Covariant derivative

In physics, 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 (also ∇_vu) which has the properties of a derivative, specified below. A vector is a geometrical object and independent of a choosen 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.

In Euclidean space and a rectangular (orthonormal) coordinate system the covariant derivative in components is simply obtained by taking the derivative of the components. In general this is not the case (see example in polar coordinates below) and extra terms appear that describe how the coordinate grid itself rotates, expands or contracts, interweaves, etc.

Any vector is known if we know its components on a choosen basis, say the vectors e_i ,i=0,1,2,... The components of the covariant derivative are known as the Christoffel symbols, so for example the linear combination (sum over k) Γ^ke_k. The covariant derivative of e_j in the direction e_i is indicated with lower indices, so

or defining the lefthand side as D_i    definition of Christoffel symbols Γ^k_{i j}

The vectors in the definition are elements of a manifold (a collection of points p with a set of smooth (differentiable) coordinates functions x^a(p), a=0,1,..., such as Euclidean space or as spacetime). In such a space, a function f that assigns real numbers to every point p in the manifold, can be considered as a function of the variables x^a(p), a=0,1,... by saying that f(x^a(p))= f(p). Curves c in a manifold can be defined as a collection of points p that depend on one parameter λ, called the curve parameter, so p=c(λ). The coordinates functions themselves define curves, the coordinate grid. A world line is another example of a curve. The derivative of f in a point p with respect to the curve parameter can be considered a vector in p, tangent to the curve in p and therefore called a tangent vector. It has components . Conversly, every vector is tangent to a curve. For example the vector in point p with components vⁱ is tangent to the curve parametrized by ''xⁱ(p) + λvⁱ

In the definition of the covariant derivative, the vectors u and v should be defined in the same point p of the space under consideration. D_vu is a vector also defined at the same point p.

The definition of the covariant derivative does not use the metric in space. However, a given metric uniquely defines the covariant derivative.

The properties of a derivative imply that D_vu depend 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 Riemann tensor can be defined in a coordinate independent way in terms of the covariant derivative. The Riemann tensor tells about the "curvature" of space and can be defined without the use of a metric.

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The rules defining the covariant derivative D (or ∇) of a vector u in the direction of the vector v is that the vector D_vu are the following properties of a differentiation. For vectors 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 as the normal differentiation of a real function in the direction of the vector v (see below). Note that D_vu is not linear in v and depends on the neighborehood of p because of the last property, the chain rule. Therefore the covariant derivative is not a tensor (which depends linearly in all its arguments) and the Christoffel symbols are not the components of a tensor.

If we express the vector u as a linear combination of the basis vectors e_i ,i=0,1,2,..., say with the coordinates u^k

u = Σ_k u^k e_k ,

one can apply the rules of the covariant diffentiation to the product in the righthand side to obtain

or

   (Covariant derivative    in components)

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:

    (Semicolon notation)

Once again this shows that the covariant derivative of a vector field is not just simply obtained by differentiating to the coordinates, but also depends on the vector v itself.

Parallel transport

With a covariant derivative it is possible to compare vectors in different (neighboring) points. This allows a description of transport of vectors. A vector u is said to be parallel transported in the direction of a vector v if D_vu = 0 , since in that case the (infinitessimal) change of the vector u in the direction of v is zero. In other words u remains the same.

A special role is played by curves that are created by transporting the tangent vector parallel to itself. They are called geodesics. A geodesic is a curve , for which the tangent vector

satisfies Duu = 0 for every point on the curve.  An example can be given in 4-dimensional spacetime for curves that are world lines. 

For a worldline the tangent vector u is the 4-velocity and its derivative is the acceleration. So D_uu = 0 one sees that geodesics are orbits in which the acceleration is zero: the worldlines of particles and observers in free fall. When a metric is introduced, it can be shown that geodesics defined in this way are also the routes between two points for which the pathlengths has a stationary point (form an extremum, the "the most straight" routes.

In components D_uu = 0 is the well known geodesic equation, writing on a basis

So D_uu = 0 in components give the important equation for geodesics     (Geodesic equation)

Note that all anti-symmetric parts in the lower indices of will cancel out in the summation, so only the symmetric parts will play a role.