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Formula Christoffel Symbols (without Torsion)

Formula: Christoffel Symbols (without Torsion)

Christoffel symbols

Christoffel symbols are used to extend a partial derivative (on a flat manifold) to a covariant derivative (on a curved manifold). The indices \( \class{red}{c}\), \(\class{blue}{a}\), \( \class{green}{b} \) and \(s\) take the values between 0 and 3 in the four-dimensional case (time + 3d space).

Here \(s\) is a summation index. Einstein summation convention is used here.

Metric tensor

The metric tensor determines the distances and angles in a curved space. In a chosen basis and in the four-dimensional case, the metric tensor is a symmetric 4x4 matrix.

Inverse metric tensor

Contravariant metric tensor is the inverse of the metric tensor. In a basis it is the inverse matrix.