Compute the Lie derivative of a tensorial expression - Maple Help

Physics[LieDerivative] - Compute the Lie derivative of a tensorial expression

 Calling Sequence LieDerivative[v, ...](T) LieDerivative(T, v, ...)

Parameters

 T - an algebraic expression, or a relation, or a list, set, Matrix or Array of them v - a contravariant vector as a tensor or tensor function, passed with or without one free spacetime contravariant index (prefixed by ~), with respect to which the derivative is being taken ... - optional, more contravariant vectors as v to perform higher order differentiation ... - optional, the last argument can be the non-covariant operator d_ to be used instead of the covariant D_

Description

 • The LieDerivative[mu] command computes the Lie derivative of a tensorial expression T - say with one contravariant and one covariant spacetime indices as in ${T}_{b}^{a}$ - according to the standard definition

${ℒ}_{v}\left({T}_{b}^{a}\right)={v}_{}^{c}{𝒟}_{c}\left({T}_{b}^{a}\right)-{T}_{b}^{c}{𝒟}_{c}\left({v}_{}^{a}\right)+{T}_{c}^{a}{𝒟}_{b}\left({v}_{}^{c}\right)$

 where Einstein's summation convention is used, $a,b$ and $c$ represent spacetime indices, $v$ is a contravariant vector field, a tensor with one index, and $𝒟$ is the covariant derivative operator D_. When the expression $T$ has no free indices, the Lie derivative is equal to only the first term of the right-hand-side, and when $T$ has more than one contravariant or covariant tensor indices, there is a term like the second one and another like the third one respectively for each contravariant and covariant free indices in $T$.
 • From this definition it is clear that the LieDerivative is a tensor with regards to the free indices of the derivand, but not with regards to the free index of the indexing vector field (${v}^{c}$ in the above), whose index appears in the definition contracted with the differentiation operator D_ (or d_). In other words, the index $c$ of ${v}^{c}$ enters the Lie derivative as a dummy. If this dummy is found in the indices of the derivand, it is automatically replaced by another spacetime dummy index. You can also pass $V$ without an index, with our with functionality, as in $V$ or $V\left(X\right)$.
 • The vector ${v}^{c}$ indexing in LieDerivative[v[~c]](T) is expected to be defined as a tensor such using Define, and ~c (entered suffixed by ~) to be a spacetime contravariant index. Because this index is a dummy index, you can also pass $v$ without indexation, possibly as a function too, as in LieDerivative[v](T) (case of a constant $v$) or LieDerivative[v(x)](T).
 • The indexation of LieDerivative can also consists of a sequence of spacetime vectors like $v$, in which case a higher order LieDerivative is computed (orderly differentiating from left to right).
 • This single differentiating vector, or a sequence of them, can also be passed after the first argument, as in other Maple differentiation commands.
 • 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

In the examples that follow, as well as in the context of tensor computations with the Physics package, Einstein's summation convention for repeated indices is used.

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

Set a system of coordinates - say X

 > $\mathrm{Setup}\left(\mathrm{coordinates}=X\right)$
 ${\mathrm{* Partial match of \text{'}coordinates\text{'} against keyword \text{'}coordinatesystems\text{'}}}$
 ${\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}}\left\{{X}{=}\left({\mathrm{x1}}{,}{\mathrm{x2}}{,}{\mathrm{x3}}{,}{\mathrm{x4}}\right)\right\}$
 ${\mathrm{Systems of spacetime Coordinates are:}}\left\{{X}{=}\left({\mathrm{x1}}{,}{\mathrm{x2}}{,}{\mathrm{x3}}{,}{\mathrm{x4}}\right)\right\}$
 $\left[{\mathrm{coordinatesystems}}{=}\left\{{X}\right\}\right]$ (2)

Define some tensors for experimentation

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

Compute the Lie derivative of a scalar $f\left(X\right)$: it is the same as the directional derivative in the direction of ${v}^{\mu }$

