Covariant transformation

See also Covariance and contravariance of vectors

In physics, a covariant transformation is a rule (specified below), that describes how certain physical entities change under a change of coordinate system. In particular the term is used for vectors and tensors. The transformation that describes the new basis vectors in terms of the old basis, 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. The inverse of a covariant transformation is a contravariant transformation. In order that a vector should be invariant under a coordinate transformation, its components must transform according to the contravariant rule. Conventionally, indices identifying the components of a vector are placed as upper indices and so are all indices of entities that transform in the same way. The summation over all indices of a product with the same lower and upper indices are invariant to a transformation.

A vector itself is a geometrical quantity, in principle, independent (invariant) of the chosen coordinate system. A vector v is given, say, in components vⁱ on a chosen basis e_i, related to a coordinate system xⁱ (the basis vectors are tangent vectors to the coordinate grid). On another basis, say ${\mathbf e}'_i$, related to a new coordinate system $x'\;^i$, the same vector v has different components $v'\;^i$ and

${\mathbf v} = \sum_i v^i {\mathbf e}_i = \sum_i {v'\;}^i {\mathbf e}'_i$

(in the so called Einstein notation the summation sign is often omitted, implying summation over the same upper and lower indices occurring in a product). With v as invariant and the ${\mathbf e}_i$ transforming covariant, it must be that the vⁱ (the set of numbers identifying the components) transform in a different way, the inverse called the contravariant transformation rule.

If, for example in a 2-dim Euclidean space, the new basis vectors are rotated anti-clockwise with respect to the old basis vectors, then it will appear in terms of the new system that the componentwise representation of the vector look as if the vector was rotated in the opposite direction, i.e. clockwise (see figure).

---

• 

  A vector v, and local tangent basis vectors {e_x,e_y} and {e_r,e_φ}.

• 

  Coordinate representations of v.

A vector v is described in a given coordinate grid (black lines) on a basis which are the tangent vectors to the (here rectangular) coordinate grid. The basis vectors are e_x and e_y. In another coordinate system (dashed and red), the new basis vectors are tangent vectors in the radial direction and perpendicular to it. These basis vectors are indicated in red as e_r and e_φ. They appear rotated anticlockwise with respect to the first basis. The covariant transformation here is thus an anticlockwise rotation.

If we view the vector v with e_φ pointed upwards, its representation in this frame appears rotated to the right. The contravariant transformation is a clockwise rotation.
.

---

Examples of covariant transformation

The derivative of a function transforms covariantly

The explicit form of a covariant transformation is best introduced with the transformation properties of the derivative of a function. Consider a scalar function f (like the temperature in a space) defined on a set of points p, identifiable in a given coordinate system $x^i,\; i=0,1,\dots$ (such a collection is called a manifold). If we adopt a new coordinates system ${x'\,}^j, j=0,1,\dots$ then for each i, the original coordinate xⁱ can be expressed as function of the new system, so ${x}^i({x'\,}^j), j=0,1,\dots$ One can express the derivative of f in new coordinates in terms of the old coordinates, using the chain rule of the derivative, as

$\frac{\partial f\;}{\partial {x'\,}^i} = \sum_j \frac{\partial f\;}{\partial {x}^j} \; \frac{\partial {x}^j\;}{\partial {x'\,}^i}$

This is the explicit form of the covariant transformation rule. The notation of a normal derivative with respect to the coordinates sometimes uses a comma, as follows

$f_{,i} \ \stackrel{\mathrm{def}}{=}\ \frac{\partial f\;}{\partial x^i}$

where the index i is placed as a lower index, because of the covariant transformation.

Basis vectors transform covariantly

A vector can be expressed in terms of basis vectors. For a certain coordinate system, we can choose the vectors tangent to the coordinate grid. This basis is called the coordinate basis.

