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Σύμβολον Chrisoffel

Christoffel symbol

Τανυστής Ricci
Σύμβολο Christoffel

Σχετικιστική Φυσική
Πεδιακές Εξισώσεις Einstein
Γενική Σχετικότητα
Βαρυτικό Πεδίο
Τανυστής Einstein
Τανυστής Ricci
Κοσμολογική Σταθερά
Θεωρία Διαστολής Σύμπαντος

- Ένας ψευδο-τανυστής.


Η ονομασία "Christoffel Symbol" σχετίζεται ετυμολογικά με το όνομα του μαθηματικού Christoffel.


In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. In a broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are also called Christoffel symbols.[1] The Christoffel symbols may be used for performing practical calculations in differential geometry. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives.

At each point of the underlying n-dimensional manifold, for any local coordinate system, the Christoffel symbol is an array with three dimensions: n × n × n. Each of the n3 components is a real number.

Under linear coordinate transformations on the manifold, it behaves like a tensor, but under general coordinate transformations, it does not. In many practical problems, most components of the Christoffel symbols are equal to zero, provided the coordinate system and the metric tensor possess some common symmetries.

In general relativity, the Christoffel symbol plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor.


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 (contra-variant and co-variant indices). The formulas hold for either sign convention, unless otherwise noted. Einstein summation convention is used in this article.


If xi, i = 1,2,...,n, is a local coordinate system on a manifold M, then the tangent vectors

define a basis of the tangent space of M at each point.

Christoffel symbols of the first kind[]

The Christoffel symbols of the first kind can be derived from the Christoffel symbols of the second kind and the metric,

Christoffel symbols of the second kind (symmetric definition)[]

The Christoffel symbols of the second kind, using the definition symmetric in i and j,[2] (sometimes Γkij ) are defined as the unique coefficients such that the equation

holds, where is the Levi-Civita connection on M taken in the coordinate direction ei.

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 semi-colon and a comma are used to set off the index that is being used for the derivative. Thus, the above is sometimes written as

By permuting the indices, and resumming, one can solve explicitly for the Christoffel symbols as a function of the metric tensor:

where the matrix is an inverse of the matrix , defined as (using the Kronecker delta, and Einstein notation for summation) . Although the Christoffel symbols are written in the same notation as tensors with index notation, they are not tensors,[3] since they do not transform like tensors under a change of coordinates; see below.

The Christoffel symbols are most typically defined in a coordinate basis, which is the convention followed here. However, the Christoffel symbols can also be defined in an arbitrary basis of tangent vectors ei by

Explicitly, in terms of the metric tensor, this is[2]

where are the commutation coefficients of the basis; that is,

where ek 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 expressions below are valid only in a coordinate basis, unless otherwise noted.

Christoffel symbols of the second kind (asymmetric definition)[]

A different definition of Christoffel symbols of the second kind is Misner et al.'s 1973 definition, which is asymmetric in i and j:[2]

Relationship to index-free notation[]

Let X and Y be vector fields with components and . Then the kth component of the covariant derivative of Y with respect to X is given by

Here, the Einstein notation is used, so repeated indices indicate summation over indices and contraction with the metric tensor serves to raise and lower indices:

Keep in mind that and that , the Kronecker delta. The convention is that the metric tensor is the one with the lower indices; the correct way to obtain from is to solve the linear equations .

The statement that the connection is torsion-free, namely that

is equivalent to the statement that the Christoffel symbol is symmetric in the lower two indices:

The index-less transformation properties of a tensor are given by pullbacks for covariant indices, and pushforwards for contravariant indices. The article on covariant derivatives provides additional discussion of the correspondence between index-free and indexed notation.

Covariant derivatives of tensors[]

The covariant derivative of a vector field is

The covariant derivative of a scalar field is just

and the covariant derivative of a covector field is

The symmetry of the Christoffel symbol now implies

for any scalar field, but in general the covariant derivatives of higher order tensor fields do not commute (see curvature tensor).

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

that is,

If the tensor field is mixed then its covariant derivative is

and if the tensor field is of type (0,2) then its covariant derivative is

Change of variable[]

Under a change of variable from to , vectors transform as

and so

where the overline denotes the Christoffel symbols in the y coordinate system. Note that the Christoffel symbol does not transform as a tensor, but rather as an object in the jet bundle.

In fact, at each point, there exist coordinate systems in which the Christoffel symbols vanish at the point.[4] These are called (geodesic) normal coordinates, and are often used in Riemannian geometry.

Applications to general relativity[]

The Christoffel symbols find frequent use in Einstein's theory of general relativity, where spacetime is represented by a curved 4-dimensional Lorentz manifold with a Levi-Civita connection. The Einstein field equations—which determine the geometry of spacetime in the presence of matter—contain the Ricci tensor, and so calculating the Christoffel symbols is essential. Once the geometry is determined, the paths of particles and light beams are calculated by solving the geodesic equations in which the Christoffel symbols explicitly appear.

Christoffel symbols, covariant derivative[]

In a smooth coordinate chart, the Christoffel symbols of the first kind are given by

and the Christoffel symbols of the second kind by

Here is the inverse matrix to the metric tensor . In other words,

and thus

is the dimension of the manifold.

Christoffel symbols satisfy the symmetry relations

or, respectively, ,

the second of which is equivalent to the torsion-freeness of the Levi-Civita connection.

The contracting relations on the Christoffel symbols are given by


where |g| is the absolute value of the determinant of the metric tensor . These are useful when dealing with divergences and Laplacians (see below).

The covariant derivative of a vector field with components is given by:

and similarly the covariant derivative of a -tensor field with components is given by:

For a -tensor field with components this becomes

and likewise for tensors with more indices.

The covariant derivative of a function (scalar) is just its usual differential:

Because the Levi-Civita connection is metric-compatible, the covariant derivatives of metrics vanish,

as well as the covariant derivatives of the metric's determinant (and volume element)

The geodesic starting at the origin with initial speed has Taylor expansion in the chart:


  1. See, for instance, Πρότυπο:Harv and Πρότυπο:Harv
  2. 2,0 2,1 2,2
  3. See, for example, Πρότυπο:Harv, page 141
  4. This is assuming that the connection is symmetric (e.g., the Levi-Civita connection). If the connection has torsion, then only the symmetric part of the Christoffel symbol can be made to vanish.

Εσωτερική Αρθρογραφία[]



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