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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.
 • The vector ${v}^{c}$ indexing in LieDerivative[v[~c]](T) is expected to be defined as a tensor 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).
 • The 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

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

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

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

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

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

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

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

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

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

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

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

 > $\mathrm{LieDerivative}\left[\mathrm{\xi }\left[\mathrm{~mu}\right]\left(X\right)\right]\left(\mathrm{g_}\left[\mathrm{\alpha },\mathrm{\beta }\right]\right)=0$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\beta }}}{}{{▿}}_{{\mathrm{\alpha }}}{}\left({{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right){+}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\mu }}}{}{{▿}}_{{\mathrm{\beta }}}{}\left({{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right){=}{0}$ (14)
 > $\mathrm{Simplify}\left(\right)$
 ${{▿}}_{{\mathrm{\beta }}}{}\left({{\mathrm{ξ}}}_{{\mathrm{\alpha }}}\right){+}{{▿}}_{{\mathrm{\alpha }}}{}\left({{\mathrm{ξ}}}_{{\mathrm{\beta }}}\right){=}{0}$ (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}\frac{{2}{}{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{r}}{}\left({2}{}{m}{-}{r}\right){}{r}{-}{2}{}{m}{}{{\mathrm{ξ}}}_{{1}}}{\left({2}{}{m}{-}{r}\right){}{r}}{=}{0}& \frac{{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\theta }}}{}{r}{-}{2}{}{{\mathrm{ξ}}}_{{2}}{+}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{r}}{}{r}}{{r}}{=}{0}& \frac{{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\phi }}}{}{r}{-}{2}{}{{\mathrm{ξ}}}_{{3}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{r}}{}{r}}{{r}}{=}{0}& \frac{\left({\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{1}}\right){}\left({2}{}{m}{-}{r}\right){}{r}{+}{2}{}{m}{}{{\mathrm{ξ}}}_{{4}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{r}}{}\left({2}{}{m}{-}{r}\right){}{r}}{\left({2}{}{m}{-}{r}\right){}{r}}{=}{0}\\ \frac{{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\theta }}}{}{r}{-}{2}{}{{\mathrm{ξ}}}_{{2}}{+}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{r}}{}{r}}{{r}}{=}{0}& {2}{}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\theta }}}{-}{2}{}\left({2}{}{m}{-}{r}\right){}{{\mathrm{ξ}}}_{{1}}{=}{0}& {\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\phi }}}{-}{2}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{3}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\theta }}}{=}{0}& {\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{2}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\theta }}}{=}{0}\\ \frac{{\left({{\mathrm{ξ}}}_{{1}}\right)}_{{\mathrm{\phi }}}{}{r}{-}{2}{}{{\mathrm{ξ}}}_{{3}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{r}}{}{r}}{{r}}{=}{0}& {\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\phi }}}{-}{2}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{3}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\theta }}}{=}{0}& {2}{}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\phi }}}{-}{2}{}\left({2}{}{m}{-}{r}\right){}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{{\mathrm{ξ}}}_{{1}}{+}{\mathrm{sin}}{}\left({2}{}{\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{2}}{=}{0}& {\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{3}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}\\ \frac{\left({\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{1}}\right){}\left({2}{}{m}{-}{r}\right){}{r}{+}{2}{}{m}{}{{\mathrm{ξ}}}_{{4}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{r}}{}\left({2}{}{m}{-}{r}\right){}{r}}{\left({2}{}{m}{-}{r}\right){}{r}}{=}{0}& {\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{2}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\theta }}}{=}{0}& {\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{3}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}& \frac{{2}{}\left({\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{4}}\right){}{{r}}^{{3}}{+}{2}{}\left({2}{}{m}{-}{r}\right){}{m}{}{{\mathrm{ξ}}}_{{1}}}{{{r}}^{{3}}}{=}{0}\end{array}\right]$ (16)

