Useful formulae

Difficulty level:   ★ ★ ★

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

Lorentz factor \gamma := (1-V^2)^{-1/2}, where V := dx/dt:

  •     \[\gamma^2-1 = V^2\gamma^2\]

  •     \[\frac{d\gamma}{dt} = V\gamma^3\frac{dV}{dt}\]

Connection coefficients: (MTW §8.5, 10.4)

  •     \[\nabla_\beta\mathbf e_\alpha = \mathbf e_\mu \Gamma^\mu_{\alpha\beta}\]

  • 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 \lambdaaffine: (MTW §8.5)

  •     \[\frac{d^2x^\mu}{d\lambda^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\lambda}\frac{dx^\beta}{d\lambda} = 0\]

Covariant derivative. Suppose f is a scalar, \mathbf u a vector, and \mathbf v and \mathbf w vector fields. Then: (MTW §8.5, 8.7, 10.3)

  •     \[\nabla_{\mathbf u}(\mathbf v+\mathbf w) = \nabla_{\mathbf u}\mathbf v + \nabla_{\mathbf u}\mathbf w\]

  •     \[\nabla_{f\mathbf u}\mathbf v = f\nabla_{\mathbf u}\mathbf v\]

  •     \[\nabla_{\mathbf u}(f\mathbf v) = f\nabla_{\mathbf u}\mathbf v + (\nabla_{\mathbf u}f)\mathbf v\]

  •     \[\nabla f = df\]

  •     \[\nabla_{\mathbf u}f = df(\mathbf u) = u^\alpha f_{,\alpha}\]

If \mathbf u is a 4-velocity, the last line is simply df/d\tau (c.f. Schutz §3.3)