covariant derivative

From Mathematics wiki

1 Introduction
2 Definition
3 Action on any tensor
- 3.1 Summary
4 Applications

Introduction

The covariant derivative is one differential operator which plays an important role in differential geometry and gives the rate of change or total derivative of a scalar field, vector field or general tensor field along some path on a (generally curved) manifold. Consider specifying a vector field in terms of some coordinate basis vectors. In generally curved space, these basis vectors change from point to point, so that finding the derivative of a vector field we must not only know how the components of the vector change, but also how the basis vectors change. Anyone who has seen expressions for $\nabla\,$ (gradient), $\nabla\times\,$ (curl) or $\nabla^2\,$ (Laplacian) in spherical, polar or other curvilinear coordinates has encountered the difficulties that arise when not dealing with a stationary coordinate basis.

Definition

A linear operator $\nabla : T^m_n \to T^m_{n+1}\,$ where $T^m_n\,$ is a $(m,n)\,$ -tensor field, that, for any two tensor fields $A\,$ and $B\,$ , satisfies the following properties:

Linearity:

$\nabla_a(c A + d B) = c \nabla_a(A) + d \nabla_a(B),\qquad \forall c, d \in \mathbb{R}\,$ ,

Satisfies the Leibniz rule:

$\nabla_a(A B) = A \nabla_a(B) + B \nabla_a(A)\,$ ,

Commutativity with regards to contraction of indices:

$\nabla_a (A^{b_1...k...b_m}_{c_1...k...c_n}) = \nabla_a A^{b_1...k...b_m}_{c_1...k...c_n}\,$ ,

Consistency with tangent vectors being directional derivatives:

$v^a \nabla_a f = \vec{v} (f)\,$ ,

alternatively, $\nabla f = \tilde{d} f\,$ in a coordinate basis.

The covariant derivative is a derivative of tensors that takes into account the curvature of the manifold on which these tensors are defined, as well as dynamics of the coordinate basis vectors. In cartesian coordinates, the covariant derivative is simply a partial derivative $\partial_a\,$ . In spherical coordinates, for example, the coordinate basis vectors change between different points, so the derivative of a vector expressed in terms of these basis vectors must take this into account.

The covariant derivative is also known as the semi-colon derivative and is written as $A_{;a} = \nabla_a A\,$ .

If we further require that, for any vectors $\vec{u}\,$ , $\vec{v}\,$ , $v^a \nabla_a \vec{u} = \nabla_{\vec{v}}\vec{u}\,$ , where $\nabla_{\vec{v}}\,$ is the affine connection, then we can completely specify the the action of $\nabla\,$ on any tensor. To do this we will also place a torsion-free condition on the connection. Write $v^a = v^\alpha e^a_\alpha\,$ , so that $v^a\,$ is a vector, whereas $v^\alpha\,$ is a function with label $\alpha\,$ , i.e., $\vec{v} = v^\alpha \vec{e}_\alpha\,$ (see abstract index notation). Then

$\nabla_{\vec{v}} \vec{u}\,$	$= v^\alpha \nabla_{\vec{e}_\alpha} ( \vec{e}_\beta u^\beta )\,$ ,
	$= v^\alpha \vec{e}_\beta \nabla_{\vec{e}_\alpha} ( u^\beta ) + v^\alpha \nabla_{\vec{e}_\alpha} ( \vec{e}_\beta )u^\beta\,$ ,
	$= v^\alpha \vec{e}_\beta \partial_\alpha u^\beta + v^\alpha u^\beta \Gamma^\gamma_{\beta\alpha} \vec{e}_\gamma\,$ ,
	$= (v^\alpha \partial_\alpha u^\beta + v^\alpha u^\gamma \Gamma^\beta_{\gamma\alpha} )\vec{e}_\beta\,$ ,

where we have defined the Christoffel symbol of the 2nd kind through

$\nabla_{\vec{e}_\alpha} ( \vec{e}_\beta) = \Gamma^\gamma_{\beta\alpha} \vec{e}_\gamma\,$ .

