What are the Christoffel 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.
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
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
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
- 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).
Date of the last change: Jena, 13.02. 2019
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