Covariant transformation

Covariance in physics has two different, but closely related meanings. Firstly, there is the covariance of theories and their underlying equations, on the other hand there are in the tensor calculus the distinction between covariant and contravariant vectorial quantities.

A theory or equation is covariant if the form of the equations remains invariant under a group of simultaneous, coordinated transformations of all variables involved. Links the equation vector quantities, and is therefore a system of equations, then there is the invariance of the equation until after a transformation of both sides of the equation system.

So, for example, the acceleration and the power to transform into the Newtonian equations of motion in the same direction as the position vectors under Galilean transformations. Therefore, the Newtonian equations of motion and thus the classical mechanics are covariant with respect to the group of Galilean transformations. In the same sense, the Einstein equations of gravitation in general relativity are covariant under arbitrary ( non-linear smooth ) Coordinate transformations and the Dirac equation of quantum electrodynamics, covariant under the group of linear Lorentz transformations.

The left side of the Klein-Gordon equation for a scalar field changes under Lorentz transformations do not, it is a special invariant or scalar.

In tensor, the contravariant shares of a tensor as the tuple of coordinates of a position vector and the covariant as the coordinates of a linear form transform. As a result, co- and contra-variant sizes after transformation exactly zero when they were before the transformation zero.

For notation: the coordinates of contravariant vectors transform as the coordinates of the position vector linear, are written with upper indices, the coordinates of covariant vectors ( or linear forms ) with lower indices. After application of the Einstein summation convention each term of an equation must have the same index position.

Co-and contra- variant

In a narrower sense of the word denotes covariant in mathematical physics quantities which so transform as differential forms. This covariant variables form a vector space on which acts a group of linear transformations.

The set of linear transformations of covariant quantities in the real numbers

Is the dual to the vector space. If we write the transformed covariant variables with a matrix as

Then defines the contravariant or contragredient transformation law of the dual space

Because of

Satisfies the contravariant transformation linking the same group as the covariant transformation.

Tensors from the times tensor product of contravariant with the times tensor product of hot -fold covariant and times.

In index notation to do at index position with below and above indices clear whether it is the components of a covariant or a contravariant vector,

That is true, show the calculation steps

Pull Index

If the contravariant transformation law of the covariant equivalent and applies to all of the transformation group

With an invertible symmetric matrix, then it is in the transformation group for

To a subgroup of the orthogonal group, which leaves the symmetric bilinear invariant. Then defines a contravariant vector, if a covariant vector. In index notation to write for the components of an abbreviation

Then vice versa

This context, the components of the covariant vector and the contravariant vector is called index drawing or even raise or lower.

If the contravariant transformation law of the covariant equivalent and applies to all of the transformation group

With an invertible, antisymmetric matrix, then it is in the transformation group for

To a subgroup of the symplectic group, which leaves the antisymmetric bilinear invariant. Then defines a contravariant vector, if a covariant vector. In index notation can be abbreviated for the components of

. Write Then vice versa

This relationship between the components of the covariant vector and the contravariant vector defines the index drawing of vectors which transform under the symplectic group.