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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[v] 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

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

 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 $▿$ 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

 > $\mathrm{Setup}\left(\mathrm{coordinates}=X\right)$
 ${}\mathrm{Systems of spacetime coordinates are:}{}\left\{X=\left(\mathrm{x1}{,}\mathrm{x2}{,}\mathrm{x3}{,}\mathrm{x4}\right)\right\}$
 $\mathrm{_______________________________________________________}$
 $\left[{\mathrm{coordinatesystems}}{=}\left\{{X}\right\}\right]$ (2)

Define some tensors for experimentation

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

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

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

 > ${\mathrm{g_}}_{\mathrm{sc}}$
 ${}\mathrm{_______________________________________________________}$
 ${}\mathrm{Systems of spacetime coordinates are:}{}\left\{X=\left(r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right)\right\}$
 ${}\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}{}\left\{X=\left(r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right)\right\}$
 $\mathrm{Setting}\mathrm{lowercaselatin_is}\mathrm{letters to represent}\mathrm{space}\mathrm{indices}$
 ${}\mathrm{The Schwarzschild metric in coordinates}{}\left[r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right]$
 $\mathrm{Parameters:}\left[m\right]$
 $\mathrm{Signature:}\left(\mathrm{- - - +}\right)$
 ${}\mathrm{_______________________________________________________}$
 ${{\mathrm{g_}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}\left(\left[\begin{array}{cccc}\frac{r}{2{}m-r}& 0& 0& 0\\ 0& -{r}^{2}& 0& 0\\ 0& 0& -{r}^{2}{}{\mathrm{sin}{}\left(\mathrm{θ}\right)}^{2}& 0\\ 0& 0& 0& \frac{r-2{}m}{r}\end{array}\right]\right)$ (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 ${\mathrm{\xi }}^{\mathrm{\alpha }}$ 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

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

The index $\mathrm{\mu }$ in ${v}^{\mathrm{\mu }}$ 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 $\mathrm{\alpha }$

 > ${\mathrm{LieDerivative}}_{{v}_{\mathrm{~alpha}}\left(X\right)}\left({\mathrm{ξ}}_{\mathrm{~alpha}}\left(X\right)\right)$
 ${{v}}_{{\mathrm{~mu}}}{}\left({X}\right){}{{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{ξ}}}_{{\mathrm{~alpha}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{ξ}}}_{{\mathrm{~mu}}}{}\left({X}\right){}{{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{v}}_{{\mathrm{~alpha}}}{}\left({X}\right){,}\left[{X}\right]\right)$ (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:

 > ${\mathrm{LieDerivative}}_{v\left(X\right)}\left({\mathrm{ξ}}_{\mathrm{~alpha}}\left(X\right)\right)$
 ${{v}}_{{\mathrm{~mu}}}{}\left({X}\right){}{{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{ξ}}}_{{\mathrm{~alpha}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{ξ}}}_{{\mathrm{~mu}}}{}\left({X}\right){}{{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{v}}_{{\mathrm{~alpha}}}{}\left({X}\right){,}\left[{X}\right]\right)$ (10)
 > ${\mathrm{LieDerivative}}_{v}\left({\mathrm{ξ}}_{\mathrm{~alpha}}\left(X\right)\right)$
 ${{v}}_{{\mathrm{~mu}}}{}{{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{ξ}}}_{{\mathrm{~alpha}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{ξ}}}_{{\mathrm{~mu}}}{}\left({X}\right){}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{\mathrm{μ}}{,}{\mathrm{ν}}}{}{{v}}_{{\mathrm{~nu}}}$ (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 ${\mathrm{\xi }}^{\mathrm{\alpha }}$. For that reason, you can also pass the first argument to LieBracket with or without the index

 > $\mathrm{LieBracket}\left({v}_{\mathrm{~mu}}\left(X\right),{\mathrm{ξ}}_{\mathrm{~alpha}}\left(X\right)\right)$
 ${{v}}_{{\mathrm{~mu}}}{}\left({X}\right){}{{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{ξ}}}_{{\mathrm{~alpha}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{ξ}}}_{{\mathrm{~mu}}}{}\left({X}\right){}{{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{v}}_{{\mathrm{~alpha}}}{}\left({X}\right){,}\left[{X}\right]\right)$ (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 ${T}_{\mathrm{\beta }}^{\mathrm{\alpha }}$ 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

