General covariance
In
Overview
A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems,
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:[3]
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 general covariance is that the Lie group GL4(R) is a fundamental "external" symmetry of the world. Other symmetries, including "internal" symmetries based on compact groups, now play a major role in fundamental physical theories.
See also
- Coordinate conditions
- Coordinate-free
- Background independence
- Differential geometry
- Diffeomorphism
- 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
Notes
- ^ More precisely, only coordinate systems related through sufficiently differentiable transformations are considered.
- ISBN 0-7167-0344-0.
References
- Ohanian, Hans C.; Ruffini, Remo (1994). Gravitation and Spacetime (2nd ed.). New York: ISBN 0-393-96501-5. See section 7.1.
External links
- Norton, J.D. (1993). "General covariance and the foundations of general relativity: eight decades of dispute" (PDF). S2CID 250902085. Archived from the original on 2017-11-24. Retrieved 2018-10-17.) ("archive" version is re-typset, 460 kbytes)
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