Covariant Differentiation and Christoffel Symbols

In General Relativity, the effects of gravitation on space make it necessary to formulate coordinate systems in space as manifolds. Roughly speaking, a manifold is a topological space in which some neighborhood of each point may possess a coordinate system consisting of real coordinate functions on the points of the neighborhood which determine the position of the points and the topology of the points in that neighborhood; that is, the space local to the neighborhood is Cartesian. Moreover the passage from one coordinate system to the another is smooth in the overlapping regions, so that the meaning of differentiable curve, function, or map is consistent when referred to either system.

So we consider the passage from one point to another in a different neighborhood with a different coordinate system. We start with the local position, the position is usually given in a vector like form as R which is not quite a vector since it does not possess all required invariance properties. In particular, the magnitude of R depends on the the origin of the coordinate system that R is expressed in. However, the time derivative of R is a vector which usefully, can represent the velocity V of a particle.

This can be written with the chain rule (using the Einstein summation convention) as:

Since coordinate systems change from point to point, so do the basis vectorswhich are the partial derivatives of the position R with respect to the coordinate surfacesSo we may write the basis vectors as:

Thus the velocity V in terms of the basis vectors using the dot notation for time differentiation is:

Acceleration is the time derivative of velocity. Since the basis vectors may change from point to point they must be differentiated as well:

Applying the chain rule again gives.

Using the Christoffel symbols to express the derivatives of the basis vectors in terms of the basis vectors themselves we have :

Note that the reciprocal basis reciprocal tois defined by the gradient of the coordinate functionssuch that The Christoffel symbols for the reciprocal basiscan be obtained differentiating the reciprocal identitywith respect to the coordinate functionsUsing the Christoffel symbols with acceleration expression gives

Changing the summation index in the second term from i to k gives the kth covariant component of the acceleration in generalized coordinates as

The term given bydescribes how the underlying coordinate system in the manifold reflects the acceleration of the particle. As an example suppose that a particle is moving at a constant speed as defined bywhere s is path length. Then for the nth coordinateby the chain rule we have

and

If we suppose that the particle is not under the influence of any external forces and substitute the above results into the expression for the acceleration components we get

This differential equation defines the kth coordinate of the path taken by a particle not under the influence of any external forces in the geometry defined by the coordinate systems of the manifold. Such paths clearly influenced the topology of the underlying manifold. For this reason this expression is known as the geodesic equation. For example, if the underlying coordinate system is Cartesian then the Christoffel symbols are zero and the path is a straight line. If the coordinate system is spherical, the path is a great circle.