Covariant derivative

Here the covariant derivative of a vector in a vector field is described, the covariant derivative of a tensor is an extension of the same concept.

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 ${\mathbf {D} }_{\mathbf {v} }{\mathbf {u} }$ (also ∇_vu) which has the properties of a derivative, specified below. A vector is a geometrical object and independent of a 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. 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 extra terms describe how the coordinate grid 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,... (here written with lower indices). Any vector is a sum over all basis vectors, the linear combination Σ_kΓ^ke_k, where Γ^k are the components (here written with upper indices). The components of the covariant derivative are known as the Christoffel symbols. The covariant derivative of e_j in the direction e_i is indicated with lower indices, so

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

or defining the lefthand side as D_i

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

   definition of Christoffel symbols Γ^k_{i j}

So a covariant derivative is defined if we know the Christiffel symbols on a given basis.

curve in polar coordinates

Example of a curve in polar coordinates in a 2-dim Euclidean space. A vector at curve parmater 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 e.g. the accelaration 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.

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The vectors u and v in the definition are defined in the same point p. Also the covariant derivative D_vu is a vector defined in 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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In order to appreciate the definition of covariant derivative, a little background on the spaces involved is needed. The points p at which the vectors are defined, are elements of a space which in the most general case is called a manifold. This is a collection of points p together with a set of smooth (differentiable) coordinates functions x^a(p), a=0,1,.... An example is Euclidean space. The covariant derivative is extensively used in general relativity where the points are elements of spacetime, another example for a manifold.

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,... simply 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, when the other coordinates are held constant. A world line is another example of a curve in spacetime. 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 $\partial f/\partial x^{a}$. Conversly, every vector is tangent to a curve. For example the vector v in point p with components vⁱ is tangent to the curve parametrized by xⁱ(p) + λvⁱ.

A vector can as well be seen as an operator v that can be applied to a function f. Consider

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

Without the function f one could write

${\mathbf {v} }={d\;}/{d\lambda }$

short for differentiation with respect to the curve to which the vector v is tangent. So v is an operator by saying that v[f] is the vector with components $\partial f/\partial x^{a}$. In these terms the base vectors are the operators ${\partial \;}/{\partial x^{i}}$ and

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

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

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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 should have 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 v[f]. 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

${\mathbf {D} }_{j}{\mathbf {u} }={\mathbf {D} }_{j}(u^{i}{\mathbf {e} }_{i})=({\mathbf {D} }_{j}u^{i}){\mathbf {e} }_{i}+u^{k}({\mathbf {D} }_{j}{\mathbf {e} }_{k})$

or

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

   (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:

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

   (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 $p(\lambda )$, for which the tangent vector $u=d/d\lambda$ satisfies

D_uu = 0

  (coordinate free Geodesic equation)

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 ${\mathbf {u} }=u^{i}{\mathbf {e} }_{i}$ on a basis

${\mathbf {D} }_{\mathbf {u} }(u^{i}{\mathbf {e} }_{i})=({\mathbf {D} }_{\mathbf {u} }u^{i}){\mathbf {e} }_{i}+(u^{j}{\mathbf {D} }_{\mathbf {u} }{\mathbf {e} }_{j})$

$={\frac {du^{i}}{d\lambda \;}}+u^{j}u^{k}{\mathbf {D} }_{k}{\mathbf {e} }_{j}$

$={\frac {d^{2}x^{i}}{d\lambda ^{2}}}\;{\mathbf {e} }_{i}+{\frac {dx^{j}}{d\lambda }}{\frac {dx^{k}}{d\lambda }}\Gamma _{jk}^{i}\;{\mathbf {e} }_{i}$

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

${\frac {d^{2}x^{i}}{d\lambda ^{2}\;}}+\Gamma _{jk}^{i}{\frac {dx^{j}}{d\lambda \;}}{\frac {dx^{k}}{d\lambda \;}}=0$   (Geodesic equation in coordinates)

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.

Metric defines a unique covariant derivative

A metric tensor, metric for short, defines a real number called the innerproduct (or dot-product) for two vector u and v. With a metric one can derive metric properties, such as length of the vector and the angle between two vectors. It is a tensor, linearly dependent on both u and v and is denoted in varies ways. For example as the tensor ${\mathbf {g} }({\mathbf {u} },{\mathbf {v} })$ or as $\mathbf {u} \cdot {\mathbf {v} }$

The components $g_{i\;j}$ are real numbers of the metric tensor applied to a basis, say ${\mathbf {e} }_{i}$, so

$g_{ij}\equiv {\mathbf {g} }({\mathbf {e} }_{i},{\mathbf {e} }_{j})\equiv {\mathbf {e} }_{i}\cdot {\mathbf {e} }_{j}$

In components on a basis ${\mathbf {e} }_{i}$, with ${\mathbf {u} }=u^{i}{\mathbf {e} }_{i}$ and ${\mathbf {v} }=v^{i}{\mathbf {e} }_{i}$

A given metric defines a unique covariant derivative by the requirement that the chain rule should be applicable to the inner product as well

 :${\mathbf {D} }_{\mathbf {w} }({\mathbf {u} }\cdot {\mathbf {v} })=({\mathbf {D} }_{\mathbf {w} }{\mathbf {u} })\cdot {\mathbf {v} }+{\mathbf {u} }\cdot ({\mathbf {D} }_{\mathbf {w} }{\mathbf {v} })$

or in components

${\mathbf {D} }_{k}({\mathbf {e} }_{i}\cdot {\mathbf {e} }_{j})=({\mathbf {D} }_{k}{\mathbf {e} }_{i})\cdot {\mathbf {e} }_{j}+{\mathbf {e} }_{i}\cdot ({\mathbf {D} }_{k}{\mathbf {e} }_{j})$

The lefthand side is simply the normal derivative, since the innerproduct is a scalar. Writing out the covariant derivative in terms of the Christoffel symbols ${\mathbf {D} }_{k}{\mathbf {e} }_{i}={\Gamma ^{a}}_{ki}{\mathbf {e} }_{a}$ and using $({\mathbf {e} }_{a}\cdot {\mathbf {e} }_{j})=g_{aj}$

${\frac {\partial g_{ij}}{\partial x^{k}}}=g_{ij,k}=\Gamma _{ki}^{a}g_{aj}+\Gamma _{ia}^{a}g_{ia}$

This gives the unique relation between the Christoffel symbols (defining the covariant derivative) and the metric tensor components.

We can invert this equation and express the Christoffel symbols with a little trick, by wring this equation three times with a handy choice of the indices

$\quad g_{ij,k}=+\Gamma _{ki}^{a}g_{aj}+\Gamma _{kj}^{a}g_{ia}$

$\quad g_{ik,j}=+\Gamma _{ji}^{a}g_{ak}+\Gamma _{jk}^{a}g_{ia}$

$-g_{jk,i}=-\Gamma _{ij}^{a}g_{ak}-\Gamma _{ik}^{a}g_{ja}$

By adding, most of the terms on the right hand side cancel and we are left with

$g_{ia}\Gamma _{kj}^{a}={\frac {1}{2}}\left({\frac {\partial g_{ij}}{\partial x^{k}}}+{\frac {\partial g_{ik}}{\partial x^{j}}}-{\frac {\partial g_{jk}}{\partial x^{i}}}\right)$

Or with the inverse of g, defined as (using the Kronecker delta function)

$g^{ki}g_{il}=\delta _{l}^{k}$

we write the Christoffel symbols as

${\Gamma ^{i}}_{kj}={\frac {1}{2}}\left(g_{ij,k}+g_{ik,j}-g_{jk,i}\right)$