This is a brief reference list of some useful and very standard formulae.
Lorentz factor , where :
Connection coefficients: (MTW §8.5, 10.4)
- 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)
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 is simply (c.f. Schutz §3.3)