Editing General covariance (section)

==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.