
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 freefall 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, nonCartesian, nonflat) 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 flatspace "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" flatspace
v
in the
direction of basis vector
e_{β}_{}  i.e., the
β component of the
flatspace 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 1forms applied to the
vector. (See Figure 2, below.)
Basis 1form 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 1form 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 1forms 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.