The LieBracket[mu] command computes the Lie bracket of the vector fields $U^{\mathrm{\mu }}$ and $V^{\mathrm{\nu }}$, defined in terms of the LieDerivative as

where Einstein's summation convention is used, $\mathrm{\mu }$ and $\mathrm{\nu }$ represent spacetime indices, $ℒ$ is the LieDerivative and ${▿}_{\mathrm{\mu }}$ is the covariant derivative operator D_. The LieBracket satisfies the Jacobi identity for commutators

When the spacetime is Galilean, so all the Christoffel symbols are zero, the operator d_ is used instead of the covariant D_. Also, when the spacetime is non-Galilean, due to the symmetry of the Christoffel symbols under permutation of their 2nd and 3rd indices, all the terms involving Christoffel symbols cancel so that a mathematically equivalent result can be obtained replacing D_ by d_. To obtain a result directly expressed using d_, pass d_ as the last argument.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Coordinates}\left(X=\mathrm{spherical}\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\left\{X\right\}$

$\mathrm{Define}\left(A\,B\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A\,B\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

$\mathrm{PDEtools}:-\mathrm{declare}\left(\left(A\,B\right)\left(X\right)\right)$

$A\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\mathrm{will\; now\; be\; displayed\; as}A$

$B\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\mathrm{will\; now\; be\; displayed\; as}B$

$\mathrm{LieBracket}\left(A\left[\mathrm{~mu}\right]\left(X\right)\,B\left[\mathrm{~nu}\right]\left(X\right)\right)$

$A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\partial }_{\mathrm{\mu }}\left(B_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)-B_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\partial }_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)$

$\mathrm{g\_}\left[\mathrm{sc}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{space}\mathrm{indices}$

$\mathrm{The\; Schwarzschild\; metric\; in\; coordinates}\left[r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]$

$\mathrm{Parameters:}\left[m\right]$

$\mathrm{Signature:}\left(\mathrm{-\; -\; -\; +}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\frac{r}{2m-r}& 0& 0& 0\\ 0& -r^2& 0& 0\\ 0& 0& -r^2{\sin \left(\mathrm{\theta }\right)}^2& 0\\ 0& 0& 0& \frac{r-2m}{r}\end{array}\right]$

$\mathrm{LieBracket}\left(A\left[\mathrm{~mu}\right]\left(X\right)\,B\left[\mathrm{~nu}\right]\left(X\right)\right)$

$A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left(B_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)-B_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)$

$\mathrm{LieDerivative}\left[A\left[\mathrm{~mu}\right]\left(X\right)\right]\left(B\left[\mathrm{~nu}\right]\left(X\right)\right)$

$A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left(B_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)-B_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)$

$\mathrm{convert}\left(\,\mathrm{d\_}\right)$

$A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left({\partial }_{\mathrm{\mu }}\left(B_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\mu }}}{B}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)-B_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left({\partial }_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\mu }}}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)$

$\mathrm{Simplify}\left(\right)$

$-{\partial }_{\mathrm{\alpha }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)B_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}+{\partial }_{\mathrm{\alpha }}\left(B_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)A_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}$

$\mathrm{Simplify}\left(-\right)$

$0$

Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

Weinberg, S. Gravitation and Cosmology: Principles and Applications of The General Theory of Relativity, John Wiley & Sons, Inc, 1972.