(Note that the differentiation index comes last on $\Gamma^\gamma_{\alpha\beta}\,$ .) Finally, restoring abstract index notation,

$\nabla_{\vec{v}} u^b\,$	$= (v^\alpha \partial_\alpha u^\beta + v^\alpha u^\gamma \Gamma^\beta_{\gamma\alpha} )e^b_\beta\,$ ,
$v^a e^\alpha_a \nabla_{\alpha} u^b\,$	$= (v^\alpha \partial_\alpha u^\beta + v^\alpha u^\gamma \Gamma^\beta_{\gamma\alpha} )e^b_\beta\,$ ,
$v^a \nabla_a u^b\,$	$= v^a e^\alpha_a ( \partial_\alpha u^\beta + u^\gamma \Gamma^\beta_{\gamma\alpha} )e^b_\beta\,$ ,
	$= v^a ( \partial_\alpha u^\beta + u^\gamma \Gamma^\beta_{\gamma\alpha} ) e^\alpha_a e^b_\beta\,$ ,

which is true for all $v^a\,$ , so we write

$\nabla_a u^b = ( \partial_\alpha u^\beta + u^\gamma \Gamma^\beta_{\gamma\alpha} ) e^\alpha_a e^b_\beta\,$ .

which shows that $\nabla\,$ has taken a tensor of type $(1,0)\,$ and returned a tensor of type $(1,1)\,$ .

Action on any tensor

Armed with this information, we can find the covariant derivative of any tensor. The method in each case will be the same: Contract the tensor with objects for which the covariant derivative is known, in such a way that the result is a scalar. Compute the covariant derivative of the product using the both the Leibniz rule for the covariant derivative and for partial derivatives, keeping in mind that the covariant derivative of a scalar is merely the gradient of that scalar.

As an example, consider the covariant derivative of a oneform $\omega_\beta\,$ , $\nabla_a\omega_\beta\,$ . Contracting $\omega_\beta\,$ with a vector $v^\beta\,$ yields a scalar, $\omega_\beta v^\beta\,$ . Thus we can compute $\nabla_\alpha (\omega_\beta v^\beta)\,$ in two ways:

Firstly, $\nabla_\alpha (\omega_\beta v^\beta) = \omega_\beta \nabla_\alpha (v^\beta) + \nabla_\alpha (\omega_\beta )v^\beta\,$ (Leibniz rule).

Since we already know the covariant derivative's action on vectors, we can expand the first term:

$\nabla_\alpha (\omega_\beta v^\beta) = \omega_\beta [\partial_\alpha v^\beta + \Gamma^\beta_{\gamma\alpha} v^\gamma] + \nabla_\alpha (\omega_\beta )v^\beta\,$ .

Secondly, $\nabla_\alpha (\omega_\beta v^\beta) = \partial_\alpha (\omega_\beta v^\beta)\,$ , which is equal to $\omega_\beta \partial_\alpha v^\beta + v^\beta \partial_\alpha \omega_\beta\,$ by the Leibniz rule.

We can now equate these two results and solve for the term we want, $\nabla_\alpha \omega_\beta\,$ :

$\omega_\beta \partial_\alpha v^\beta + v^\beta \partial_\alpha \omega_\beta\,$	$= \omega_\beta [\partial_\alpha v^\beta + \Gamma^\beta_{\gamma\alpha} v^\gamma] + \nabla_\alpha (\omega_\beta )v^\beta\,$ ,
$\nabla_\alpha (\omega_\beta )v^\beta\,$	$= v^\beta \partial_\alpha \omega_\beta - \omega_\beta \Gamma^\beta_{\gamma\alpha} v^\gamma\,$ ,
	$= v^\beta \partial_\alpha \omega_\beta - \omega_\gamma \Gamma^\gamma_{\beta\alpha} v^\beta\,$ ,

and since $v^\beta\,$ is arbitrary, we have

$\nabla_\alpha (\omega_\beta ) = \partial_\alpha \omega_\beta - \Gamma^\gamma_{\beta\alpha}\omega_\gamma \,$ .

Abstractly,

$\nabla_a \omega_b = (\partial_\alpha \omega_\beta - \Gamma^\gamma_{\beta\alpha}\omega_\gamma ) e^\alpha_a e^\beta_b\,$ .

This method generalizes to any tensor, for example:

$\nabla_a T_{bc}^{de} = (\partial_\alpha T_{\beta\gamma}^{\delta\epsilon} - \Gamma^\kappa_{\beta\alpha} T_{\kappa\gamma}^{\delta\epsilon} - \Gamma^\kappa_{\gamma\alpha} T_{\beta\kappa}^{\delta\epsilon} + \Gamma^\delta_{\kappa\alpha} T_{\beta\gamma}^{\kappa\epsilon} + \Gamma^\epsilon_{\kappa\alpha} T_{\beta\gamma}^{\delta\kappa}) e^\alpha_a e^\beta_b e^\gamma_c e_\delta^d e_\epsilon^e\,$

For each raised index we contract with a lowered index on the Christoffel symbol, and for each lowered index we contract with a raised index on the Christoffel symbol, whilst taking a negative sign.

