The Christoffel symbols are tensor-like objects derived from a Riemannian metric . They are used to study the geometry of the metric and appear, for example, in the geodesic equation. There are two closely related kinds of Christoffel symbols, the first kind , and the second kind . Christoffel symbols of the second kind are also known as affine connections (Weinberg 1972, p. 71) or connection coefficients (Misner et al. 1973, p. 210).

It is always possible to pick a coordinate system on a Riemannian manifold such that the Christoffel symbol vanishes at a chosen point. In general relativity, Christoffel symbols are "gravitational forces," and the preferred coordinate system referred to above would be one attached to a body in free fall.

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by Eugene Salamin

Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco, CA: W. H. Freeman, 1973.

Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972.

CITE THIS AS:

Rowland, Todd; Salamin, Eugene; and Weisstein, Eric W. "Christoffel Symbol." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ChristoffelSymbol.html