 > $\mathrm{LieDerivative}[v[\mathrm{~mu}]\left(X\right)]\left(f\left(X\right)\right)$
 ${{v}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({X}\right){}\left({{\partial }}_{{\mathrm{\mu }}}{}\left({f}{}\left({X}\right)\right)\right)$ (4)

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_):

 > ${\mathrm{g_}}_{\mathrm{sc}}$
 ${\mathrm{Systems of spacetime Coordinates are:}}\left\{{X}{=}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right)\right\}$
 ${\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}}\left\{{X}{=}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right)\right\}$
 ${\mathrm{The Schwarzschild metric in coordinates}}\left[{r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right]$
 ${\mathrm{Parameters:}}\left[{m}\right]$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\left[\begin{array}{cccc}\frac{{r}}{{-}{r}{+}{2}{}{m}}& {0}& {0}& {0}\\ {0}& {-}{{r}}^{{2}}& {0}& {0}\\ {0}& {0}& {-}{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}& {0}\\ {0}& {0}& {0}& \frac{{r}{-}{2}{}{m}}{{r}}\end{array}\right]$ (5)

Use the declare facility of PDEtools to avoid redundant display of functionality and have derivatives displayed with compact indexed notation

 > $\mathrm{PDEtools}:-\mathrm{declare}\left(\left(v,\mathrm{ξ},T\right)\left(X\right)\right)$
 ${v}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right){}{\mathrm{will now be displayed as}}{}{v}$
 ${\mathrm{ξ}}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right){}{\mathrm{will now be displayed as}}{}{\mathrm{ξ}}$
 ${T}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}{,}{t}\right){}{\mathrm{will now be displayed as}}{}{T}$ (6)

The Lie derivatives of the covariant and contravariant vectors ${\mathrm{xi}}_{\mathrm{alpha}}$ and ${\xi }^{\alpha }$ differ in the sign of the second term, but not just in that: note the different ways in which the index $\mathrm{\mu }$ is contracted

 > $\mathrm{LieDerivative}[v[\mathrm{~mu}]\left(X\right)]\left(\mathrm{ξ}[\mathrm{α}]\left(X\right)\right)$
 ${{v}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{\mathrm{ξ}}}_{{\mathrm{\alpha }}}\right)\right){+}{{\mathrm{ξ}}}_{{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\alpha }}}{}\left({{v}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)\right)$ (7)
 > $\mathrm{LieDerivative}[v[\mathrm{~mu}]\left(X\right)]\left(\mathrm{ξ}[\mathrm{~alpha}]\left(X\right)\right)$
 ${{v}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)\right){-}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{v}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)\right)$ (8)

The index $\mathrm{\mu }$ in ${v}^{\mu }$ enters the Lie derivative as a dummy. If this dummy is found in the indices of the derivand, it is automatically replaced by another spacetime index. For example: pass $V$ and $\mathrm{xi}$ with the same index $\mathrm{alpha}$

 > $\mathrm{LieDerivative}[v[\mathrm{~alpha}]\left(X\right)]\left(\mathrm{ξ}[\mathrm{~alpha}]\left(X\right)\right)$
 ${{v}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)\right){-}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{v}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)\right)$ (9)

You can also pass $v$ without indices, with or without functionality depending on whether $v$ is constant, in which case only the part involving the Christoffel symbols remains after computing the covariant derivative:

 > $\mathrm{LieDerivative}[v\left(X\right)]\left(\mathrm{ξ}[\mathrm{~alpha}]\left(X\right)\right)$
 ${{v}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)\right){-}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{v}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)\right)$ (10)
 > $\mathrm{LieDerivative}[v]\left(\mathrm{ξ}[\mathrm{~alpha}]\left(X\right)\right)$
 ${{v}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)\right){-}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{\mathrm{\mu }}{,}{\mathrm{\nu }}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{\mathrm{\mu }}}\phantom{{,}}\phantom{{\mathrm{\nu }}}}{}{{v}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}$ (11)

When the derivand is a contravariant vector field, the Lie derivative is equal to the the LieBracket between the indexing vector field $v$ in the above) and the derivand ${\xi }^{\alpha }$. For that reason, you can also pass the first argument to LieBracket with or without the index

 > $\mathrm{LieBracket}\left(v[\mathrm{~mu}]\left(X\right),\mathrm{ξ}[\mathrm{~alpha}]\left(X\right)\right)$
 ${{v}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)\right){-}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{v}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)\right)$ (12)

So in LieBracket the free index of the first vector field is also a dummy and the index of the second one is the free index of the result.

