**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):

*Connection coefficients*: (MTW §8.5, 10.4)

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)

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)

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):

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