Christoffel symbols

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group. As a result, such a manifold is necessarily a Riemannian manifold. The Christoffel symbols provide a concrete representation of the connection of Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.
In general, there are an infinite number of metric connections for a given metric tensor; however, there is a unique connection that is free of torsion, the Levi-Civita connection. It is common in physics and general relativity to work almost exclusively with the Levi-Civita connection, by working in coordinate frames where the torsion vanishes. For example, in Euclidean spaces, the Christoffel symbols describe how the local coordinate bases change from point to point.
At each point of the underlying -dimensional manifold, for any local coordinate system around that point, the Christoffel symbols are denoted for. Each entry of this array is a real number. Under linear coordinate transformations on the manifold, the Christoffel symbols transform like the components of a tensor, but under general coordinate transformations they do not. Most of the algebraic properties of the Christoffel symbols follow from their relationship to the affine connection; only a few follow from the fact that the structure group is the orthogonal group .
Christoffel symbols are used for performing practical calculations. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In general relativity, the connection plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the are zero.
The Christoffel symbols are named for Elwin Bruno Christoffel.

Note

The definitions given below are valid for both Riemannian manifolds and pseudo-Riemannian manifolds, such as those of general relativity, with careful distinction being made between upper and lower indices. The formulas hold for either sign convention, unless otherwise noted.
Einstein summation convention is used in this article, with vectors indicated by bold font. The connection coefficients of the Levi-Civita connection expressed in a coordinate basis are called Christoffel symbols.

Preliminary definitions

Given a manifold, an atlas consists of a collection of charts for each open cover. Such charts allow the standard vector basis on to be pulled back to a vector basis on the tangent space of. This is done as follows. Given some arbitrary real function, the chart allows a gradient to be defined:
This gradient is commonly called a pullback because it "pulls back" the gradient on to a gradient on. The pullback is independent of the chart. In this way, the standard vector basis on pulls back to a standard vector basis on. This is called the "coordinate basis", because it explicitly depends on the coordinates on. It is sometimes called the "local basis".
This definition allows a common abuse of notation. The were defined to be in one-to-one correspondence with the basis vectors on. The notation serves as a reminder that the basis vectors on the tangent space came from a gradient construction. Despite this, it is common to "forget" this construction, and just write vectors on such that. The full range of commonly used notation includes the use of arrows and boldface to denote vectors:
where is used as a reminder that these are defined to be equivalent notation for the same concept. The choice of notation is according to style and taste, and varies from text to text.
The coordinate basis provides a vector basis for vector fields on. Commonly used notation for vector fields on include
The upper-case, without the vector-arrow, is particularly popular for index-free notation, because it both minimizes clutter and reminds that results are independent of the chosen basis, and, in this case, independent of the atlas.
The same abuse of notation is used to push forward one-forms from to. This is done by writing or or. The one-form is then. This is soldered to the basis vectors as. Note the careful use of upper and lower indexes, to distinguish contravariant and covariant vectors.
The pullback induces a metric tensor on. Several styles of notation are commonly used:
where both the centerdot and the angle-bracket denote the scalar product. The last form uses the tensor, which is understood to be the "flat-space" metric tensor. For Riemannian manifolds, it is the Kronecker delta. For pseudo-Riemannian manifolds, it is the diagonal matrix having signature. The notation serves as a reminder that pullback really is a linear transform, given as the gradient, above. The index letters live in while the index letters live in the tangent manifold.
The matrix inverse of the metric tensor is given by
This is used to define the dual basis:
Some texts write for, so that the metric tensor takes the particularly beguiling form. This is commonly done so that the symbol can be used unambiguously for the vierbein.

Definition in Euclidean space

In Euclidean space, the general definition given below for the Christoffel symbols of the second kind can be proven to be equivalent to:
Christoffel symbols of the first kind can then be found via index lowering:
Rearranging, we see that :
In words, the arrays represented by the Christoffel symbols track how the basis changes from point to point. If the derivative does not lie on the tangent space, the right expression is the projection of the derivative over the tangent space. Symbols of the second kind decompose the change with respect to the basis, while symbols of the first kind decompose it with respect to the dual basis. In this form, it is easy to see the symmetry of the lower or last two indices:
and
from the definition of and the fact that partial derivatives commute.
The same numerical values for Christoffel symbols of the second kind also relate to derivatives of the dual basis, as seen in the expression:
which we can rearrange as:

General definition

The Christoffel symbols come in two forms: the first kind, and the second kind. The definition of the second kind is more basic, and thus is presented first.

Christoffel symbols of the second kind (symmetric definition)

The Christoffel symbols of the second kind are the connection coefficients—in a coordinate basis—of the Levi-Civita connection.
In other words, the Christoffel symbols of the second kind are defined as the unique coefficients such that
where is the Levi-Civita connection on taken in the coordinate direction and where is a local coordinate basis. Since this connection has zero torsion, and holonomic vector fields commute we have
Hence in this basis the connection coefficients are symmetric:
For this reason, a torsion-free connection is often called symmetric.
The Christoffel symbols can be derived from the vanishing of the [|covariant derivative] of the metric tensor :
As a shorthand notation, the nabla symbol and the partial derivative symbols are frequently dropped, and instead a semicolon and a comma are used to set off the index that is being used for the derivative. Thus, the above is sometimes written as
Using that the symbols are symmetric in the lower two indices, one can solve explicitly for the Christoffel symbols as a function of the metric tensor by permuting the indices and resumming:
where is the inverse of the matrix, defined as . Although the Christoffel symbols are written in the same notation as tensors with index notation, they do not transform like tensors under [|a change of coordinates].

Contraction of indices

Contracting the upper index with either of the lower indices leads to
where is the determinant of the metric tensor. This identity can be used to evaluate the divergence of vectors and the covariant derivatives of tensor densities. Also

Christoffel symbols of the first kind

The Christoffel symbols of the first kind can be derived either from the Christoffel symbols of the second kind and the metric,
or from the metric alone,
As an alternative notation one also finds
It is worth noting that.

Connection coefficients in a nonholonomic basis

The Christoffel symbols are most typically defined in a coordinate basis, which is the convention followed here. In other words, the name Christoffel symbols is reserved only for coordinate frames. However, the connection coefficients can also be defined in an arbitrary basis of tangent vectors by
Explicitly, in terms of the metric tensor, this is
where are the commutation coefficients of the basis; that is,
where are the basis vectors and is the Lie bracket. The standard unit vectors in spherical and cylindrical coordinates furnish an example of a basis with non-vanishing commutation coefficients. The difference between the connection in such a frame, and the Levi-Civita connection is known as the contorsion tensor.