The Lie derivative of a ${T}_{\beta }^{\alpha }$ contains three terms: first there is the term equivalent to a directional derivative, then another with a minus sign related to the contravariant index, then another one related to the covariant index

 > $\mathrm{LieDerivative}[v[\mathrm{~mu}]\left(X\right)]\left(T[\mathrm{~alpha},\mathrm{β}]\left(X\right)\right)$
 ${{v}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{T}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{\mathrm{\beta }}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{\mathrm{\beta }}}}\right)\right){-}{{T}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}{\mathrm{\beta }}}^{\phantom{{}}{\mathrm{\mu }}\phantom{{\mathrm{\beta }}}}{}\left({{𝒟}}_{{\mathrm{\mu }}}{}\left({{v}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)\right){+}{{T}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{\mathrm{\mu }}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{\mathrm{\mu }}}}{}\left({{𝒟}}_{{\mathrm{\beta }}}{}\left({{v}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)\right)$ (13)

The Lie derivative is essentially the change in form of the derivand under transformations generated by the indexing vector field ($v$ in the examples above). In turn the Killing vectors generate transformations that leave the form of the spacetime metric g_ invariant. So equating to zero the Lie Derivative of the metric (currently Schwarzschild) results in a system of partial differential equations defining the Killing vectors

 > $\mathrm{LieDerivative}[\mathrm{ξ}[\mathrm{~mu}]\left(X\right)]\left({\mathrm{g_}}_{\mathrm{α},\mathrm{β}}\right)=0$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\beta }}}{}\left({{𝒟}}_{{\mathrm{\alpha }}}{}\left({{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)\right){+}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\mu }}}{}\left({{𝒟}}_{{\mathrm{\beta }}}{}\left({{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)\right){=}{0}$ (14)
 > $\mathrm{Simplify}\left(\right)$
 ${{𝒟}}_{{\mathrm{\beta }}}{}\left({{\mathrm{ξ}}}_{{\mathrm{\alpha }}}\right){+}{{𝒟}}_{{\mathrm{\alpha }}}{}\left({{\mathrm{ξ}}}_{{\mathrm{\beta }}}\right){=}{0}$ (15)

The array of Killing equations behind this tensorial expression can be obtained with TensorArray, KillingVectors or the Library command TensorComponents:

