Кристоффель символдары: метрикалық байланыс және геометриялық қолданыстары

Array of numbers describing a metric connection

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. 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. 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. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group O(p, q). As a result, such a manifold is necessarily a (pseudo )Riemannian manifold. The Christoffel symbols provide a concrete representation of the connection of (pseudo )Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols. 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. 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. For example, in Euclidean spaces, the Christoffel symbols describe how the local coordinate bases change from point to point. At each point of the underlying n dimensional manifold, for any local coordinate system around that point, the Christoffel symbols are denoted Γ^(i)jk for 1=i, j, k = 1, 2, , n. Each entry of this n × n × n array is a real number. 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. Most of the algebraic properties of the Christoffel symbols follow from their relationship to the affine connection; only a few follow from the fact that the structure group is the orthogonal group O(m, n) (or the Lorentz group O(3, 1) for general relativity). Christoffel symbols are used for performing practical calculations. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In general relativity, the connection plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γ^(i)jk are zero. The Christoffel symbols are named for Elwin Bruno Christoffel (1829–1900).

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, with vectors indicated by bold font. The connection coefficients of the Levi Civita connection (or pseudo Riemannian connection) expressed in a coordinate basis are called Christoffel symbols.

The Christoffel symbols come in two forms: the first kind, and the second kind. The definition of the second kind is more basic, and thus is presented first.

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.

Given a spherical coordinate system, which describes points on the Earth surface (approximated as an ideal sphere). For a point x, R is the distance to the Earth core (usually approximately the Earth radius). θ and φ are the latitude and longitude. Positive θ is the northern hemisphere. 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). At any location, the tangent directions are (up), (north) and (east) you can also use indices 1,2,3. The related metric tensor has only diagonal elements (the squared vector lengths). This is an advantage of the coordinate system and not generally true.