# Useful formulae

Difficulty level:   ★ ★ ★

This is a brief reference list of some formulae I have found useful.

Lorentz factor: , where is the magnitude of 3-velocity in geometric units (c=1):

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Connection coefficients: (MTW §8.5, 10.4)

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

Geodesic equation for affine parameter : (MTW §8.5)

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If the parameter is not affine, a more general formula is needed.

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 equates to (c.f. Schutz §3.3)

Coordinate basis and dual basis: These are dual as bases (Schutz §3.3):

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but not dual as individual vectors, i.e. and are not dual in general, instead:

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