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

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.

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

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

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

$\mathrm{Setup}\left(\mathrm{coordinates}=X\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

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

$\left[\mathrm{coordinatesystems}=\left\{X\right\}\right]$

$\mathrm{Define}\left(v\,\mathrm{\xi }\,T\right)$

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

$\left\{T\,v\,\mathrm{\ξ}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

$\mathrm{LieDerivative}\left[v\left[\mathrm{~mu}\right]\left(X\right)\right]\left(f\left(X\right)\right)$

$v_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right){\partial }_{\mathrm{\mu }}\left(f\left(X\right)\right)$

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

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

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

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

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

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

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

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

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

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

$\mathrm{PDEtools}:-\mathrm{declare}\left(\left(v\,\mathrm{\xi }\,T\right)\left(X\right)\right)$

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

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

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

$\mathrm{LieDerivative}\left[v\left[\mathrm{~mu}\right]\left(X\right)\right]\left(\mathrm{\xi }\left[\mathrm{\alpha }\right]\left(X\right)\right)$

$v_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left({\mathrm{\ξ}}_{\mathrm{\alpha }}\right)+{\mathrm{\ξ}}_{\mathrm{\mu }}{▿}_{\mathrm{\alpha }}\left(v_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)$

$\mathrm{LieDerivative}\left[v\left[\mathrm{~mu}\right]\left(X\right)\right]\left(\mathrm{\xi }\left[\mathrm{~alpha}\right]\left(X\right)\right)$

$v_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left({\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)-{\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left(v_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)$

$\mathrm{LieDerivative}\left[v\left[\mathrm{~alpha}\right]\left(X\right)\right]\left(\mathrm{\xi }\left[\mathrm{~alpha}\right]\left(X\right)\right)$

$v_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left({\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)-{\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left(v_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)$

$\mathrm{LieDerivative}\left[v\left(X\right)\right]\left(\mathrm{\xi }\left[\mathrm{~alpha}\right]\left(X\right)\right)$

$v_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left({\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)-{\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left(v_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)$

$\mathrm{LieDerivative}\left[v\right]\left(\mathrm{\xi }\left[\mathrm{~alpha}\right]\left(X\right)\right)$

$v_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left({\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)-{\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\alpha }}\mathrm{\mu },\mathrm{\nu }}^{\phantom{}\mathrm{\alpha }\phantom{\mathrm{\mu },\mathrm{\nu }}}{v}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

$\mathrm{LieBracket}\left(v\left[\mathrm{~mu}\right]\left(X\right)\,\mathrm{\xi }\left[\mathrm{~alpha}\right]\left(X\right)\right)$

$v_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left({\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)-{\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left(v_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)$

$\mathrm{LieDerivative}\left[v\left[\mathrm{~mu}\right]\left(X\right)\right]\left(T\left[\mathrm{~alpha},\mathrm{\beta }\right]\left(X\right)\right)$

$v_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{▿}_{\mathrm{\mu }}\left(T_{\phantom{}\phantom{\mathrm{\alpha }}\mathrm{\beta }}^{\phantom{}\mathrm{\alpha }\phantom{\mathrm{\beta }}}\right)-T_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\beta }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\beta }}}{▿}_{\mathrm{\mu }}\left(v_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)+T_{\phantom{}\phantom{\mathrm{\alpha }}\mathrm{\mu }}^{\phantom{}\mathrm{\alpha }\phantom{\mathrm{\mu }}}{▿}_{\mathrm{\beta }}\left(v_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)$

$\mathrm{LieDerivative}\left[\mathrm{\xi }\left[\mathrm{~mu}\right]\left(X\right)\right]\left(\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\beta }\right]\right)=0$

$g_{\mathrm{\mu },\mathrm{\beta }}{▿}_{\mathrm{\alpha }}\left({\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)+g_{\mathrm{\alpha },\mathrm{\mu }}{▿}_{\mathrm{\beta }}\left({\mathrm{\ξ}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)=0$

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

${▿}_{\mathrm{\beta }}\left({\mathrm{\ξ}}_{\mathrm{\alpha }}\right)+{▿}_{\mathrm{\alpha }}\left({\mathrm{\ξ}}_{\mathrm{\beta }}\right)=0$

