Properties of the Lie derivative

1. It is a linear operator

   Given two generic tensors ${\boldsymbol{\it Y}}$, and ${\boldsymbol{\it Z}}$, the

   $${\mathcal{L}}_{_{\bf V}}(\tau Y^{\alpha \nu}+\sigma Z^{\beta \nu}... ...l{L}}_{_{\bf V}}Y^{\alpha \nu}+ \sigma{\mathcal{L}}_{_{\bf V}}Z^{\beta \nu} .$$ (53)

   
   
   where $\tau$ and $\sigma$ are two constant real coefficients.

2. It follows the Leibniz rule

   Given two generic tensors ${\boldsymbol{\it Z}}$, and ${\boldsymbol{\it U}}$, then

   $${\mathcal{L}}_{_{\bf V}}(Z^{\mu \nu} U_{\alpha\beta})={\mathcal{L... ...mu \nu})U_{\alpha\beta}+ Z^{\mu \nu}{\mathcal{L}}_{_{\bf V}}U_{\alpha\beta} .$$ (54)

   
   

3. It is ``type-preserving''

   Given a generic $${\left (\begin{array}{c} m \ n \\ \end{array} \right) }$$-form, its Lie derivative is still a $${\left (\begin{array}{c} m \ n \\ \end{array} \right) }$$-form

   $${\mathcal{L}}_{_{\bf V}} \left [ \left ( \begin{array}{c} m \ n ... ...ight ]= \left ( \begin{array}{c} m \ n \ \end{array} \right) \mathrm{form} .$$ (55)

   
   
   .

4. Directional derivative

   When acted upon a scalar function $\Phi(x^{\alpha })$, the Lie derivative provides a directional derivative

   $${\mathcal{L}}_{_{\bf V}} \Phi= \Phi_{,\nu} V^\nu .$$ (56)

   
   

5. It commutes with contraction

   Given a generic tensor ${\boldsymbol {\it T}}$ then

   $$\delta^\nu_{ \mu}({\mathcal{L}}_{_{\bf V}}T^\mu_{  \nu})= {\mathc... ... V}}(\delta^\nu_{ \mu}T^\mu_{  \nu})= {\mathcal{L}}_{_{\bf V}}T^\nu_{  \nu} .$$ (57)

   
   

6. It is equivalent to the commutator operator

   Given two generic tensors ${\boldsymbol{\it Y}}$, and ${\boldsymbol{\it X}}$ (hereafter just two 4-vectors), then

   $${\mathcal{L}}_{\bf {X}} Y^\mu=Y^\mu_{  ,\nu}X^\nu-X^\mu_{  ,\nu}Y^\nu = [\bf {X},\bf {Y}]^\mu .$$ (58)