Compute the Lie bracket of two vector fields using algebraic tensor notation - Maple Programming Help

Physics[LieBracket] - Compute the Lie bracket of two vector fields using algebraic tensor notation

 Calling Sequence LieBracket(U, V, ...)

Parameters

 U, V - two contravariant vectors, as tensors functions with one free spacetime contravariant index (prefixed by ~); the first vector can also be passed without its index ... - optional, a third argument can be the non-covariant operator d_ to be used instead of the covariant D_

Description

 • 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

$\mathrm{LieBracket}\left({U}_{}^{\mu },{V}_{}^{\nu }\right)={ℒ}_{U}\left({V}_{}^{\nu }\right)={U}_{}^{\mu }{𝒟}_{\mu }\left({V}_{}^{\nu }\right)-{V}_{}^{\mu }{𝒟}_{\mu }\left({U}_{}^{\nu }\right)$

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

$\mathrm{LieBracket}\left({A}_{}^{\mu },\mathrm{LieBracket}\left({B}_{}^{\nu },{C}_{}^{\rho }\right)\right)+\mathrm{LieBracket}\left({B}_{}^{\mu },\mathrm{LieBracket}\left({C}_{}^{\nu },{A}_{}^{\rho }\right)\right)+\mathrm{LieBracket}\left({C}_{}^{\mu },\mathrm{LieBracket}\left({A}_{}^{\nu },{B}_{}^{\rho }\right)\right)=0$

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (1)

Set a system of coordinates - say X

 > $\mathrm{Coordinates}\left(X=\mathrm{spherical}\right)$
 ${\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}}\left\{{X}{=}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right)\right\}$
 ${\mathrm{Systems of spacetime Coordinates are:}}\left\{{X}{=}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right)\right\}$
 $\left\{{X}\right\}$ (2)

Define two tensors for experimentation

 > $\mathrm{Define}\left(A,B\right)$
 ${\mathrm{Defined objects with tensor properties}}$
 $\left\{{A}{,}{B}{,}{{\mathrm{\gamma }}}_{{\mathrm{\mu }}}{,}{{\mathrm{\sigma }}}_{{\mathrm{\mu }}}{,}{{X}}_{{\mathrm{\mu }}}{,}{{\partial }}_{{\mathrm{\mu }}}{,}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{\delta }}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{\epsilon }}}_{{\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\mu }}{,}{\mathrm{\nu }}}\right\}$ (3)

Use the declare facility of PDEtools to avoid redundant display of functionality

 > $\mathrm{PDEtools}:-\mathrm{declare}\left(\left(A,B\right)\left(X\right)\right)$
 ${A}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right){}{\mathrm{will now be displayed as}}{}{A}$
 ${B}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right){}{\mathrm{will now be displayed as}}{}{B}$ (4)

Compute the Lie bracket of ${A}^{\mu }$ and ${B}^{\nu }$

 > $\mathrm{LieBracket}\left(A[\mathrm{~mu}]\left(X\right),B[\mathrm{~nu}]\left(X\right)\right)$
 ${{A}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{\partial }}_{{\mathrm{\mu }}}{}\left({{B}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)\right){-}{{B}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{\partial }}_{{\mathrm{\mu }}}{}\left({{A}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)\right)$ (5)

Because the spacetime at this point in the worksheet is flat, the output above involves d_, not the covariant D_. Set the spacetime to any nongalilean value, for instance (see g_):

 > ${\mathrm{g_}}_{\mathrm{sc}}$
 ${\mathrm{The Schwarzschild metric in coordinates}}\left[{r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right]$
 ${\mathrm{Parameters:}}\left[{m}\right]$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\left[\begin{array}{cccc}\frac{{r}}{{-}{r}{+}{2}{}{m}}& {0}& {0}& {0}\\ {0}& {-}{{r}}^{{2}}& {0}& {0}\\ {0}& {0}& {-}{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}& {0}\\ {0}& {0}& {0}& \frac{{r}{-}{2}{}{m}}{{r}}\end{array}\right]$ (6)

Compute the Lie bracket again

 > $\mathrm{LieBracket}\left(A[\mathrm{~mu}]\left(X\right),B[\mathrm{~nu}]\left(X\right)\right)$
 ${{A}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{B}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)\right){-}{{B}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{A}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)\right)$ (7)

Note that the Lie bracket between two vector fields is equal to the LieDerivative of the second one with respect to the first one, that you can pass with or without its index to both LieBracket and LieDerivative:

 > $\mathrm{LieDerivative}[A[\mathrm{~mu}]\left(X\right)]\left(B[\mathrm{~nu}]\left(X\right)\right)$
 ${{A}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{B}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)\right){-}{{B}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{A}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)\right)$ (8)

Due to the symmetry properties of Christoffel symbols, this expression can also be expressed replacing the covariant derivative operator D_ by d_. To see that, rewrite the covariant derivatives in terms of Christoffel symbols and the d_ operator

 > $\mathrm{convert}\left(,\mathrm{d_}\right)$
 ${{A}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{\partial }}_{{\mathrm{\mu }}}{}\left({{B}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right){+}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}{\mathrm{\beta }}{,}{\mathrm{\mu }}}^{\phantom{{}}{\mathrm{\nu }}\phantom{{\mathrm{\beta }}}\phantom{{,}}\phantom{{\mathrm{\mu }}}}{}{{B}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}\right){-}{{B}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{\partial }}_{{\mathrm{\mu }}}{}\left({{A}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right){+}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}{\mathrm{\alpha }}{,}{\mathrm{\mu }}}^{\phantom{{}}{\mathrm{\nu }}\phantom{{\mathrm{\alpha }}}\phantom{{,}}\phantom{{\mathrm{\mu }}}}{}{{A}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)$ (9)
 > $\mathrm{Simplify}\left(\right)$
 ${-}{{B}}_{{\mathrm{\mu }}}{}\left({{\partial }}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{A}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)\right){+}{{A}}_{{\mathrm{\mu }}}{}\left({{\partial }}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{B}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)\right)$ (10)

This expression is the same one obtained above in (5) for a flat spacetime but for the dummy used for repeated index:

 > $\mathrm{Simplify}\left(-\right)$
 ${0}$ (11)
 >