Christoffel Symbols

In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. In a broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are also called Christoffel symbols. The Christoffel symbols may be used for performing practical calculations in differential geometry. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives.

At each point of the underlying n-dimensional manifold, for any local coordinate system, the Christoffel symbol is an array with three dimensions: n × n × n. Each of the n3 components is a real number.

Under linear coordinate transformations on the manifold, it behaves like a tensor, but under general coordinate transformations, it does not. In many practical problems, most components of the Christoffel symbols are equal to zero, provided the coordinate system and the metric tensor possess some common symmetries.

In general relativity, the Christoffel symbol plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor.