#### Autoparallel Curves, the Levi-Civita Connection,and the Geodesic Equation

David K. Zhang December 29, 2020
Definition: Let be a smooth manifold, and let be a smooth curve. The velocity vector of at a point is the tangent vector defined by
Definition: Let be a smooth manifold with affine connection , and let be a smooth curve. We say that is autoparallel if

Geometrically speaking, autoparallel curves are those which are “as straight as possible” (i.e., never deviating from their current direction of travel) and move at uniform speed. Indeed, the condition stipulates that they have constant velocity. Of course, we don't have a formal notion of speed or angle in the absence of a metric, so this is only a vague metaphor for now.

If we write the condition in terms of a coordinate basis, we obtain:

Strictly speaking, the intermediate expressions in this calculation don't make any sense. For example, what does mean? The velocity vector is only specified along the image of the curve so there's nothing to differentiate if doesn't travel in the direction.

The proper way to make this mathematically sensible is to introduce the notion of a covariant derivative along a curve as opposed to a vector field It turns out that a unique such operation is induced by any affine connection and it enjoys properties so strikingly similar to that we get the right answer by blindly computing as though we were working with vector fields.

For details of this construction, see page 101 of Lee's Introduction to Riemannian Manifolds, second edition.

Modulo this technical disclaimer, this calculation proves (or at least strongly suggests) the following statement.

Theorem: Let be a smooth manifold with affine connection A smooth curve is autoparallel if and only if it satisfies the autoparallel equation

in some coordinate chart around each of its points.

Although the above formulation of the autoparallel equation is chart-dependent, the property of being autoparallel is not, since the quantity is a legitimate vector field along If it vanishes in one chart, it vanishes in every chart.

Definition: Let be a Riemannian manifold, and let be a smooth curve. The arc-length of is the quantity

This notion of arc-length for smooth curves serves as a crucial link between the notion of an affine connection and a metric A priori, given a smooth manifold we can separately specify a connection and a metric on that have absolutely nothing to do with each other. But when we remember that allows us to tell which curves are straight (i.e., autoparallel), while lets us measure arc-lengths of curves, we might want to say that the shortest curve between two given points ought to be straight. By checking whether straight and shortest curves coincide, we can determine if is compatible with

We will later see that, for any given (pseudo-)Riemannian metric there exists a unique torsion-free connection which is compatible with in this sense. This is called the Levi-Civita connection. In light of this fact, most physicists treat spacetime manifolds as being defined by the metric structure and implicitly work with the connection that the given metric induces. However, it should be kept in mind that is only one of infinitely many possible choices for an affine connection. Indeed, there have been attempts to define physical theories involving torsion (though none of these have yet reached mainstream status).

For now, let us further investigate the notion of a shortest curve. More precisely, we would like to study the stationary curves of the arc-length functional defined above. Choose a coordinate chart and define:

To form the EulerLagrange equation for the arc-length functional, we compute the following quantities:

In the final step, we use the symmetry of the metric to combine terms and cancel the factor of two in the denominator. Before computing the final expression needed to form the EulerLagrange equation, we first calculate an intermediate result.

At this point, it is becoming too cumbersome to carry function arguments everywhere. We will henceforth drop them, writing in place of and in place of This should not cause confusion, since the metric and the curve are always evaluated at the same location. Under this convention, our preceding results become:

We now have an opportunity to make a substantial simplification. If we declare that we are only interested in curves which are parameterized by arc-length (and it is always possible to re-parameterize a smooth curve to make this so), then we may assume Under this assumption, we can write:

We thus obtain the EulerLagrange equation:

We now take a step that presently appears artificial, but will turn out to have geometric significance. Namely, we symmetrize the term

By plugging this expression for into the EulerLagrange equation and moving all terms to the same side, we obtain:

In order to isolate we contract both sides with noting that

We are now poised to make a remarkable observation. This equation has precisely the form of an autoparallel equation if we define the following connection coefficients:

Of course, to be sure that these connection coefficients specify a well-defined affine connection on we need to verify that they satisfy the transformation law

between any two coordinate charts and This is straightforward to check, but we will take this result on faith and christen our newly-discovered compatible connection.

Definition: The Levi-Civita connection on a (pseudo-)Riemannian manifold is the affine connection whose connection coefficients in each coordinate chart are defined by

The work we have done so far provides the following characterization of the Levi-Civita connection:

Theorem: Let be a (pseudo-)Riemannian manifold. The Levi-Civita connection on is the unique affine connection on which is torsion-free and whose autoparallel curves are stationary points of the arc-length functional.

The autoparallel curves of are so important that they have their own distinguished name.

Definition: Let be a (pseudo-)Riemannian manifold. A smooth curve is called a geodesic if it is autoparallel with respect to the Levi-Civita connection on or equivalently, if it satisfies the geodesic equation

in some coordinate chart around each of its points.

In fact, itself is so important that we will rarely have occasion to use any other connection on a (pseudo-)Riemannian manifold.

Notation: Whenever a connection or connection coefficients are mentioned on a Riemannian manifold assume that they are the Levi-Civita connection unless otherwise specified.

Suppose now that we would like to repeat the derivation of the geodesic equation without making any assumptions about the parameterization of In this case, our previous simple expression for no longer applies. Instead, we have this monstrosity:

To make these expressions more manageable, we will abbreviate as Keep in mind that this “dot product” notation hides an implicit dependence on the metric. When we form the EulerLagrange equation, we will multiply both sides by to clear denominators.

As before, we contract both sides with This allows us to rewrite the left-hand side in terms of the Levi-Civita connection coefficients using the previously-discussed symmetrization trick.

Another application of the symmetrization trick yields the identity If we introduce the abbreviation then we can write:

This is the equivalent of the geodesic equation for curves with variable-speed parameterizations. Notice that it does not have the form of an autoparallel equation, since such curves are not autoparallel in general.

Note that the variable-speed geodesic equation does not constitute a system of independent second-order differential equations as runs from 1 to This system is rank-deficient, since if we contract both sides with , we find the trivial identity

Thus, if we were to try to solve an initial value problem for the variable-speed geodesic equation, we would find one free parameter at each time step, corresponding to the speed of the parameterization.

© David K. Zhang 2016 2021