To illustrate the transformation properties, consider again the set of points p, identifiable in a given coordinate system xⁱ where $i=0,1,\dots$ (manifold). A scalar function f, that assigns a real number to every point p in this space, is a function of the coordinates $f\;(x^0,x^1,\dots)$. A curve is a one-parameter collection of points c, say with curve parameter λ, c(λ). A tangent vector v to the curve is the derivative dc / dλ along the curve with the derivative taken at the point p under consideration. Note that we can see the tangent vector v as an operator (the Directional derivative) which can be applied to a function

${\mathbf v}[f] \ \stackrel{\mathrm{def}}{=}\ \frac{df}{d\lambda}= \frac{d\;\;}{d\lambda} f(c(\lambda))$

The parallel between the tangent vector and the operator can also be worked out in coordinates

${\mathbf v}[f] = \sum_i \frac{dx^i}{d\lambda} \frac{\partial f}{\partial x^i}$

or in terms of operators $\partial/\partial x^i$

${\mathbf v} = \frac{dx^i}{d\lambda} \frac{\partial \;\;}{\partial x^i} = \frac{dx^i}{d\lambda} {\mathbf e}_i$

where we have written ${\mathbf e}_i = \partial/\partial x^i$, the tangent vectors to the curves which are simply the coordinate grid itself.

If we adopt a new coordinates system ${x'\,}^i, \;i=0,1,\dots$ then for each i, the old coordinate xⁱ can be expressed as function of the new system, so $x^i({x'\,}^j), j=0,1,\dots$ Let ${\mathbf e}'_i = {\partial\;}/{\partial {x'\,}^i}$ be the basis, tangent vectors in this new coordinates system. We can express ${\mathbf e}_i$ in the new system by applying the chain rule on x. As a function of coordinates we find the following transformation

${\mathbf e}'_i = \frac{\partial\;\;\;}{\partial {x'\,}^i} = \frac{\partial x^j\;}{\partial {x'\,}^i} \frac{\partial\;\;\;}{\partial x^j} = \frac{\partial x^j\;\;}{\partial {x'\,}^i\;} {\mathbf e}_j$

which indeed is the same as the covariant transformation for the derivative of a function.

Contravariant transformation

The components of a (tangent) vector transform in a different way, called contravariant transformation. Consider a tangent vector v and call its components vⁱ on a basis ${\mathbf e}_i$. On another basis ${\mathbf e}'\,_i$ we call the components ${v'\,}^i$, so

${\mathbf v} = \sum_i v^i {\mathbf e}_i = \sum_i {v'\,}^i {\mathbf e}'\,_i$

in which

$v^i = \frac{dx^i}{d\lambda\;} \;\mbox{ and }\; {v'\,}^i = \frac{d{x'\,}^i}{d\lambda\;\;}$

If we express the new components in terms of the old ones, then

${v'\,}^i = \frac{d{x'\,}^i}{d\lambda\;\;} = \frac{\partial {x'\,}^i}{\partial x^j} \frac{dx^j}{d\lambda\;\;} = \frac{\partial {x'\,}^i}{\partial x^j} {v}^j$

This is the explicit form of a transformation called the contravariant transformation and we note that it is different and just the inverse of the covariant rule. In order to distinguish them from the covariant (tangent) vectors, the index is placed on top.

Differential forms transform contravariantly

An example of a contravariant transformation is given by a differential form df. For f as a function of coordinates xⁱ, df can be expressed in terms of dxⁱ. The differentials dx transform according to the contravariant rule since

$d{x'\,}^i = \sum_j \frac{\partial {x'\,}^i}{\partial {x}^j\;\;} {dx}^j$

Dual properties

Entities that transform covariantly (like basis vectors) and the ones that transform contravariantly (like components of a vector and differential forms) are "almost the same" and yet they are different. They have "dual" properties. What is behind this, is mathematically known as the dual space that always goes together with a given linear vector space.

Take any vector space T. A function f on T is called linear if, for any vectors v, w and scalar α:

$f({\mathbf v}+{\mathbf w}) = f({\mathbf v}) + f({\mathbf w})$

$f(\alpha {\mathbf v}) = \alpha f({\mathbf v})$

A simple example is the function which assigns a vector the value of one of its components (called a projection function). It has a vector as argument and assigns a real number, the value of a component.

