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 In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

coordinate transformation In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...

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. 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 In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...

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 field In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...

s. 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 In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...

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 In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 paper, "On the Electrodynamics of Moving Bodies", the theory is presen ...

; 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 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. For example, in the framework of special relativity, the Maxwell equations h ...

should also apply to accelerated relative motions, and he used the newly developed tool of

tensor calculus In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...

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 General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physi ...

. 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 Since the 19th century, some physicists, notably Albert Einstein, have attempted to develop a single theoretical framework that can account for all the fundamental forces of nature – a unified field theory. Classical unified field theories are at ...

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 ''Reports on Progress in Physics'' is a peer reviewed journal published by IOP Publishing. The editor-in-chief as of 2022 is Subir Sachdev (Harvard University Harvard University is a Private university, private Ivy League research universi ...

, volume = 56 , pages = 791–858 , publisher =

IOP Publishing IOP Publishing (previously Institute of Physics Publishing) is the publishing company of the Institute of Physics. It provides publications through which scientific research is distributed worldwide, including journals, community websites, maga ...

, 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