In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. The Christoffel symbols may be used for performing practical calculations in differential geometry. Unfortunately, the calculations are usually quite lengthy and complex, and require careful attention to detail. By contrast, the index-less, formal notation for the Levi-Civita connection is terse, and allows theorems to be stated in an elegant way, but requires more advanced techniques for practical calculations. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space and time. ... Elwin Bruno Christoffel - Wikipedia /**/ @import /skins-1. ... Johann Wolfgang von Goethe 1829 was a common year starting on Thursday (see link for calendar). ... 1900 (MCM) was an exceptional common year starting on Monday of the Gregorian calendar, but a leap year starting on Saturday of the Julian calendar. ... In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

Preliminaries

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. In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... It has been suggested that this article or section be merged into Covariant transformation. ... It has been suggested that this article or section be merged into Covariant transformation. ... In some physics textbooks and articles, certain quantities are defined with the opposite sign from that which is used in other publications. ...

Definition

The Christoffel symbols can be derived from the vanishing of the covariant derivative of the metric tensor : In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

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 Nabla is a symbol, shown as . ...

By permuting the indices, and resumming, one can solve explicitly for the Christoffel symbols:

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. Indeed, they do not transform like tensors under a change of coordinates; see below. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ... The following is a component-based classical treatment of tensors. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

NB. Note that most authors choose to define the Christoffel symbols in a holonomic coordinate basis, which is the convention followed here. In anholonomic coordinates, the Christoffel symbols take the more complex form In mathematics, the term holonomic may occur with several different meanings. ...

where are the commutation coefficients of the basis; that is, In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...

where e_k are the basis vectors and is the Lie bracket. An example of an anholonomic basis with non-vanishing commutation coefficients are the unit vectors in spherical and cylindrical coordinates. In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... // Vector fields in cylindrical coordinates Vectors are defined in cylindrical coordinates by (ρ,φ,z), where ρ is the length of the vector projected onto the X-Y-plane, φ is the angle of the projected vector with the positive X-axis (0 ≤ φ < 2π), z is the regular z-coordinate. ...

The expressions below are valid only in a holonomic basis, unless otherwise noted.

Relationship to index-less 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 Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ...

Some older physics books occasionally write dx in place of X, and place it after the equation, rather than before. 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: In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ...

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 . In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

The statement that the connection is torsion-free, namely that // Mathmatics In mathematics, the term torsion has several meanings, mostly unrelated to each other. ...

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. This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, the push forward (or pushforward) of a smooth map F : M &#8594; N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...

Covariant derivatives of tensors

The covariant derivative of a vector field is In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...

The covariant derivative of a scalar field is just

and the covariant derivative of a covector field is A one-form, also called a covector, is a linear function which maps each vector in a vector space to a real number, such that the mapping is invariant with respect to coordinate transformations of the vector space. ...

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). In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

The covariant derivative of a type (2,0) tensor field is In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

that is,

If the tensor field is mixed then its covariant derivative is In tensor analysis, a mixed tensor is a tensor which is neither covariant nor contravariant. ...

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 frame. Note that the Christoffel symbol does not transform as a tensor, but rather as an object in the jet bundle. In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. ...

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. General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... In physics, spacetime is a mathematical model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ... In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. ... In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... This article or section is in need of attention from an expert on the subject. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... This article is in need of attention from an expert on the subject. ...

See also

• Basic introduction to the mathematics of curved spacetime

This article is on the minimal body of mathematics necessary to understand general relativity. ...

References

• Lev Davidovich Landau and Evgeny Mikhailovich Lifshitz, The Classical Theory of Fields, Fourth Revised English Edition, Course of Theoretical Physics, Volume 2, (1951) Pergamon Press, Oxford; ISBN 0-08-025072-6. See chapter 10, paragraphs 85,86 and 87.
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin/Cummings Publishing, London; ISBN 0-8053-0102-X. See chapter 2, paragraph 2.7.1
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. See chapter 8, paragraph 8.5