What are the Christoffel symbols

Christmas symbols

 

In differential geometry they are Christmas symbols, after Elwin Bruno Christoffel (1829–1900), auxiliary quantities for describing the covariant derivation on Riemannian manifolds. Its defining property consists in the requirement that the covariant derivative of the metric tensor vanishes. The main theorem of Riemannian geometry ensures that they are uniquely determined by this definition.

In the general theory of relativity, the Christoffel symbols enable the description of the movement of particles in a gravitational field that are not affected by any other external forces.

Einstein's summation convention is used in this article.

Christoffelsymbols related to an area

In classical differential geometry, the Christoffel symbols were defined for the first time for curved surfaces in three-dimensional Euclidean space. So be an oriented regular surface and a parameterization of . The vectors and form a base of the tangent plane, and with the normal vector to the tangential plane is called. So form the vectors a base of the . The Christoffelsymbols , are regarding the parameterization then defined by the following system of equations:

You write For , For and For , For etc., the defining equations can be summarized as

write. Because of Black's theorem, we have , this means, , and from this follows the symmetry of the Christoffelsymbols, what and means. The coefficients are the coefficients of the second fundamental form.

Be a curve with respect to the Gaussian parametric representation , then the tangential part of its second derivative is through

given. By solving the differential equation system so one finds the geodesics on the surface.

general definition

The Christoffs symbols defined in the previous section can be generalized to manifolds. So be a -dimensional differentiable manifold with a connection. Regarding a card is obtained by means of a base of the tangent space and thus also a local framework of the tangential bundle. For all indices and are then the Christoffelsymbols by

Are defined. The Symbols thus form a system of functions which depend on the point of the manifold (but this system forms none Tensor, see below).

You can also use the Christoffelsymbols for a local framework which is not induced by a card, by

define.

properties

Covariant derivation of vector fields

In the following, as in the previous section, a local frame induced by a map and any local framework.

Be Vector fields with the in local representations and . Then the covariant derivative of in the direction of :

Here designated the directional derivative of the component function in the direction

Choose a local framework that from a card is induced, and one chooses for the vector field specifically the basis vector field so you get

or for the -th component

One also writes for it in the index calculus for tensors or while taking the ordinary derivative of after -th coordinate as designated. It should be noted, however, that not only the component is derived, but that it is the -th component of the covariant derivative of the entire vector field acts. The above equation is then written as

or.

If you choose for and the tangent vector a curve and is has a 2-dimensional manifold, so the same local representation regarding the Christoffel symbols as from 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 related to the local framework given.

  • In this case, the Christoffel symbols are symmetrical, that is, it applies for all and .
  • You can see the Christoffelsymbols
from the metric tensor win.
In this case, the symbols of Christ considered here are also called Christmas symbols of the second kind. As Christmas symbols of the first kind become the expressions
designated.
Older notations, especially used in general relativity, are
for the Christoffelsymbols of the first kind as well
for the Christoffelsymbols of the second kind, whereby, as usual in general relativity, Greek letters are used for the indices (Latin indices, on the other hand, are only reserved for a special part, the so-called space-like parts).

Application to tensor fields

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

The covariant derivative of a scalar field is

The covariant derivative of a vector field is

and with a covector field, i.e. a (0,1) -tensor field you get

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

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

and for a (0,2) -tensor field you get

Only the sums or differences occurring here, but not the Christoffel symbols themselves, have the tensor properties (e.g. the correct transformation behavior).

Based on an article in: Wikipedia.de
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Date of the last change: Jena, 13.02. 2019