Covariant Derivative of a Vector

The directional derivative depends on the coordinate system.  In an arbitrary coordinate system, the directional derivative is also known as the coordinate derivative, and it's written



The covariant derivative is the directional derivative with respect to locally flat coordinates at a particular point.  It's what would be measured by an observer in free-fall at that point.  It is written



I'd like to show, visually, how to find the covariant derivative in an arbitrary coordinate system.

Preliminaries:  The Christoffel Symbols

The Christoffel symbols relate the coordinate derivative to the covariant derivative.  There is more than one way to define them; we take the simplest and most intuitive approach here.  Given basis vectors e_α we define them to be:



where x^γ is a coordinate in a locally flat (Cartesian) coordinate system.  In other words, we move into a locally flat coordinate system, and evaluate the coordinate derivative of our original (general, non-Cartesian, non-flat) basis vectors with respect to those (flat) coordinates.  This is the standard "simple" definition, as used in [Schutz1].  Other more general definitions are possible but they don't help with picturing the covariant derivative.

In plain English, the Christoffel symbols measure how the basis vectors change as we move in a particular direction along a geodesic, where a geodesic is a line which is straight in locally flat coordinates.  Once we know how the basis vectors change, then we can use this information to "correct" the coordinate derivative which we obtained using that basis, in order to obtain the underlying flat-space "covariant" derivative.

The Covariant Derivative of a Vector

In curved space, the covariant derivative is the "coordinate derivative" of the vector, plus the change in the vector caused by the changes in the basis vectors.  To see what it must be, consider a basis B = {e_α} defined at each point on the manifold and a vector field v^α which has constant components in basis B.  Look at the directional derivative in the direction of basis vector γ.  The coordinate derivative is zero, because the components are constant:



Next, we want to find the β component of the covariant derivative taken in the direction of x^γ.

To do this, we want to imagine the change to the "real" flat-space v in the direction of basis vector e_β -- i.e., the β component of the flat-space directional derivative -- taken in the direction γ.

Imagine any basis vector -- say, e_α.  Picture it as we slide along in the γ direction.   e_α will change, because the coordinates are not flat.  We're interested in how much e_α changes in the β direction.  See figure 1.

Figure 1:


We need to add the change in e_α to the new "value" of v^β ... but we need to scale it, by the amount of v which lies along the direction of e_α.

In other words, the part of v which lies in the α direction will contribute a change to the β component of v which is proportional to v^α times the change in the β component of the α basis vector.  In English, this is confusing.  Hopefully, looking at Figure 1 will make it clear why this must be so.

All the basis vectors will contribute to the result in the same way, so we must have



If we generalize this a little, by taking the derivative of V in the direction of another vector, u, then we just need to sum over the components of u, scaling the pieces by the appropriate component of u as we go, and we obtain the familiar formula

The Components of a Vector

Figure 1 on this page brings to light an interesting side issue.  It's common to think of the components of a vector as the values obtained by projecting the vector onto the basis vectors.

This is extremely misleading!

In a metric space, when using an arbitrary basis, the components of the vector are the values of the basis 1-forms applied to the vector.  (See Figure 2, below.)

Basis 1-form ω^α is perpendicular to all basis vectors other than e_α.  However, it is not necessarily parallel to e_α.  In Figure 1 above, note that "New e_α" isn't perpendicular to "e_β".  But basis 1-form ω^α must be perpendicular to e_β!  In consequence, the α component of a vector in this basis isn't equal to the projection of the vector onto e_α.  In Figure 2 we've explicitly shown the basis 1-forms in an attempt to make clearer what's actually going on.

Figure 2:







Page created on 8/23/04.  Very minor changes on 11/20/06.