Christoffel symbols

In the differential geometry are the Christoffel symbols, Elwin Bruno Christoffel after (1829-1900), auxiliary variables to describe the covariant derivative on Riemannian manifolds. Your definitional property consists in the requirement that the covariant derivative of the metric tensor vanishes. The fundamental theorem of Riemannian geometry ensures that they are uniquely determined by this definition.

In general relativity, the Christoffel symbols allow the description of the motion of particles in a gravitational field, acting on no other external forces. It can be both for massive and massless particles. Masselos is used as a synonym for particles with vanishingly small rest mass.

In this article, the Einstein summation convention is used.

Christoffel symbols with respect to a surface

In classical differential geometry, the Christoffel symbols were first defined for curved surfaces in three-dimensional Euclidean space. Let an oriented regular surface and a parametrization of. And the vectors form a basis of the tangential plane, and the normal vector is referred to the tangential plane. Thus, the vectors form a basis of. The Christoffel symbols with respect to the parameterisation then defined by the following system of equations:

If you write for, for and for, for, etc., as can be the defining equations collectively as

. Write Due to the set of black is considered, that is, and it follows the symmetry of the Christoffel symbols and what means. The coefficients are the coefficients of the second fundamental form.

Be a curve with respect to the Gaussian parameter representation, the tangential part of their second derivative is by

Given. Thus, by solving the system of differential equations to find the geodesics on the surface.

General definition

The Christoffel symbols defined in the previous section can be generalized to manifolds. So be one -dimensional differentiable manifold with a connection. For a map is obtained by means of a base of the tangent space and thus a local frame of the tangent bundle. For all indices, and then the Christoffel symbols by

Defined. Thus, the symbols form a system of functions, which from the point of diversity depend ( but this system does not form a tensor, see below).

Similarly, one can the Christoffel symbols is not which also induces a local frame by a map, by

Define.

Properties

Covariant derivative of vector fields

Hereinafter referred to, as in the previous section, a local frame, which is induced by a card and any local frame.

Be the vector fields in local and representations. Then for the covariant derivative of in the direction of: The directional derivative of the function components referred to in the direction of

If you choose a local frame which is induced by a map, and you select for the vector field specially the basis vector field, we obtain

Or for the -th component

The index calculus for tensors to write it or even while denotes the ordinary derivative of the -th coordinate as claimed. However, it is to be noted that not only the component is derived, but that it is the ith component of the covariant derivative of the whole vector field. The above equation then written as

Or

If you choose to and the tangent vector of a curve and is a 2 -dimensional manifold, so has the same local representation with respect to the Christoffel symbols as in the first section.

Christoffel symbols in ( pseudo-) Riemannian manifolds

Be a Riemannian or pseudo - Riemannian manifold and the Levi- Civita connection. The Christoffel symbols are given with respect to the local frame.

• In this case, the Christoffel symbols are symmetric, that is, it applies to all and.
• You can browse the Christoffel symbols obtained from the metric tensor. In this case, it is called the Christoffel symbols considered here also Christoffel symbols of the second kind Christoffel symbols of the first kind as the expressions referred to. Older, used especially in general relativity theory notations are for the Christoffel symbols of the first kind and for the Christoffel symbols of the second kind are Here, as in general relativity common for the indices used Greek letters (Latin indices are there, however, only to a particular part, the so-called space-like proportions reserved).

Application to tensor fields

The covariant derivative can be generalized to arbitrary tensor fields of vector fields. Here, too, in the coordinate representation of the Christoffel symbols. In this section, the index calculus described above is used throughout. As usual in the theory of relativity, the indices are denoted by lowercase Greek letters.

The covariant derivative of a scalar field

The covariant derivative of a vector field

And at a Kovektorfeld, ie a (0,1) - tensor field obtained

The covariant derivative of a (2,0) - tensor field

For a (1,1) - tensor field it is

And for a (0,2) - tensor field obtained

Only the sums occurring here or differences, but not the Christoffel symbols themselves, the tensor possess (eg, the correct transformation behavior).