 > ${\mathrm{LieDerivative}}_{{v}_{\mathrm{~mu}}\left(X\right)}\left({T}_{\mathrm{~alpha},\mathrm{β}}\left(X\right)\right)$
 ${{v}}_{{\mathrm{~mu}}}{}\left({X}\right){}{{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{T}}_{{\mathrm{~alpha}}{,}{\mathrm{β}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{T}}_{{\mathrm{~mu}}{,}{\mathrm{β}}}{}\left({X}\right){}{{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{v}}_{{\mathrm{~alpha}}}{}\left({X}\right){,}\left[{X}\right]\right){+}{{T}}_{{\mathrm{~alpha}}{,}{\mathrm{μ}}}{}\left({X}\right){}{{\mathrm{D_}}}_{{\mathrm{β}}}{}\left({{v}}_{{\mathrm{~mu}}}{}\left({X}\right){,}\left[{X}\right]\right)$ (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

 > ${\mathrm{LieDerivative}}_{{\mathrm{ξ}}_{\mathrm{~mu}}\left(X\right)}\left({\mathrm{g_}}_{\mathrm{α},\mathrm{β}}\right)=0$
 ${{\mathrm{g_}}}_{{\mathrm{μ}}{,}{\mathrm{β}}}{}{{\mathrm{D_}}}_{{\mathrm{α}}}{}\left({{\mathrm{ξ}}}_{{\mathrm{~mu}}}{}\left({X}\right){,}\left[{X}\right]\right){+}{{\mathrm{g_}}}_{{\mathrm{α}}{,}{\mathrm{μ}}}{}{{\mathrm{D_}}}_{{\mathrm{β}}}{}\left({{\mathrm{ξ}}}_{{\mathrm{~mu}}}{}\left({X}\right){,}\left[{X}\right]\right){=}{0}$ (14)
 > $\mathrm{Simplify}\left(\right)$
 ${{\mathrm{D_}}}_{{\mathrm{β}}}{}\left({{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){,}\left[{X}\right]\right){+}{{\mathrm{D_}}}_{{\mathrm{α}}}{}\left({{\mathrm{ξ}}}_{{\mathrm{β}}}{}\left({X}\right){,}\left[{X}\right]\right){=}{0}$ (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}\frac{2{}\left(\frac{\partial }{\partial r}{}{\mathrm{ξ}}_{1}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}\left(2{}m-r\right){}r-2{}m{}{\mathrm{ξ}}_{1}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)}{\left(2{}m-r\right){}r}=0& \frac{\left(\frac{\partial }{\partial \mathrm{θ}}{}{\mathrm{ξ}}_{1}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}r-2{}{\mathrm{ξ}}_{2}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\left(\frac{\partial }{\partial r}{}{\mathrm{ξ}}_{2}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}r}{r}=0& \frac{\left(\frac{\partial }{\partial \mathrm{φ}}{}{\mathrm{ξ}}_{1}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}r-2{}{\mathrm{ξ}}_{3}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\left(\frac{\partial }{\partial r}{}{\mathrm{ξ}}_{3}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}r}{r}=0& \frac{\left(\frac{\partial }{\partial t}{}{\mathrm{ξ}}_{1}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}\left(2{}m-r\right){}r+2{}m{}{\mathrm{ξ}}_{4}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\left(\frac{\partial }{\partial r}{}{\mathrm{ξ}}_{4}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}\left(2{}m-r\right){}r}{\left(2{}m-r\right){}r}=0\\ \frac{\left(\frac{\partial }{\partial \mathrm{θ}}{}{\mathrm{ξ}}_{1}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}r-2{}{\mathrm{ξ}}_{2}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\left(\frac{\partial }{\partial r}{}{\mathrm{ξ}}_{2}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}r}{r}=0& 2{}\left(\frac{\partial }{\partial \mathrm{θ}}{}{\mathrm{ξ}}_{2}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right)-2{}\left(2{}m-r\right){}{\mathrm{ξ}}_{1}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)=0& \frac{\partial }{\partial \mathrm{φ}}{}{\mathrm{ξ}}_{2}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)-2{}\mathrm{cot}{}\left(\mathrm{θ}\right){}{\mathrm{ξ}}_{3}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\frac{\partial }{\partial \mathrm{θ}}{}{\mathrm{ξ}}_{3}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)=0& \frac{\partial }{\partial t}{}{\mathrm{ξ}}_{2}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\frac{\partial }{\partial \mathrm{θ}}{}{\mathrm{ξ}}_{4}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)=0\\ \frac{\left(\frac{\partial }{\partial \mathrm{φ}}{}{\mathrm{ξ}}_{1}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}r-2{}{\mathrm{ξ}}_{3}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\left(\frac{\partial }{\partial r}{}{\mathrm{ξ}}_{3}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}r}{r}=0& \frac{\partial }{\partial \mathrm{φ}}{}{\mathrm{ξ}}_{2}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)-2{}\mathrm{cot}{}\left(\mathrm{θ}\right){}{\mathrm{ξ}}_{3}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\frac{\partial }{\partial \mathrm{θ}}{}{\mathrm{ξ}}_{3}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)=0& 2{}\left(\frac{\partial }{\partial \mathrm{φ}}{}{\mathrm{ξ}}_{3}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right)-2{}\left(2{}m-r\right){}{\mathrm{sin}{}\left(\mathrm{θ}\right)}^{2}{}{\mathrm{ξ}}_{1}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+2{}\mathrm{sin}{}\left(\mathrm{θ}\right){}\mathrm{cos}{}\left(\mathrm{θ}\right){}{\mathrm{ξ}}_{2}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)=0& \frac{\partial }{\partial t}{}{\mathrm{ξ}}_{3}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\frac{\partial }{\partial \mathrm{φ}}{}{\mathrm{ξ}}_{4}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)=0\\ \frac{\left(\frac{\partial }{\partial t}{}{\mathrm{ξ}}_{1}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}\left(2{}m-r\right){}r+2{}m{}{\mathrm{ξ}}_{4}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\left(\frac{\partial }{\partial r}{}{\mathrm{ξ}}_{4}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}\left(2{}m-r\right){}r}{\left(2{}m-r\right){}r}=0& \frac{\partial }{\partial t}{}{\mathrm{ξ}}_{2}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\frac{\partial }{\partial \mathrm{θ}}{}{\mathrm{ξ}}_{4}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)=0& \frac{\partial }{\partial t}{}{\mathrm{ξ}}_{3}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)+\frac{\partial }{\partial \mathrm{φ}}{}{\mathrm{ξ}}_{4}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)=0& \frac{2{}\left(\frac{\partial }{\partial t}{}{\mathrm{ξ}}_{4}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)\right){}{r}^{3}+2{}\left(2{}m-r\right){}m{}{\mathrm{ξ}}_{1}{}\left(r,\mathrm{θ},\mathrm{φ},t\right)}{{r}^{3}}=0\end{array}\right]$ (16)

