | isbn=0-393-96501-5}} 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)

[[Category:General relativity]]
[[Category:Differential geometry]]
[[Category:Diffeomorphisms]]