These equations can be solved using pdsolve

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

 > $\mathrm{KillingVectors}\left(\mathrm{\xi }\right)$
 $\left[{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{0}{,}{0}{,}{1}\right]{,}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){,}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){,}{0}\right]{,}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){,}{-}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){,}{0}\right]{,}{{\mathrm{ξ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{=}\left[{0}{,}{0}{,}{1}{,}{0}\right]\right]$ (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{ξ}}}_{{1}}\right)}_{{r}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{1}{,}{1}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{1}{,}{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}{,}{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}{,}{3}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{r}}{=}{0}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{1}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{1}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{1}{,}{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}{,}{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}{,}{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}{,}{3}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{2}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{2}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{2}{,}{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}{,}{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}{,}{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}{,}{3}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{=}{0}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{3}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{3}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{3}{,}{4}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{1}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{1}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{1}{,}{4}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{r}}{=}{0}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{2}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{2}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{2}{,}{4}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{3}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{3}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{3}{,}{4}}}{}{{\mathrm{ξ}}}_{{\mathrm{\alpha }}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}{,}{2}{}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{4}}{-}{2}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}{4}{,}{4}}^{\phantom{{}}{\mathrm{\alpha }}\phantom{{4}{,}{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({2}{}{m}{-}{r}\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}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{1}}{+}\frac{{2}{}{m}{}{{\mathrm{ξ}}}_{{4}}}{{r}{}\left({2}{}{m}{-}{r}\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({2}{}{m}{-}{r}\right){}{{\mathrm{ξ}}}_{{1}}{=}{0}{,}{\left({{\mathrm{ξ}}}_{{2}}\right)}_{{\mathrm{\phi }}}{-}{2}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{3}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{2}}{+}{\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 }}}{-}{2}{}{\mathrm{cot}}{}\left({\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{3}}{+}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{2}{}{\left({{\mathrm{ξ}}}_{{3}}\right)}_{{\mathrm{\phi }}}{-}{2}{}\left({2}{}{m}{-}{r}\right){}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{{\mathrm{ξ}}}_{{1}}{+}{\mathrm{sin}}{}\left({2}{}{\mathrm{\theta }}\right){}{{\mathrm{ξ}}}_{{2}}{=}{0}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{3}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{1}}{+}\frac{{2}{}{m}{}{{\mathrm{ξ}}}_{{4}}}{{r}{}\left({2}{}{m}{-}{r}\right)}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{r}}{=}{0}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{2}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\theta }}}{=}{0}{,}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{3}}{+}{\left({{\mathrm{ξ}}}_{{4}}\right)}_{{\mathrm{\phi }}}{=}{0}{,}{2}{}{\stackrel{{\mathbf{.}}}{{\mathrm{ξ}}}}_{{4}}{+}\frac{{2}{}\left({2}{}{m}{-}{r}\right){}{m}{}{{\mathrm{ξ}}}_{{1}}}{{{r}}^{{3}}}{=}{0}\right]$ (20)
 > $\mathrm{pdsolve}\left(\right)$
 $\left\{{{\mathrm{ξ}}}_{{1}}{=}{0}{,}{{\mathrm{ξ}}}_{{2}}{=}\left(\mathrm{c__2}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){+}\mathrm{c__3}{}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right)\right){}{{r}}^{{2}}{,}{{\mathrm{ξ}}}_{{3}}{=}\frac{{{r}}^{{2}}{}\left({\mathrm{sin}}{}\left({2}{}{\mathrm{\theta }}\right){}\left(\mathrm{c__2}{}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){-}\mathrm{c__3}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right)\right){-}\mathrm{c__4}{}\left({-}{1}{+}{\mathrm{cos}}{}\left({2}{}{\mathrm{\theta }}\right)\right)\right)}{{2}}{,}{{\mathrm{ξ}}}_{{4}}{=}\frac{\left({r}{-}{2}{}{m}\right){}\mathrm{c__1}}{{r}}\right\}$ (21)

References

 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.