Covariant and Lie Derivatives

Notation

g_{\mu\nu} is the metric, and \Gamma^\mu_{\alpha\beta} = \frac{1}{2} g^{\mu\nu} ( \partial_\alpha g_{\beta\nu} + \partial_\beta g_{\alpha\nu} - \partial_\nu g_{\alpha\beta}) are the Christoffel symbols.

\nabla_\mu is the covariant derivative, and \partial_\mu is the partial derivative with respect to x^\mu.

S is a scalar, V^\mu is a contravariant vector, and W_\mu is a covariant vector.

\rho is a scalar density of weight 1, and \rho^w is a scalar density of weight w. (Note that \sqrt{-g} is a density of weight 1, where g is the determinant of the metric. It follows that \rho/\sqrt{-g} is a scalar.)

Covariant Derivatives

Covariant derivatives are defined by

(1)   \begin{eqnarray*}   \nabla_\alpha S & = & \partial_\alpha S \\   \nabla_\alpha V^\mu & = & \partial_\alpha V^\mu + \Gamma^\mu_{\alpha\nu} V^\nu \\   \nabla_\alpha W_\mu & = & \partial_\alpha W_\mu - \Gamma^\nu_{\alpha\mu} W_\nu \\   \nabla_\alpha \rho & = & \partial_\alpha \rho - \rho \Gamma^\beta_{\alpha\beta}       = \sqrt{-g} \,\partial_\alpha (\rho/\sqrt{-g}) \\   \nabla_\alpha \rho^w & = & \partial_\alpha \rho^w - w \rho^w \Gamma^\beta_{\alpha\beta}       = \sqrt{-g}^w \,\partial_\alpha (\rho^w/\sqrt{-g}^w)  \end{eqnarray*}

The extension to tensors of different types is straightforward. Note the useful result for the divergence of a vector field:

    \[ \sqrt{-g} \nabla_\alpha V^\alpha = \partial_\alpha (\sqrt{-g} V^\alpha) \]

Lie Derivatives

Lie derivatives with respect to a vector field \beta^\mu are defined by

(2)   \begin{eqnarray*} {\mathcal L}_\beta S & = & \beta^\alpha \partial_\alpha S \\ {\mathcal L}_\beta V^\mu & = & \beta^\alpha \partial_\alpha V^\mu - V^\alpha \partial_\alpha \beta^\mu \\ {\mathcal L}_\beta W_\mu & = & \beta^\alpha \partial_\alpha W_\mu + W_\alpha \partial_\mu \beta^\alpha \\ {\mathcal L}_\beta \rho & = & \partial_\alpha(\rho \beta^\alpha) \\ {\mathcal L}_\beta \rho^w & = & \beta^\alpha \partial_\alpha \rho^w + w\rho^w \partial_\alpha \beta^\alpha \end{eqnarray*}

The extension to tensors of different type is straightforward. Lie derivatives do not rely on the presence of a metric for their definitions. When a metric is present, they can be written in terms of covariant derivatives by simply replacing \partial with \nabla in the formulas above. Note the useful result for the Lie derivative of the metric tensor:

    \[ {\mathcal L}_\beta g_{\mu\nu} = \nabla_\mu \beta_\nu + \nabla_\nu \beta_\mu \]

A Killing vector field K^\mu satisfies {\mathcal L}_K g_{\mu\nu} = 2\nabla_{(\mu}K_{\nu)} = 0.