Covariant derivative

From Academic Kids

In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold.

There is no real difference between the covariant derivative and the connection concept except for the style in which they are introduced.

In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.

Here we give a traditional index-notation introduction to the covariant derivative (also known as the tensor derivative) of a vector with respect to a vector field; the covariant derivative of a tensor is an extension of the same concept.

Everywhere in this article we use Einstein notation. It is assumed that the reader is familiar with concept of a differentiable manifold and in particular with tangent vectors.

Contents

General concept

The covariant derivative <math>\nabla<math> (also written as D) of a vector u in the direction of the vector v is a rule that defines a third vector called <math>\nabla_{\mathbf v} {\mathbf u}<math> (also Dvu) which has the properties of a derivative, specified below. A vector is a geometrical object and independent of a chosen basis (coordinate system). Upon fixing a coordinate system, this derivative transforms under a change of coordinates "in the same way" as the vector itself (covariant transformation), hence the name.

In the case of Euclidean space with an orthonormal coordinate system, one tends to define the derivative of a vector field in terms of the difference between two vectors at two nearby points. In such a system one translates one of the vectors to the origin of the other, keeping it parallel. The obtained covariant derivative on Euclidean space can simply be obtained by taking the derivative of the components.

In the general case, however, one must take into account the change of the coordinate system. In a curved space, such as the surface of the Earth (regarded as a sphere), the translation is not well defined and its analog, parallel transport, depends on the path along which the vector is translated. For example, in polar coordinates in a two dimensional Euclidean plane, the derivative contains extra terms that describe how the coordinate grid itself "rotates". In other cases the extra terms describe how the coordinate grid expands, contracts, twists, interweaves, etc.

Missing image
Path_in_polarcoordinates.png
curve in polar coordinates

Here is an example of a curve in polar coordinates in a 2-dim Euclidean space. A vector at curve parameter t (say the acceleration, not shown) is expressed in a coordinate system <math>({\mathbf e}_r, {\mathbf e}_{\theta})<math>, where <math>{\mathbf e}_r<math> and <math>{\mathbf e}_{\theta}<math> are unit tangent vectors for the polar coordinates, serving as a basis to decompose a vector in terms of radial and tangential components. At a slightly later time, the new basis in polar coordinates appears slightly rotated with respect to the first set. The covariant derivative of the basis vectors (the Christoffel symbols) serve to express this change.

(It is probably better not to think of t as a time parameter, at least for applications in general relativity. It is simply an arbitrary parameter varying smoothly and monotonically along the path.)

Missing image
Parallel_transport_on_globe.png
parallel transport of vector on a globe

Another example: A vector e on a globe on the equator in Q is directed to the north. Suppose we parallel transport the vector first along the equator until P and then (keeping it parallel to itself) drag it along a meridian to the pole N and (keeping the direction there) subsequently transport it along another meridian back to Q. Then we notice that the parallel-transported vector along a closed circuit does not return as the same vector; instead, it has another orientation. This would not happen in Euclidean space and is caused by the curvature of the surface of the globe. The same effect can be noticed if we drag the vector along an infinitesimally small closed surface subsequently along two directions and then back. The infinitesimal change of the vector is a measure of the curvature.


Notes

The vectors u and v in the definition are defined at the same point p. Also the covariant derivative <math>\nabla_{\mathbf v}{\mathbf u} <math> is a vector defined at p.

The definition of the covariant derivative does not use the metric in space. However, a given metric uniquely defines a special covariant derivative called the Levi-Civita connection.

The properties of a derivative imply that <math>\nabla_{\mathbf v} {\mathbf u}<math> depends on the surrounding of point p in the same way as e.g. the derivative of a scalar function along a curve in a given point p depends on the surroundings of p. Therefore, the covariant derivative is not a tensor.

The information on the surroundings of a point p in the covariant derivative can be used to define parallel transport of a vector. Also the curvature, torsion and geodesics can be defined only in terms of the covariant derivative.

Occasionally the term "covariant derivative" refers to a derivative of sections of a general vector bundle along a tangent vector of the base; see subsection "Vector bundles" in "Connection form".

Formal definition

Functions

Given a function <math>f<math>, the covariant derivative <math>\nabla_{\mathbf v}f<math> coincides with the normal differentiation of a real function in the direction of the vector v, usually denoted by <math>{\mathbf v}f<math> and by <math>df({\mathbf v})<math>.