 > $\mathrm{TensorArray}\left(\right)$
 $\left[\begin{array}{cccc}{2}{}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{r}}{-}\frac{{2}{}{m}{}{{\mathrm{ξ}}}_{{1}}}{{r}{}\left({-}{r}{+}{2}{}{m}\right)}{=}{0}& {\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\theta }}}{-}\frac{{2}{}{{\mathrm{ξ}}}_{{2}}}{{r}}{+}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{r}}{=}{0}& {\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\phi }}}{-}\frac{{2}{}{{\mathrm{ξ}}}_{{3}}}{{r}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{r}}{=}{0}& {\left({{\mathrm{ξ}}}_{{1}}\right)}_{{t}}{+}\frac{{2}{}{m}{}{{\mathrm{ξ}}}_{{4}}}{{r}{}\left({-}{r}{+}{2}{}{m}\right)}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{r}}{=}{0}\\ {\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\theta }}}{-}\frac{{2}{}{{\mathrm{ξ}}}_{{2}}}{{r}}{+}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{r}}{=}{0}& {2}{}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\theta }}}{-}{2}{}\left({-}{r}{+}{2}{}{m}\right){}{{\mathrm{ξ}}}_{{1}}{=}{0}& {\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\phi }}}{-}\frac{{2}{}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{3}}}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\theta }}}{=}{0}& {\left({{\mathrm{ξ}}}_{{2}}\right)}_{{t}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\theta }}}{=}{0}\\ {\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\phi }}}{-}\frac{{2}{}{{\mathrm{ξ}}}_{{3}}}{{r}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{r}}{=}{0}& {\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\phi }}}{-}\frac{{2}{}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{3}}}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\theta }}}{=}{0}& {2}{}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\phi }}}{-}{2}{}\left({-}{r}{+}{2}{}{m}\right){}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{{\mathrm{ξ}}}_{{1}}{+}{2}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{2}}{=}{0}& {\left({{\mathrm{ξ}}}_{{3}}\right)}_{{t}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}\\ {\left({{\mathrm{ξ}}}_{{1}}\right)}_{{t}}{+}\frac{{2}{}{m}{}{{\mathrm{ξ}}}_{{4}}}{{r}{}\left({-}{r}{+}{2}{}{m}\right)}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{r}}{=}{0}& {\left({{\mathrm{ξ}}}_{{2}}\right)}_{{t}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\theta }}}{=}{0}& {\left({{\mathrm{ξ}}}_{{3}}\right)}_{{t}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}& {2}{}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{t}}{+}\frac{{2}{}\left({-}{r}{+}{2}{}{m}\right){}{m}{}{{\mathrm{ξ}}}_{{1}}}{{{r}}^{{3}}}{=}{0}\end{array}\right]$ (16)

These equations can be solved using pdsolve

 > $\mathrm{pdsolve}\left(\right)$
 $\left\{{{\mathrm{ξ}}}_{{1}}{=}{0}{,}{{\mathrm{ξ}}}_{{2}}{=}\left({\mathrm{_C2}}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){+}{\mathrm{_C3}}{}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right)\right){}{{r}}^{{2}}{,}{{\mathrm{ξ}}}_{{3}}{=}{{r}}^{{2}}{}\left({-}\frac{{\mathrm{_C4}}{}{\mathrm{cos}}{}\left({2}{}{\mathrm{\theta }}\right)}{{2}}{+}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}\left({\mathrm{_C2}}{}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){-}{\mathrm{_C3}}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right)\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){+}\frac{{\mathrm{_C4}}}{{2}}\right){,}{{\mathrm{ξ}}}_{{4}}{=}\frac{\left({r}{-}{2}{}{m}\right){}{\mathrm{_C1}}}{{r}}\right\}$ (17)

Alternatively, the same equations components of the tensorial expression (15) can be obtained directly with KillingVectors that in addition performs a reduction to involutive form (canonical form) for the DE system, and can as well compute and solve the equations all in one go

 > $\mathrm{KillingVectors}\left(\mathrm{ξ}\right)$
 $\left[{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{0}{,}{0}{,}{1}\right]{,}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){,}\frac{{\mathrm{cos}}{}\left({\mathrm{\phi }}\right)}{{\mathrm{tan}}{}\left({\mathrm{\theta }}\right)}{,}{0}\right]{,}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){,}{-}\frac{{\mathrm{sin}}{}\left({\mathrm{\phi }}\right)}{{\mathrm{tan}}{}\left({\mathrm{\theta }}\right)}{,}{0}\right]{,}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{0}{,}{1}{,}{0}\right]\right]$ (18)

Yet another manner of obtaining the same result is using Physics:-Library:-TensorComponents - that computes similar to TensorArray - but returns a list of equations instead, where the ordering is ascending with respect to value of the free indices -

