Useful formulae

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

Lorentz factor: \gamma := (1-\beta^2)^{-1/2}, where \beta := dx/dt is the magnitude of 3-velocity in geometric units (c=1):

  •     \[\gamma^2-1 = \beta^2\gamma^2\]

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

Connection coefficients: (MTW §8.5, 10.4)

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

Recall \nabla_\beta := \nabla_{\mathbf e_\beta}. In a coordinate basis, the \Gamma are termed Christoffel symbols. In a non-coordinate basis, they include nonzero “commutation coefficients” (MTW §8.4)

Geodesic equation for affine parameter \lambda: (MTW §8.5)

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

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

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 equates to df/d\tau (c.f. Schutz §3.3)

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

  •     \[dx^\alpha(\partial_\beta}) = \delta^\alpha_\beta\]

but not dual as individual vectors, i.e. \partial_\alpha and dx^\alpha are not dual in general, instead:

  •     \[(dx^\alpha)^\sharp = g^{\mu\alpha}\partial_\mu\]

  •     \[(\partial_{x^\alpha})^\flat = g_{\mu\alpha}dx^\mu\]