These equations can be solved using pdsolve

 > $\mathrm{pdsolve}\left(\right)$
 $\left\{{{\mathrm{ξ}}}_{{1}}{}\left({X}\right){=}{0}{,}{{\mathrm{ξ}}}_{{2}}{}\left({X}\right){=}\left(\mathrm{c__2}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right){+}\mathrm{c__3}{}{\mathrm{cos}}{}\left({\mathrm{φ}}\right)\right){}{{r}}^{{2}}{,}{{\mathrm{ξ}}}_{{3}}{}\left({X}\right){=}\left({-}\frac{{1}}{{2}}{}\mathrm{c__4}{}{\mathrm{cos}}{}\left({2}{}{\mathrm{θ}}\right){+}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}\left(\mathrm{c__2}{}{\mathrm{cos}}{}\left({\mathrm{φ}}\right){-}\mathrm{c__3}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)\right){+}\frac{{1}}{{2}}{}\mathrm{c__4}\right){}{{r}}^{{2}}{,}{{\mathrm{ξ}}}_{{4}}{}\left({X}\right){=}\frac{\left({r}{-}{2}{}{m}\right){}\mathrm{c__1}}{{r}}\right\}$ (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

 > $\mathrm{KillingVectors}\left(\mathrm{ξ}\right)$
 $\left[{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{0}{,}{0}{,}{1}\right]{,}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){,}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){,}{0}\right]{,}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){,}{-}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){,}{0}\right]{,}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{0}{,}{1}{,}{0}\right]\right]$ (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{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{r}\right)\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{1}{,}{1}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{\mathrm{θ}}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{1}{,}{2}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{r}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{\mathrm{φ}}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{1}{,}{3}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{r}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{t}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{1}{,}{4}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{r}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{\mathrm{θ}}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{1}{,}{2}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{r}\right){=}{0}{,}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{\mathrm{θ}}\right)\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{2}{,}{2}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{\mathrm{φ}}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{2}{,}{3}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{\mathrm{θ}}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{t}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{2}{,}{4}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{\mathrm{θ}}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{\mathrm{φ}}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{1}{,}{3}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{r}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{\mathrm{φ}}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{2}{,}{3}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{\mathrm{θ}}\right){=}{0}{,}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{\mathrm{φ}}\right)\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{3}{,}{3}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{t}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{3}{,}{4}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{\mathrm{φ}}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{t}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{1}{,}{4}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{r}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{t}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{2}{,}{4}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{\mathrm{θ}}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{t}\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{3}{,}{4}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{\mathrm{φ}}\right){=}{0}{,}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{t}\right)\right){-}{2}{}{{\mathrm{Christoffel}}}_{{\mathrm{~alpha}}{,}{4}{,}{4}}{}{{\mathrm{ξ}}}_{{\mathrm{α}}}{}\left({X}\right){=}{0}\right]$ (19)

