Covariant derivative videos and info

Given 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 of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination Γ^ke_k. To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field e_j along e_i.

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

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

In words: the covariant derivative is the normal derivative along the coordinates with correction terms which tell how the coordinates change.

The covariant derivative of a type (r,s) tensor field along

e c

is given by the expression:

Or, in words: take the partial derivative of the tensor and add: a $+\Gamma^{a_i}{}_{dc}$ for every upper index

a i

, and a $-\Gamma^{d}{}_{b_ic}$ for every lower index

b i

If instead of a tensor, one is trying to differentiate a tensor density (of weight +1), then you also add a term

If it is a tensor density of weight W, then multiply that term by W. For example, $\sqrt{-g}$ is a scalar density (of weight +1), so we get:

where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation. Incidentally, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is always zero.

Notation

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 partial derivative is indicated by a comma. In this notation we write the same as:

Once again this shows that the covariant derivative of a vector field is not just simply obtained by differentiating to the coordinates

v i, j

, but also depends on the vector v itself through

v k Γ i k j

In some older texts (notably Adler, Bazin & Schiffer, Introduction to General Relativity), the covariant derivative is denoted by a double pipe:

Derivative along curve

Since the covariant derivative $\nabla_XT$ of a tensor field

T

at a point

p

depends only on value of the vector field

X

p

one can define the covariant derivative along a smooth curve

γ(t)

in a manifold:

Note that the tensor field

T

only needs to be defined on the curve

γ(t)

for this definition to make sense.

In particular, $\dot{\gamma}(t)$ is a vector field along the curve

γ

itself. If $\nabla_{\dot\gamma(t)}\dot\gamma(t)$ vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a certain metric then the geodesics for the connection are precisely the geodesics of the metric that are parametrised by arc length.

The derivative along a curve is also used to define the parallel transport along the curve.

Sometimes the covariant derivative along a curve is called absolute or intrinsic derivative.

Relation to Lie derivative

A covariant derivative introduces an extra geometric structure on a manifold which allows vectors in neighboring tangent spaces to be compared. This extra structure is necessary because there is no canonical way to compare vectors from different vector spaces, as is necessary for this generalization of the directional derivative. There is however another generalization of directional derivatives which is canonical: the Lie derivative. The Lie derivative evaluates the change of one vector field along the flow of another vector field. Thus, one must know both vector fields in an open neighborhood. The covariant derivative on the other hand introduces its own change for vectors in a given direction, and it only depends on the vector direction at a single point, rather than a vector field in an open neighborhood of a point. In other words, the covariant derivative is linear (over C^∞(M)) in the direction argument, while the Lie derivative is linear in neither argument.

Note that the antisymmetrized covariant derivative ∇_uv - ∇_vu, and the Lie derivative L_uv differ by the torsion of the connection, so that if a connection is symmetric, then its antisymmetrization is the Lie derivative.

Notes

^ Levi-Civita, T. and Ricci, G. "Méthodes de calcul différential absolu et leurs applications", Math. Ann. B, 54 (1900) 125-201.
^ Riemann, G.F.B., "Über die Hypothesen, welche der Geomtrie zu Grunde liegen", Gesammelte Mathematische Werke (1866); reprint, ed. Weber, H.: Dover, New York, 1953.
^ Christoffel, E.B., "Über die Transformation der homogenen Differentialausdrücke zweiten Grades," J. für die Reine und Angew. Math. 70 (1869), 46-70.
^ cf. with Cartan, E. "Sur les variétés à connexion affine et la theorie de la relativité généralisée", Annales, Ecole Normale 40 (1923), 325-412.
^ Koszul, J. L. "Homologie et cohomologie des algebres de Lie", Bulletin de la Société Mathématique 78 (1950) 65-127.
^ The covariant derivative is also denoted variously by $\partial$ _vu, D_vu, or other notations.
^ In many applications, it may be better not to think of t as corresponding to time, at least for applications in general relativity. It is simply regarded as an abstract parameter varying smoothly and monotonically along the path.

Diemazz

Notation

Derivative along curve

Relation to Lie derivative

Notes

References

See also