Christoffel symbols

Christoffel symbolChristoffel coefficientsChristoffel connectionChristoffel symbol of the second kindconnection coefficientsRicci rotation coefficientsecond kind
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.wikipedia
164 Related Articles

Metric connection

Riemannian connectioncompatibility with the metriccompatible
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.
Following standard practice, one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second Bianchi identity.

Affine connection

connectionaffineaffine connections
The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface.
Correction terms were introduced by Elwin Bruno Christoffel (following ideas of Bernhard Riemann) in the 1870s so that the (corrected) derivative of one vector field along another transformed covariantly under coordinate transformations — these correction terms subsequently came to be known as Christoffel symbols.

Covariant derivative

covariant differentiationtensor derivativecovariant differential
In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric.
Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) observed that the Christoffel symbols used to define the curvature could also provide a notion of differentiation which generalized the classical directional derivative of vector fields on a manifold.

Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
In general, there are an infinite number of metric connections for a given metric tensor; however, there is a unique connection that is free of torsion, the Levi-Civita connection.
The components of this connection with respect to a system of local coordinates are called Christoffel symbols.

Geodesic

geodesicsgeodesic flowgeodesic equation
In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric.
:where are the Christoffel symbols of the metric.

General relativity

general theory of relativitygeneral relativity theoryrelativity
It is common in physics and general relativity to work almost exclusively with the Levi-Civita connection, by working in coordinate frames (called holonomic coordinates) where the torsion vanishes.
:where s is a scalar parameter of motion (e.g. the proper time), and are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) which is symmetric in the two lower indices.

Torsion tensor

torsiontorsion-freetorsion form
In general, there are an infinite number of metric connections for a given metric tensor; however, there is a unique connection that is free of torsion, the Levi-Civita connection.
Here are the Christoffel symbols defining the connection.

Curvilinear coordinates

curvilinearcurvilinear coordinate systemcurvilinear coordinate
For example, in Euclidean spaces, the Christoffel symbols describe how the local coordinate bases change from point to point.
;Christoffel symbols of the first kind:

Elwin Bruno Christoffel

ChristoffelElwin ChristoffelChristoffel, Elwin Bruno
The Christoffel symbols are named for Elwin Bruno Christoffel (1829–1900).
In the same paper he introduced the Christoffel symbols and which express the components of the Levi-Civita connection with respect to a system of local coordinates.

Normal coordinates

geodesic normal coordinatesnormal coordinate systemnormal ball
These are called (geodesic) normal coordinates, and are often used in Riemannian geometry.
In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations.

Geodesics in general relativity

geodesicgeodesicsnull geodesic
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.
:where s is a scalar parameter of motion (e.g. the proper time), and are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) symmetric in the two lower indices.

Proofs involving covariant derivatives

proofProofs involving Christoffel symbols
This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols.

Ricci calculus

tensor index notationabsolute differential calculusantisymmetrization of indices
Christoffel symbols of the first kind can then be found via index lowering:
is a Christoffel symbol of the second kind.

Contorsion tensor

contorsion
The difference between the connection in such a frame, and the Levi-Civita connection is known as the contorsion tensor.
In metric geometry, the contorsion tensor expresses the difference between a metric-compatible affine connection with Christoffel symbol and the unique torsion-free Levi-Civita connection for the same metric.

List of formulas in Riemannian geometry

formulas in Riemannian geometryreturn to article
In a smooth coordinate chart, the Christoffel symbols of the first kind are given by

Ricci curvature

Ricci tensorRicci curvature tensorTrace-free Ricci tensor
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.
In terms of the Riemann curvature tensor and the Christoffel symbols, one has

Tensor

tensorsorderclassical treatment of tensors
Under linear coordinate transformations on the manifold, the Christoffel symbols transform like the components of a tensor, but under general coordinate transformations (diffeomorphisms) they do not.
The Christoffel symbols also belong to the holors.

Lie derivative

Lie bracketLie commutatorcommuting vector fields
is the Lie bracket.
Alternatively, if we are using a torsion-free connection (e.g., the Levi Civita connection), then the partial derivative \partial_a can be replaced with the covariant derivative which means replacing with (by abuse of notation) where the are the Christoffel coefficients.

Deriving the Schwarzschild solution

Example computation of Christoffel symbolshereSchwarzschild solution
Using the metric above, we find the Christoffel symbols, where the indices are.

Introduction to the mathematics of general relativity

Basic introduction to the mathematics of curved spacetimenon-Euclidean geometry of curved space-time
For example, Christoffel symbols cannot be tensors themselves if the coordinates don't change in a linear way.

Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives.
The above expression can be written using Christoffel symbols:

Gauss–Codazzi equations

Gauss-Codazzi equationsGauss–Codazzi equationGauss–Codazzi equations (relativity)
It is possible to express the second partial derivatives of r using the Christoffel symbols and the second fundamental form.

Mathematics

mathematicalmathmathematician
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.

Physics

physicistphysicalphysicists
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.