Expanding the sum over the repeated indices,

 > $\mathrm{SumOverRepeatedIndices}\left(\right)$
 $\left[{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{r}\right)\right){-}\frac{{2}{}{m}{}{{\mathrm{ξ}}}_{{1}}{}\left({X}\right)}{{r}{}\left({2}{}{m}{-}{r}\right)}{=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{\mathrm{θ}}\right){-}\frac{{2}{}{{\mathrm{ξ}}}_{{2}}{}\left({X}\right)}{{r}}{+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{r}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{\mathrm{φ}}\right){-}\frac{{2}{}{{\mathrm{ξ}}}_{{3}}{}\left({X}\right)}{{r}}{+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{r}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{t}\right){+}\frac{{2}{}{m}{}{{\mathrm{ξ}}}_{{4}}{}\left({X}\right)}{{r}{}\left({2}{}{m}{-}{r}\right)}{+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{r}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{\mathrm{θ}}\right){-}\frac{{2}{}{{\mathrm{ξ}}}_{{2}}{}\left({X}\right)}{{r}}{+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{r}\right){=}{0}{,}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{\mathrm{θ}}\right)\right){-}{2}{}\left({2}{}{m}{-}{r}\right){}{{\mathrm{ξ}}}_{{1}}{}\left({X}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{\mathrm{φ}}\right){-}{2}{}{\mathrm{cot}}{}\left({\mathrm{θ}}\right){}{{\mathrm{ξ}}}_{{3}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{\mathrm{θ}}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{t}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{\mathrm{θ}}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{\mathrm{φ}}\right){-}\frac{{2}{}{{\mathrm{ξ}}}_{{3}}{}\left({X}\right)}{{r}}{+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{r}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{\mathrm{φ}}\right){-}{2}{}{\mathrm{cot}}{}\left({\mathrm{θ}}\right){}{{\mathrm{ξ}}}_{{3}}{}\left({X}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{\mathrm{θ}}\right){=}{0}{,}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{\mathrm{φ}}\right)\right){-}{2}{}\left({2}{}{m}{-}{r}\right){}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{{\mathrm{ξ}}}_{{1}}{}\left({X}\right){+}{2}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{{\mathrm{ξ}}}_{{2}}{}\left({X}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{t}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{\mathrm{φ}}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{1}}{}\left({X}\right){,}{t}\right){+}\frac{{2}{}{m}{}{{\mathrm{ξ}}}_{{4}}{}\left({X}\right)}{{r}{}\left({2}{}{m}{-}{r}\right)}{+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{r}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{2}}{}\left({X}\right){,}{t}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{\mathrm{θ}}\right){=}{0}{,}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{3}}{}\left({X}\right){,}{t}\right){+}{\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{\mathrm{φ}}\right){=}{0}{,}{2}{}\left({\mathrm{diff}}{}\left({{\mathrm{ξ}}}_{{4}}{}\left({X}\right){,}{t}\right)\right){+}\frac{{2}{}\left({2}{}{m}{-}{r}\right){}{m}{}{{\mathrm{ξ}}}_{{1}}{}\left({X}\right)}{{{r}}^{{3}}}{=}{0}\right]$ (20)
 > $\mathrm{pdsolve}\left(\right)$
 $\left\{{{\mathrm{ξ}}}_{{1}}{}\left({X}\right){=}{0}{,}{{\mathrm{ξ}}}_{{2}}{}\left({X}\right){=}\left(\mathrm{c__2}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right){+}\mathrm{c__3}{}{\mathrm{cos}}{}\left({\mathrm{φ}}\right)\right){}{{r}}^{{2}}{,}{{\mathrm{ξ}}}_{{3}}{}\left({X}\right){=}\left({-}\frac{{1}}{{2}}{}\mathrm{c__4}{}{\mathrm{cos}}{}\left({2}{}{\mathrm{θ}}\right){+}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}\left(\mathrm{c__2}{}{\mathrm{cos}}{}\left({\mathrm{φ}}\right){-}\mathrm{c__3}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)\right){+}\frac{{1}}{{2}}{}\mathrm{c__4}\right){}{{r}}^{{2}}{,}{{\mathrm{ξ}}}_{{4}}{}\left({X}\right)\right\}$