# Useful formulae

Difficulty level:   ★ ★ ★

This is a brief reference list of some useful and very standard formulae.

Lorentz factor , where :

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Connection coefficients: (MTW §8.5, 10.4)

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• In a coordinate basis, they are called Christoffel symbols. In a non-coordinate basis, they include nonzero “commutation coefficients” (MTW §8.4)

Geodesic equation for parameter affine: (MTW §8.5)

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Covariant derivative. Suppose is a scalar, a vector, and and vector fields. Then: (MTW §8.5, 8.7, 10.3)

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If is a 4-velocity, the last line is simply (c.f. Schutz §3.3)