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

$\left[\begin{array}{cccc}\frac{2{\left({\mathrm{\ξ}}_1\right)}_r\left(2m-r\right)r-2m{\mathrm{\ξ}}_1}{\left(2m-r\right)r}=0& \frac{{\left({\mathrm{\ξ}}_1\right)}_{\mathrm{\theta }}r-2{\mathrm{\ξ}}_2+{\left({\mathrm{\ξ}}_2\right)}_{r}r}{r}=0& \frac{{\left({\mathrm{\ξ}}_1\right)}_{\mathrm{\phi }}r-2{\mathrm{\ξ}}_3+{\left({\mathrm{\ξ}}_3\right)}_{r}r}{r}=0& \frac{\left({\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_1\right)\left(2m-r\right)r+2m{\mathrm{\ξ}}_4+{\left({\mathrm{\ξ}}_4\right)}_r\left(2m-r\right)r}{\left(2m-r\right)r}=0\\ \frac{{\left({\mathrm{\ξ}}_1\right)}_{\mathrm{\theta }}r-2{\mathrm{\ξ}}_2+{\left({\mathrm{\ξ}}_2\right)}_{r}r}{r}=0& 2{\left({\mathrm{\ξ}}_2\right)}_{\mathrm{\theta }}-2\left(2m-r\right){\mathrm{\ξ}}_1=0& {\left({\mathrm{\ξ}}_2\right)}_{\mathrm{\phi }}-2\cot \left(\mathrm{\theta }\right){\mathrm{\ξ}}_3+{\left({\mathrm{\ξ}}_3\right)}_{\mathrm{\theta }}=0& {\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_2+{\left({\mathrm{\ξ}}_4\right)}_{\mathrm{\theta }}=0\\ \frac{{\left({\mathrm{\ξ}}_1\right)}_{\mathrm{\phi }}r-2{\mathrm{\ξ}}_3+{\left({\mathrm{\ξ}}_3\right)}_{r}r}{r}=0& {\left({\mathrm{\ξ}}_2\right)}_{\mathrm{\phi }}-2\cot \left(\mathrm{\theta }\right){\mathrm{\ξ}}_3+{\left({\mathrm{\ξ}}_3\right)}_{\mathrm{\theta }}=0& 2{\left({\mathrm{\ξ}}_3\right)}_{\mathrm{\phi }}-2\left(2m-r\right){\sin \left(\mathrm{\theta }\right)}^2{\mathrm{\ξ}}_1+\sin \left(2\mathrm{\theta }\right){\mathrm{\ξ}}_2=0& {\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_3+{\left({\mathrm{\ξ}}_4\right)}_{\mathrm{\phi }}=0\\ \frac{\left({\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_1\right)\left(2m-r\right)r+2m{\mathrm{\ξ}}_4+{\left({\mathrm{\ξ}}_4\right)}_r\left(2m-r\right)r}{\left(2m-r\right)r}=0& {\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_2+{\left({\mathrm{\ξ}}_4\right)}_{\mathrm{\theta }}=0& {\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_3+{\left({\mathrm{\ξ}}_4\right)}_{\mathrm{\phi }}=0& \frac{2\left({\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_4\right)r^3+2\left(2m-r\right)m{\mathrm{\ξ}}_1}{r^3}=0\end{array}\right]$

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

$\left\{{\mathrm{\ξ}}_1=0\,{\mathrm{\ξ}}_2=\left(\mathrm{c\_\_2}\sin \left(\mathrm{\phi }\right)+\mathrm{c\_\_3}\cos \left(\mathrm{\phi }\right)\right)r^2\,{\mathrm{\ξ}}_3=\frac{r^2\left(\sin \left(2\mathrm{\theta }\right)\left(\mathrm{c\_\_2}\cos \left(\mathrm{\phi }\right)-\mathrm{c\_\_3}\sin \left(\mathrm{\phi }\right)\right)-\mathrm{c\_\_4}\left(-1+\cos \left(2\mathrm{\theta }\right)\right)\right)}{2}\,{\mathrm{\ξ}}_4=\frac{\left(r-2m\right)\mathrm{c\_\_1}}{r}\right\}$

