In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.
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.
[5][6] The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold.
For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives.
Einstein summation convention is used in this article, with vectors indicated by bold font.
The full range of commonly used notation includes the use of arrows and boldface to denote vectors: where
Note the careful use of upper and lower indexes, to distinguish contravarient and covariant vectors.
In Euclidean space, the general definition given below for the Christoffel symbols of the second kind can be proven to be equivalent to:
and the fact that partial derivatives commute (as long as the manifold and coordinate system are well behaved).
The same numerical values for Christoffel symbols of the second kind also relate to derivatives of the dual basis, as seen in the expression:
In other words, the name Christoffel symbols is reserved only for coordinate (i.e., holonomic) frames.
However, the connection coefficients can also be defined in an arbitrary (i.e., nonholonomic) basis of tangent vectors ui by
The standard unit vectors in spherical and cylindrical coordinates furnish an example of a basis with non-vanishing commutation coefficients.
When we choose the basis Xi ≡ ui orthonormal: gab ≡ ηab = ⟨Xa, Xb⟩ then gmk,l ≡ ηmk,l = 0.
Applying the parallel transport rule for the two arbitrary vectors and relabelling dummy indices and collecting the coefficients of
This is same as the equation obtained by requiring the covariant derivative of the metric tensor to vanish in the General definition section.
is equivalent to the statement that—in a coordinate basis—the Christoffel symbol is symmetric in the lower two indices:
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.
, the potential function, exists then the contravariant components of the generalized force per unit mass are
When Cartesian coordinates can be adopted (as in inertial frames of reference), we have an Euclidean metrics, the Christoffel symbol vanishes, and the equation reduces to Newton's second law of motion.
Given a spherical coordinate system, which describes points on the Earth surface (approximated as an ideal sphere).
To simplify the derivatives, the angles are given in radians (where d sin(x)/dx = cos(x), the degree values introduce an additional factor of 360 / 2 pi).
The related metric tensor has only diagonal elements (the squared vector lengths).
) change, seen from an outside perspective (e.g. from space), but given in the tangent directions of the actual location (rows: R, θ, φ).
Cartesian points exist and Christoffel Symbols vanish as time passes, therefore, in cylindrical coordinates:
Christoffel symbols being calculated from the metric tensor, the equations can be derived and expressed from the principle of least action.
The Euler-Lagrange equation is applied to a functional related to the path of an object in a spherical coordinate system, Given
The differential equation provides the mathematical conditions that must be satisfied for this optimal path.