Vector fields

A covariant derivative <math>\nabla<math> of a vector field <math>{\mathbf u}<math> in the direction of the vector <math>{\mathbf v} <math> denoted <math>\nabla_{\mathbf v} {\mathbf u}<math> is defined by the following properties for any vector fields u, v, w and scalar functions f and g:

  1. <math>\nabla_{\mathbf v} {\mathbf u}<math> is algebraically linear in <math>{\mathbf v}<math> so <math>\nabla_{f{\mathbf v}+g{\mathbf w}} {\mathbf u}=f\nabla_{\mathbf v} {\mathbf u}+g\nabla_{\mathbf w} {\mathbf u}<math>
  2. <math>\nabla_{\mathbf v} {\mathbf u}<math> is additive in <math>{\mathbf u}<math> so <math>\nabla_{\mathbf v}({\mathbf u}+{\mathbf w})=\nabla_{\mathbf v} {\mathbf u}+\nabla_{\mathbf v} {\mathbf w}<math>
  3. <math>\nabla_{\mathbf v} {\mathbf u}<math> obeys the product rule, i.e. <math>\nabla_{\mathbf v} f{\mathbf u}=f\nabla_{\mathbf v} {\mathbf u}+{\mathbf u}\nabla_{\mathbf v}f<math> where <math>\nabla_{\mathbf v}f<math> is defined above.

Note that <math>\nabla_{\mathbf v} {\mathbf u}<math> at point p depends on the value of v at p and on values of u in a neighbourhood of p because of the last property, the product rule. That means that the covariant derivative is not a tensor.

Covector fields

Given a field of covectors (or 1-form) <math>\alpha<math>, its covariant derivative <math>\nabla_{\mathbf v}\alpha<math> can be defined using the following identity which is satisfied for all vector fields u

<math>\nabla_{\mathbf v}(\alpha({\mathbf u}))=(\nabla_{\mathbf v}\alpha)({\mathbf u})+\alpha(\nabla_{\mathbf v}{\mathbf u}).<math>

The covariant derivative of a covector field along a vector field v is again a covector field.

Tensor fields

Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary tensor fields using the following identities where <math>\varphi<math> and <math>\psi<math> are any two tensors:

<math>\nabla_{\mathbf v}(\varphi\otimes\psi)=(\nabla_{\mathbf v}\varphi)\otimes\psi+\varphi\otimes(\nabla_{\mathbf v}\psi),<math>

and if <math>\varphi<math> and <math>\psi<math> are tensor fields of the same tensor bundle then

<math>\nabla_{\mathbf v}(\varphi+\psi)=\nabla_{\mathbf v}\varphi+\nabla_{\mathbf v}\psi.<math>

The covariant derivative of a tensor field along a vector field v is again a tensor field of the same type.

Coordinate description

Given coordinate functions <math>x^i,\ i=0,1,2,...<math>, any tangent vector can be described by its components in the basis <math>e_i={\partial\over\partial x^i}<math>. The covariant derivative is a vector and so can be expressed as a sum over all basis vectors as a linear combination Γkek, where Γk are the components (see Einstein notation). To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field ej along ei.

<math> \nabla_{{\mathbf e}_i} {\mathbf e}_j = \Gamma^k_{i j} {\mathbf e}_k,<math>

the coefficients Γki j are called Christoffel symbols. Then using the rules in the definition, we find that for general vector fields <math>{\mathbf v}= v^ie_i<math> and <math>{\mathbf u}= u^ie_i<math> we get

<math> \nabla_{\mathbf v} {\mathbf u} = (v^i u^j \Gamma^k_{i j}+v^i{\partial u^k\over\partial x^i}){\mathbf e}_k,<math>

the first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field u. In particular

<math>\nabla_{{\mathbf e}_j} {\mathbf u}=\nabla_j {\mathbf u} = \left( \frac{\partial u^i}{\partial x^j} + u^k \Gamma^i_{jk} \right) {\mathbf e}_i <math>

In words: the covariant derivative is the normal derivative along the coordinates along with correction terms which tell how the coordinates change. In textbooks on physics, the covariant derivative is sometimes simply stated in terms of its components in this equation.

Often a notation is used in which the covariant derivative is given with a semicolon, while a normal derivative is indicated by a comma. In this notation we write the same as:

<math>
         \nabla_j {\mathbf v} \equiv v^i_{;\;j} \;\;\;\;\;\;
         v^i_{;\;j}  = 
         v^i_{,\;j} + v^k\Gamma^i_{k \;j}

<math> Once again this shows that the covariant derivative of a vector field is not just simply obtained by differentiating to the coordinates <math> v^i_{,\;j}<math>, but also depends on the vector v itself through <math> v^k\Gamma^i_{k \;j}<math>.

See also:

  1. Connection
  2. Connection form
  3. Levi-Civita connection
  4. Christoffel symbolsde:Kovariante Ableitung
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