All such scalar-valued linear functions together form a vector space, called the dual space of T. One can easily see that, indeed, the sum f+g is again a linear function for linear f and g, and that the same holds for scalar multiplication αf.

Given a basis ${\mathbf e}_i$ for T, we can define a basis, called the dual basis for the dual space in a natural way by taking the set of linear functions mentioned above: the projection functions. So those functions ω that produce the number 1 when they are applied to one of the basis vector ${\mathbf e}_i$. For example ω⁰ gives a 1 on ${\mathbf e}_0$ and zero elsewhere. Applying this linear function ω⁰ to a vector ${\mathbf v} =v^i {\mathbf e}_i$, gives (using its linearity)

$\omega^0({\mathbf v}) = \omega^0(v^i {\mathbf e}_i) = v^i \omega^0({\mathbf e}_i) = v^0$

so just the value of the first coordinate. For this reason it is called the projection function.

There are as many dual basis vectors ωⁱ as there are basis vectors ${\mathbf e}_i$, so the dual space has the same dimension as the linear space itself. It is "almost the same space", except that the elements of the dual space (called dual vectors) transform covariant and the elements of the tangent vector space transform contravariantly.

Sometimes an extra notation is introduced where the real value of a linear function σ on a tangent vector u is given as

$\sigma [{\mathbf u}] := \langle \sigma, {\mathbf u}\rangle$

where $\langle\sigma, {\mathbf u}\rangle$ is a real number. This notation emphasizes the bilinear character of the form. it is linear in σ since that is a linear function and it is linear in u since that is an element of a vector space.

Co- and contravariant tensor components

Without coordinates

With the aid of the section of dual space, a tensor of type (r,s) is simply defined as a real-valued multilinear function of r dual vectors and s vectors in a point p. So a tensor is defined in a point. It is a linear machine: feed it with vectors and dual vectors and it produces a real number. Since vectors (and dual vectors) are defined independent of coordinate system, this definition of a tensor is also free of coordinates and does not depend on the choice of a coordinate system. This is the main importance of tensors in physics.

The notation of a tensor is

$T(\sigma, \ldots ,\rho, {\mathbf u}, \ldots, {\mathbf v}) \;\mbox{ or as }\; { T^{\sigma \ldots \rho} }_{ {\mathbf u} \ldots {\mathbf v}}$

for dual vectors (differential forms) ρ, σ and tangent vectors ${\mathbf u}, {\mathbf v}$. In the second notation the distinction between vectors and differential forms is more obvious.

With coordinates

Because a tensor depends linearly on its arguments, it is completely determined if one knows the values on a basis $\omega^i \ldots \omega^j$ and ${\mathbf e}_k \ldots {\mathbf e}_l$

$T(\omega^i,\ldots,\omega^j, {\mathbf e}_k \ldots {\mathbf e}_l) = {T^{i\ldots j}}_{k\ldots l}$

The numbers ${T^{i\ldots j}}_{k\ldots l}$ are called the components of the tensor on the chosen basis.

If we choose another basis (which are a linear combination of the original basis), we can use the linear properties of the tensor and we will find that the tensor components in the upper indices transform as dual vectors (so contravariant), whereas the lower indices will transform as the basis of tangent vectors and are thus covariant. For a tensor of rank 2, we can easily verify that

$A'_{i j} = \frac{\partial x^l}{\partial {x'\,}^i} \frac{\partial x^m}{\partial {x'\,}^j} A_{l m}$ covariant tensor

${A'\,}^{i j} = \frac{\partial {x'\,}^i}{\partial x^l} \frac{\partial {x'\,}^j}{\partial x^m} A^{l m}$ contravariant tensor

For a mixed co- and contravariant tensor of rank 2

${A'\,}^i_j= \frac {\partial {x'\,}^i} {\partial x^l} \frac {\partial x^m} {\partial {x'\,}^j} A^l_m$ mixed co- and contravariant tensor