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The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation. Conventionally, indices identifying the basis vectors are placed as lower indices and so are all entities that transform in the same way.
Covariant and contravariant components of a vector when the basis is not orthogonal. The general formulation of covariance and contravariance refers to how the components of a coordinate vector transform under a change of basis (passive transformation).
The covariant derivative is required to transform, under a change in coordinates, by a covariant transformation in the same way as a basis does (hence the name). In the case of Euclidean space , one usually defines the directional derivative of a vector field in terms of the difference between two vectors at two nearby points.
The transformations between frames are all arbitrary (invertible and differentiable) coordinate transformations. The covariant quantities are scalar fields, vector fields, tensor fields etc., defined on spacetime considered as a manifold. Main example of covariant equation is the Einstein field equations.
From the physical viewpoint, general covariant transformations are treated as particular reference frame transformations in general relativity. In mathematics, general covariant transformations are defined as particular automorphisms of so-called natural fiber bundles.
The covariant derivative of a vector field with components is given by: ; = = + and ... Note that this transformation formula is for the mean curvature vector, ...
Both covariant and contravariant four-vectors can be Lorentz covariant quantities. Local Lorentz covariance , which follows from general relativity , refers to Lorentz covariance applying only locally in an infinitesimal region of spacetime at every point.
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.