 > $\mathrm{Library}:-\mathrm{TensorComponents}\left(\right)$
 $\left[{2}{}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{r}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{1}{,}{1}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{1}}\phantom{{,}}\phantom{{1}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\theta }}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{1}{,}{2}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{1}}\phantom{{,}}\phantom{{2}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{r}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\phi }}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{1}{,}{3}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{1}}\phantom{{,}}\phantom{{3}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{r}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{t}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{1}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{1}}\phantom{{,}}\phantom{{4}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{r}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\theta }}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{1}{,}{2}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{1}}\phantom{{,}}\phantom{{2}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{r}}{=}{0}{,}{2}{}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\theta }}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{2}{,}{2}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{2}}\phantom{{,}}\phantom{{2}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\phi }}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{2}{,}{3}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{2}}\phantom{{,}}\phantom{{3}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{t}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{2}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{2}}\phantom{{,}}\phantom{{4}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\phi }}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{1}{,}{3}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{1}}\phantom{{,}}\phantom{{3}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{r}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\phi }}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{2}{,}{3}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{2}}\phantom{{,}}\phantom{{3}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{2}{}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\phi }}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{3}{,}{3}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{3}}\phantom{{,}}\phantom{{3}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{t}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{3}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{3}}\phantom{{,}}\phantom{{4}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{t}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{1}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{1}}\phantom{{,}}\phantom{{4}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{r}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{t}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{2}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{2}}\phantom{{,}}\phantom{{4}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{t}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{3}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{3}}\phantom{{,}}\phantom{{4}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}{,}{2}{}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{t}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{4}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{4}}\phantom{{,}}\phantom{{4}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{=}{0}\right]$ (19)

Expanding the sum over the repeated indices,

 > $\mathrm{SumOverRepeatedIndices}\left(\right)$
 $\left[{2}{}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{r}}{-}\frac{{2}{}{m}{}{{\mathrm{ξ}}}_{{1}}}{{r}{}\left({-}{r}{+}{2}{}{m}\right)}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\theta }}}{-}\frac{{2}{}{{\mathrm{ξ}}}_{{2}}}{{r}}{+}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{r}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\phi }}}{-}\frac{{2}{}{{\mathrm{ξ}}}_{{3}}}{{r}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{r}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{t}}{+}\frac{{2}{}{m}{}{{\mathrm{ξ}}}_{{4}}}{{r}{}\left({-}{r}{+}{2}{}{m}\right)}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{r}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\theta }}}{-}\frac{{2}{}{{\mathrm{ξ}}}_{{2}}}{{r}}{+}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{r}}{=}{0}{,}{2}{}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\theta }}}{-}{2}{}\left({-}{r}{+}{2}{}{m}\right){}{{\mathrm{ξ}}}_{{1}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\phi }}}{-}\frac{{2}{}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{3}}}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{t}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\phi }}}{-}\frac{{2}{}{{\mathrm{ξ}}}_{{3}}}{{r}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{r}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\phi }}}{-}\frac{{2}{}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{3}}}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{2}{}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\phi }}}{-}{2}{}\left({-}{r}{+}{2}{}{m}\right){}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{{\mathrm{ξ}}}_{{1}}{+}{2}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{2}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{t}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{t}}{+}\frac{{2}{}{m}{}{{\mathrm{ξ}}}_{{4}}}{{r}{}\left({-}{r}{+}{2}{}{m}\right)}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{r}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{t}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{t}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}{,}{2}{}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{t}}{+}\frac{{2}{}\left({-}{r}{+}{2}{}{m}\right){}{m}{}{{\mathrm{ξ}}}_{{1}}}{{{r}}^{{3}}}{=}{0}\right]$ (20)
 > $\mathrm{pdsolve}\left(\right)$
 $\left\{{{\mathrm{ξ}}}_{{1}}{=}{0}{,}{{\mathrm{ξ}}}_{{2}}{=}\left({\mathrm{_C2}}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){+}{\mathrm{_C3}}{}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right)\right){}{{r}}^{{2}}{,}{{\mathrm{ξ}}}_{{3}}{=}{{r}}^{{2}}{}\left({-}\frac{{\mathrm{_C4}}{}{\mathrm{cos}}{}\left({2}{}{\mathrm{\theta }}\right)}{{2}}{+}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}\left({\mathrm{_C2}}{}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){-}{\mathrm{_C3}}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right)\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){+}\frac{{\mathrm{_C4}}}{{2}}\right){,}{{\mathrm{ξ}}}_{{4}}{=}\frac{\left({r}{-}{2}{}{m}\right){}{\mathrm{_C1}}}{{r}}\right\}$ (21)
 >