General covariance

Source: Wikipedia, the free encyclopedia.

In

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,

electrodynamics
is one theory that has such a formulation.

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:[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

Notes

  1. ^ More precisely, only coordinate systems related through sufficiently differentiable transformations are considered.
  2. ISBN 978-1-4008-8868-9. Extract of page 367
  3. ISBN 0-7167-0344-0
    .

References

External links
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