Christmas symbols

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In differential geometry , the Christoffel symbols , according to Elwin Bruno Christoffel (1829–1900), are auxiliary quantities for describing the covariant derivation on manifolds . They indicate by how much vector components change when they are shifted parallel along a curve.

In the general theory of relativity , the Christoffel symbols serve to derive the Riemann curvature tensor .

Christmas symbols of a plane

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

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

write. On the basis of Black's theorem , that is, and from this follows the symmetry of the Christoffel symbols, that is and . The coefficients , and are the coefficients of the second fundamental form .

If a curve is related to the Gaussian parametric representation , then the tangential part of its second derivative is through

given. By solving the differential equation system 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 . With regard to a map , one obtains a base of the tangent space and thus also a local reper (base field) of the tangential bundle . For all indices and then the Christoffelsymbols are through

Are defined. The symbols thus form a system of functions which depend on the point of the manifold (but this system does not form a tensor , see below).  

The Christoffelsymbols can also be used for an n-leg , i.e. H. a local base that is not directly determined by a map, according to

define, whereby here and in the following the summation symbols are omitted according to Einstein's summation convention .

properties

Covariant derivation of vector fields

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

Let be vector fields with those in local representations and . Then for the covariant derivative of in the direction of :

The application of derivation to the component function refers to this .

If one chooses a local frame that is induced by a map and one specifically chooses the basic vector field for the vector field , one obtains

or for the -th component

In the index calculus for tensors one also writes or , while the partial derivative is called as . It should be noted, however, that not only the component is derived here , but that it is the -th component of the covariant derivation of the entire vector field . The above equation is then written as

or.

Is chosen for and the tangent 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.

Christoffelsymbols in Riemannian and pseudo-Riemannian manifolds

Let be a Riemannian or pseudo-Riemannian manifold and the Levi-Civita connection . Let the local frame be that induced by a map .

Here you can see the Christoffelsymbols

from the metric tensor , using Greek letters for the space-time indices , as is usual in general relativity . In this case, the Christoffel symbols are symmetrical, that is, it applies to all and . These symbols of Christ are also called symbols of Christ of the second kind .

As Christoffel symbols of the first kind , the terms

designated.

Older notations, especially used in general relativity, are for the Christoffels symbols of the first kind

as well as for the Christoffel symbols of the second kind

Application to tensor fields

The covariant derivation can be generalized from vector fields to any tensor fields. Here, too, the symbols of Christ appear in the coordinate representation. The index calculus described above is used throughout this section. As is customary in the theory of relativity, the indices are designated 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 , one obtains

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

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

and for a (0,2) -tensor field one obtains

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

literature

  • Manfredo Perdigão do Carmo: Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., Upper Saddle River NJ 1976, ISBN 0-13-212589-7 .
  • Manfredo Perdigão do Carmo: Riemannian Geometry. Birkhäuser, Boston et al. 1992, ISBN 0-8176-3490-8 .
  • John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 .

Individual evidence

  1. Eric Weisstein : Christoffel Symbols of the Second Kind (Wolfram Mathworld)
  2. Bruce Kusse, Erik Westwig: Christoffel Symbols and covariant derivatives (page 5, formula F.24)