Finally, if we require that the covariant derivative be torsion free, which means that covariant derivatives of a scalar field commute, then

$\nabla_a \nabla_b (f) = \nabla_b \nabla_a (f)\,$

We can expand either side since $\nabla_\alpha(f) = \partial_\alpha f\,$ are the components of a oneform (gradient). Then

$\partial_\alpha \partial_\beta f - \Gamma^\gamma_{\beta\alpha}\partial_\gamma f = \partial_\beta \partial_\alpha f - \Gamma^\gamma_{\alpha\beta}\partial_\gamma f\,$

Since partial derivatives commute, the torsion free condition requires that $\Gamma^\gamma_{\alpha\beta} = \Gamma^\gamma_{\beta\alpha}\,$ .

More formally, the torsion free condition requires that $(\nabla_\vec{u} \nabla_{\vec{v}} - \nabla_\vec{v} \nabla_{\vec{u}}) f = [\vec{u},\vec{v}] f\qquad \forall f\,$ where $[\vec{u},\vec{v}]\,$ is the Lie bracket or commutator of $\vec{u}\,$ and $\vec{v}\,$ , and $f\,$ is a function. Here $\nabla_\vec{u} \nabla_{\vec{v}} A\,$ is short-hand for $v^a\nabla_a (u^b\nabla_b A)\,$ . The commutator is such that if $u = \frac{d}{d\lambda}\,$ and $v = \frac{d}{d\mu}\,$ then $[\vec{u},\vec{v}]f = \frac{d^2 f}{d\lambda d\mu} - \frac{d^2 f}{d\mu d\lambda}\,$ . If $\lambda\,$ and and $\mu\,$ are coordinates then the derivatives become partial derivatives, which commute, and the commutator vanishes. Above we only considered a special case of the torsion free condition applied to scalar fields in order to discover the constraint imposed on $\Gamma^\gamma_{\alpha\beta}\,$ .

Note that the original definition of the covariant derivative was made without fixing its effect on any general tensors. However, by requiring that it behave like the affine connection, and by requiring that the connection be torsion free, we were able to completely specify the covariant derivative. The result is what is usually meant by the term "covariant derivative", but it is not the only one. Normally, though, the metric induces a unique metric compatible connection, which in turn specifies the covariant derivative uniquely.

Summary

$\nabla_a f = (\partial_\alpha f)e_a^\alpha\,$ ,
$\nabla_a u^b = (\partial_\alpha u^\beta + \Gamma^\beta_{\gamma\alpha}) e_a^\alpha e_\beta^b\,$ ,
$\nabla_a \omega_b = (\partial_\alpha \omega_\beta - \Gamma^\gamma_{\beta\alpha}\omega_\gamma ) e^\alpha_a e^\beta_b\,$ .
$\nabla_a T_{bc}^{de} = (\partial_\alpha T_{\beta\gamma}^{\delta\epsilon} - \Gamma^\kappa_{\beta\alpha} T_{\kappa\gamma}^{\delta\epsilon} - \Gamma^\kappa_{\gamma\alpha} T_{\beta\kappa}^{\delta\epsilon} + \Gamma^\delta_{\kappa\alpha} T_{\beta\gamma}^{\kappa\epsilon} + \Gamma^\epsilon_{\kappa\alpha} T_{\beta\gamma}^{\delta\kappa}) e^\alpha_a e^\beta_b e^\gamma_c e_\delta^d e_\epsilon^e\,$

Using the covariant derivative we can succinctly express various concepts in differential geometry. For example, consider parallel transporting the vector $u^b\,$ along the curve $\gamma(\lambda)\,$ with tangent vector $v^a = (d/d\lambda)^a\,$ . Using the affine connection we would require that $\nabla_\vec{v} \vec{u} = 0\,$ , i.e. the vector remains parallel to it's original direction as it is moved along $\gamma\,$ . Using the covariant derivative we can write the equivalent expression $v^a\nabla_a u^b = 0\,$ . Thus for a given curve, once the Christoffel symbols are known, this expression results in a set of differential equations that describe how the components of $u\,$ change as it is parallel transported along the curve.

We can also obtain the geodesic equations which describe curves that are geodesics, or curves between two points for which the arc length is an extremum (local minimum/maximum). These curves can be thought of as "straight lines" in curved space, for example great circles on the surface of a 2-sphere. It can be shown that a geodesic is a curve that parallel transports its own tangent vector. Thus for a curve $\gamma\,$ with an affine parameter $\lambda\,$ with tangent vector $v^a = (d/d\lambda)^a\,$ , we require that $v^a\nabla_a v^b = 0\,$ .