$\mathrm{KillingVectors}\left(\mathrm{\xi }\right)$

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

$\mathrm{Library}:-\mathrm{TensorComponents}\left(\right)$

$\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]$

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

$\left[2{\left({\mathrm{\ξ}}_1\right)}_r-\frac{2m{\mathrm{\ξ}}_1}{r\left(2m-r\right)}=0\,{\left({\mathrm{\ξ}}_1\right)}_{\mathrm{\theta }}-\frac{2{\mathrm{\ξ}}_2}{r}+{\left({\mathrm{\ξ}}_2\right)}_r=0\,{\left({\mathrm{\ξ}}_1\right)}_{\mathrm{\phi }}-\frac{2{\mathrm{\ξ}}_3}{r}+{\left({\mathrm{\ξ}}_3\right)}_r=0\,{\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_1+\frac{2m{\mathrm{\ξ}}_4}{r\left(2m-r\right)}+{\left({\mathrm{\ξ}}_4\right)}_r=0\,{\left({\mathrm{\ξ}}_1\right)}_{\mathrm{\theta }}-\frac{2{\mathrm{\ξ}}_2}{r}+{\left({\mathrm{\ξ}}_2\right)}_r=0\,2{\left({\mathrm{\ξ}}_2\right)}_{\mathrm{\theta }}-2\left(2m-r\right){\mathrm{\ξ}}_1=0\,{\left({\mathrm{\ξ}}_2\right)}_{\mathrm{\phi }}-2\cot \left(\mathrm{\theta }\right){\mathrm{\ξ}}_3+{\left({\mathrm{\ξ}}_3\right)}_{\mathrm{\theta }}=0\,{\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_2+{\left({\mathrm{\ξ}}_4\right)}_{\mathrm{\theta }}=0\,{\left({\mathrm{\ξ}}_1\right)}_{\mathrm{\phi }}-\frac{2{\mathrm{\ξ}}_3}{r}+{\left({\mathrm{\ξ}}_3\right)}_r=0\,{\left({\mathrm{\ξ}}_2\right)}_{\mathrm{\phi }}-2\cot \left(\mathrm{\theta }\right){\mathrm{\ξ}}_3+{\left({\mathrm{\ξ}}_3\right)}_{\mathrm{\theta }}=0\,2{\left({\mathrm{\ξ}}_3\right)}_{\mathrm{\phi }}-2\left(2m-r\right){\sin \left(\mathrm{\theta }\right)}^2{\mathrm{\ξ}}_1+\sin \left(2\mathrm{\theta }\right){\mathrm{\ξ}}_2=0\,{\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_3+{\left({\mathrm{\ξ}}_4\right)}_{\mathrm{\phi }}=0\,{\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_1+\frac{2m{\mathrm{\ξ}}_4}{r\left(2m-r\right)}+{\left({\mathrm{\ξ}}_4\right)}_r=0\,{\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_2+{\left({\mathrm{\ξ}}_4\right)}_{\mathrm{\theta }}=0\,{\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_3+{\left({\mathrm{\ξ}}_4\right)}_{\mathrm{\phi }}=0\,2{\stackrel{\mathbf{.}}{\mathrm{\ξ}}}_4+\frac{2\left(2m-r\right)m{\mathrm{\ξ}}_1}{r^3}=0\right]$

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

$\left\{{\mathrm{\ξ}}_1=0\,{\mathrm{\ξ}}_2=\left(\mathrm{c\_\_2}\sin \left(\mathrm{\phi }\right)+\mathrm{c\_\_3}\cos \left(\mathrm{\phi }\right)\right)r^2\,{\mathrm{\ξ}}_3=\frac{r^2\left(\sin \left(2\mathrm{\theta }\right)\left(\mathrm{c\_\_2}\cos \left(\mathrm{\phi }\right)-\mathrm{c\_\_3}\sin \left(\mathrm{\phi }\right)\right)-\mathrm{c\_\_4}\left(-1+\cos \left(2\mathrm{\theta }\right)\right)\right)}{2}\,{\mathrm{\ξ}}_4=\frac{\left(r-2m\right)\mathrm{c\_\_1}}{r}\right\}$

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.