Derivative Rules

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Notation

g_{ab} is the metric, and \Gamma^a_{bc} = \frac{1}{2} g^{ad} ( \partial_b g_{dc} + \partial_c g_{bd} - \partial_d g_{bc}) are the Christoffel symbols.

\nabla_a is the covariant derivative, and \partial_a is the partial derivative with respect to x^a.

S is a scalar.

V^c is a contravariant vector.

W_c 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


\begin{array}{rcl}
   \nabla_a S & = & \partial_a S \\
   \nabla_a V^c & = & \partial_a V^c + \Gamma^c_{ab} V^b \\
   \nabla_a W_c & = & \partial_a W_c - \Gamma^b_{ac} W_b \\
   \nabla_a \rho & = & \partial_a \rho - \rho \Gamma^c_{ac} \\
     & = & \sqrt{|g|} \,\partial_a (\rho/\sqrt{|g|}) \\
   \nabla_a \rho^w & = & \partial_a \rho^w - w \rho^w \Gamma^c_{ac} \\
     & = & \sqrt{|g|}^w \,\partial_a (\rho^w/\sqrt{|g|}^w) 
\end{array}

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

\sqrt{|g|} \nabla_a V^a = \partial_a (\sqrt{|g|} V^a)

Lie Derivatives

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


\begin{array}{rcl}
   {\mathcal L}_\beta S & = & \beta^a \partial_a S \\
   {\mathcal L}_\beta V^c & = & \beta^a \partial_a V^c - V^a \partial_a \beta^c \\
   {\mathcal L}_\beta W_c & = & \beta^a \partial_a W_c + W_a \partial_c \beta^a \\
   {\mathcal L}_\beta \rho & = & \partial_a (\rho \beta^a) \\
   {\mathcal L}_\beta \rho^w & = & \beta^a \partial_a\rho^w + w \rho^w \partial_a\beta^a  \\
\end{array}

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 as follows:


\begin{array}{rcl}
   {\mathcal L}_\beta S  & = & \beta^a \nabla_a S  \\
   {\mathcal L}_\beta V^c & = & \beta^a \nabla_a V^c - V^a \nabla_a \beta^c  \\
   {\mathcal L}_\beta W_c & = & \beta^a \nabla_a W_c + W_a \nabla_c \beta^a  \\
   {\mathcal L}_\beta \rho & = &  \nabla_a (\rho\beta^a)  \\
   {\mathcal L}_\beta \rho^w & = & \beta^a \nabla_a\rho^w + w \rho^w \nabla_a\beta^a  
\end{array}

Note the useful result for the Lie derivative of the metric tensor:

{\mathcal L}_\beta  g_{ab} = \nabla_a\beta_b + \nabla_b \beta_a
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