General covariance

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In theoretical physics, general covariance (also known as diffeomorphism covariance) is the invariance of physical laws under arbitrary coordinate transformations.

In the context of Einstein's general theory of relativity, a physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems, and is implemented by expressing the equations of physics in terms of tensor fields. All known fundamental physical theories, such as electrodynamics, have a generally covariant formulation.

Armed with general covariance Einstein was then able to extend the global Lorentz covariance in Special Relativity (which applies to all inertial frames) to a local Lorentz covariance in General Relativity (which applies to all frames, inertial and non-inertial). The Lorentzian metric can be locally reduced everywhere to the Minkowski metric under a coordinate transformation

The general principle of relativity as used in GR, is that the laws of physics should be formulated in the same way in all reference frames. In special relativity, some (inertial, i.e. non-accelerating) reference frames have a preferred status which does not apply in general relativity.

1 Remarks
2 External links
3 Reference
4 See also

[edit] Remarks

There are many erroneous statements concerning General Covariance and General Relativity. These may be summarized by quotes from a standard doctorate level textbook on General Relativity, "Gravitation" by Charles W. Misner (University of Maryland), Kip S. Thorne (Cal Tech) and John Wheeler (Princeton).

Page 431, Section 17.7

Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, co-ordinate 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.'

Page 302, sec 12.5

The principle of covariance has no forcible content.' 'Any physical theory originally written in a special coordinate system can be recast in geometric, co-ordinate-free language. Newton theory is a good example, with its equivalent geometric and standard formulation (Box 12.4)

The above statements may be understood by the appreciation that coordinate systems and real physical frames are conceptually not the same. A coordinate system is a virtual mathematical construct that by its very nature can be tautologically used to describe any specific point with reference to any system of coordinates. For example, it is clear that the fender of a car is independent of the coordinate system used to describe its shape, even if that coordinate system is rotating and the car (frame) is stationary. However, it is not clear, prior, whether or not the physics of a physically rotating frame (car) is independent of such physical rotation. A physical frame does what it does, irrespective of any coordinates systems used to describe it, but this is not covariance.