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In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. [1] 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.
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
In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold , one can additionally arrange that the metric tensor is the Kronecker delta at the point p , and that the first ...
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel.Levi-Civita, [1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols [2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
On an n-dimensional Riemannian manifold, the geodesic equation written in a coordinate chart with coordinates is: + = where the coordinates x a (s) are regarded as the coordinates of a curve γ(s) in and are the Christoffel symbols.
[citation needed] 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.
In other words, the basis vectors of the coordinates may vary in time at fixed positions, or they may vary with position at fixed times, or both. When equations of motion are expressed in terms of any non-inertial coordinate system (in this sense), extra terms appear, called Christoffel symbols.
This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are derived from the Levi-Civita connection, which is the unique metric compatible, torsion-free connection on a Riemannian Manifold. In general, there is no restriction: the spin connection may also contain torsion.