General covariance

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In theoretical physics, general covariance (also known as diffeomorphism covariance) is the invariance of the form of physical laws under arbitrary coordinate transformations. More precisely, this means that physical laws take the same mathematical form in all coordinate systems whose metric can be locally reduced everywhere to the Minkowski form ( Minkowski metric) under a coordinate transformation. The principle of general covariance was formulated by Einstein who wanted to extend the Lorentz covariance in Special Relativity to non-inertial frames. All known physical theories such as electrodynamics must necessarily have a generally covariant formulation.

The general principle of relativity as used in Einstein's general theory of relativity is that the laws of physics must take the same form in all reference frames. This is an extension of the special principle of relativity.

The “general principle” was defined by Ernst Mach as the principle that all forms of motion between bodies can be said to be purely relative.

Modern textbooks tend to define the general principle as being the principle that all relationships between frames of reference can be said to be purely relative.

This change in emphasis from physical “bodies” to more abstract “frames” allows a simpler derivation of a general theory of relativity, avoiding some of the complicating factors that are expected to arise at small scales with real bodies (e.g. quantum mechanics).

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