Covariant Derivative of a 1-Form

We worked out what the covariant derivative of a vector must be here.  Now we're going to try to see, visually, how the covariant derivative of a 1-form must relate to the coordinate derivative of the 1-form.

The Covariant Derivative of a 1-Form

Again, we want to find the difference between the coordinate (directional) derivative of a 1-form in a particular coordinate system, and the coordinate derivative, in the same direction, evaluated in flat space.

Imagine a 1-form field, ω(P), with zero derivative at each point in flat space.  In other words, the 1-form ω is "really" the same at every point on the manifold, even though it may appear to vary from place to place in some particular coordinate system, S.  We want to find the directional derivative evaluated using the S's basis, which we'll call B.  The difference we're looking for, which is the difference between the covariant derivative and the coordinate derivative, will be the value of that coordinate derivative, negated.

We'll start by looking at the derivative in the direction of basis vector e_β.

The α component of the 1-form is the value of the 1-form applied to basis vector e_α.

The α component of the derivative is, therefore, the derivative of the value of the 1-form applied to the basis vector e_α, taken along the vector e_β.  So we will look at how ω_ξe_α^ξ changes as we move in the β direction.

Figure 1:


In Figure 1, basis vector e_α is shown before and after we move one unit in the direction of e_β.  Basis vector e_α has stretched and rotated in the α and γ directions.  Studying the diagram should make clear what the coordinate derivative must be; the following text is a description in words and equations of what's going on.

The blue lines are the 1-form (see 1-forms for notes on their representation).  The number of blue lines crossed by any vector is the value of the 1-form applied to that vector.

Since the vector shown is the α basis vector, the number of blue lines it crosses is the α component of the 1-form.

The difference in the number of lines crossed by the red versus gray basis vectors is the change in the α component of the 1-form.  The number of blue lines crossed by the thin gray line labeled "Net Change in e_α" is the change in the α component of ω.

The green lines show the change in the component value broken down, itself, into components.  The basis vector actually stretched in the α direction, and then grew in the γ direction.  The total change in its value is clearly the sum of those two changes (as we can see from the picture).

That's it -- the rest is just details!  But they're details we do need to fill in, so let us soldier on.

In English, to reiterate, the diagram shows us what the value of the coordinate derivative with respect to x_β must be, when the basis vectors are changing but the covariant (flat-space) derivative is actually zero.

The length Δα represents the "flat-space" derivative of e_α with respect to β in the α direction.  The contribution of this change to the derivative of ω is obtained by multiplying that length by ω_α, since that tells us how many additional lines of ω e_α will cross as a result of the change -- and we can also read the number of new crossings directly from the diagram (it's about 1).

The length Δγ represents the "flat-space" derivative of e_α with respect to β in the γ direction.   The contribution of this change to the derivative of ω is obtained by multiplying that length by ω_γ, since that tells us how many additional lines of ω basis vector e_α crosses as a result this change -- and we can also read the number of new crossings directly from the diagram (it's about 2.7).

So, remembering that the Christoffel symbols measure how fast the basis vectors are changing relative to a "flat-space" basis, the directional derivative we will actually measure due to the parts shown in the diagram will be (no implied sum just yet!):



For spaces of higher dimension than 2, we can see that the α component of the directional derivative in the direction β, for a constant 1-form field, must be (now we take an implied sum, over γ):



So, the actual covariant derivative must be the coordinate derivative, minus that value.  When we sum across all components of a general vector to get the directional derivative with respect to that vector, we obtain:



which is the formula typically derived by non-visual (but more rigorous) means in relativity texts.