Covariant derivative

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 style in which they are introduced.

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, the reader supposed to be familiar with concept of differentiable manifold (in particular tangent vector).

General concept

The covariant derivative D (also written as ∇) of a vector u in the direction of the vector v is a rule that defines a third vector called ${\mathbf {D} }_{\mathbf {v} }{\mathbf {u} }$ (also ∇_vu) which has the properties of a derivative, specified below. A vector is a geometrical object and independent of a chosen basis (coordinate system). In terms of a coordinate system, this derivative transforms under a change of coordinate system "in the same way" as the vector itself (covariant transformation), hence the name.

One would tend to define the derivative of a vector field in a proces that involves the difference between two vectors in two nearby points. In Euclidean space and orthonormal coordinate 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, the translation is not well defined and its analog, parallel transport, is depending on the path of moving the vector. 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.

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 $({\mathbf {e} }_{r},{\mathbf {e} }_{\theta })$,

where ${\mathbf {e} }_{r}$ and ${\mathbf {e} }_{\theta }$ 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.)

parallel transport of vector on a globe

Yet an other 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 will notice that the parallel-transported vector along a close circuit doesnot return as the same vector. 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 infinitesimal small close 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 D_vu 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 the covariant derivative called Levi-Civita connection.

The properties of a derivative imply that D_vu 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 as well geodesics can be defined only in terms of the covariant derivative.

Ocasionally covariant derivative refer to derivative of sections of a general vector bundle along tangent vector of the base. The definitions in this case are very similar, but we omit dicussing it here.

Formal definition

Functions

Given a function $f$ the covariant derivative ${\mathbf {D} }_{\mathbf {v} }f$ coinsides with the normal differentiation of a real function in the direction of the vector v, usually denoted by ${\mathbf {v} }f$ and by $df({\mathbf {v} })$.

Vector fields

The rules defining the covariant derivative D (or ∇) of a vector field u in the direction of the vector v is that the vector D_vu should have the following properties of a differentiation: For any vector fields u, v, w and scalar functions f and g these are

1. D_vu is algebraically linear in v so  D_{f v + g w}u = f D_vu + g D_wu
2. D_vu is additive in u so                    D_v(u + w) = D_vu + D_vw
3. D_vu obeys the "chain rule"             D_v(f u) = f (D_vu) + (D_v f )u

where D_v f is defined is defined above. Note that D_vu 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 chain rule. That means that the covariant derivative is not a tensor.

Covector fields

Given a field of covectors (or 1-form) $\alpha$ its covariant derivative ${\mathbf {D} }_{\mathbf {v} }\alpha$ can be defined using the following identity which is satisfyed for all vector fields u

${\mathbf {D} }_{\mathbf {v} }(\alpha ({\mathbf {u} }))=({\mathbf {D} }_{\mathbf {v} }\alpha )({\mathbf {u} })+\alpha ({\mathbf {D} }_{\mathbf {v} }{\mathbf {u} }).$

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

Tensor fields

Once the covariant derivative is defined for fields of vectors and covectors it cand be defined for arbitrary tensor fields using the following identities, given any two tensors $\varphi$ and $\psi$

${\mathbf {D} }_{\mathbf {v} }(\varphi \otimes \psi )=({\mathbf {D} }_{\mathbf {v} }\varphi )\otimes \psi +\varphi \otimes ({\mathbf {D} }_{\mathbf {v} }\psi ),$

and if $\varphi$ and $\psi$ are tensor fields of the same tensor bundle then

${\mathbf {D} }_{\mathbf {v} }(\varphi +\psi )={\mathbf {D} }_{\mathbf {v} }\varphi +{\mathbf {D} }_{\mathbf {v} }\psi .$

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

Coordinate description

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

${\mathbf {D} }_{{\mathbf {e} }_{i}}{\mathbf {e} }_{j}=\Gamma _{ij}^{k}{\mathbf {e} }_{k},$

the coeficients Γ^k_{i j} are called Christoffel symbols. Then using the rules in the definition, for general vector fields ${\mathbf {v} }=v^{i}e_{i}$ and ${\mathbf {u} }=u^{i}e_{i}$ we get

${\mathbf {D} }_{\mathbf {v} }{\mathbf {u} }=(v^{i}u^{j}\Gamma _{ij}^{k}+v^{i}{\partial u^{k} \over \partial x^{i}}){\mathbf {e} }_{k},$

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

${\mathbf {D} }_{{\mathbf {e} }_{j}}{\mathbf {u} }={\mathbf {D} }_{j}{\mathbf {u} }=\left({\frac {\partial u^{i}}{\partial x^{j}}}+u^{k}\Gamma _{jk}^{i}\right){\mathbf {e} }_{i}$

In words: the covariant derivative is the normal derivative along the coordinates plus correction terms that tells you how the coordinates changes. 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:

${\mathbf {D} }_{j}{\mathbf {v} }\equiv v_{;\;j}^{i}\;\;\;\;\;\;v_{;\;j}^{i}=v_{,\;j}^{i}+v^{k}\Gamma _{k\;j}^{i}$

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