KillingVectors computes and attempts to solve the Killing equations, that is, the PDE system

where $▿$ is the covariant derivative operator D_, and $V$ entering $V_{\alpha }$ is the name representing the Killing vectors, provided, with or without a spacetime index, and ${ℒ}_V\left(g_{\mathrm{\alpha },\mathrm{\nu }}\right)$ is the LieDerivative of the spacetime metric with respect to $V_{\mathrm{\mu }}$. In a spacetime of dimension $n$ there are at most $\frac{1}{2}n\left(n\+1\right)$ independent Killing vectors, where the maximum happens for a Minkowski spacetime. As it is the case of the other commands in Physics, KillingVectors operates in tensorial notation. To perform the same operation using differential forms notation see DifferentialGeometry[Tensor][KillingVectors].

When the name $V$ is provided without an index, the equations computed are satisfied by the contravariant components $V^{\mathrm{\alpha }}$. To request the equations or solutions for the covariant components $\mathrm{V\_\_alpha}$, pass it with a covariant index, say as in V[alpha]. Note that when the spacetime is non-galilean, while the system may succeed in computing solutions for the covariant components, depending on the metric g_ it may fail in computing solutions for the contravariant ones, or vice-versa.

By default, KillingVectors automatically computes the equations and attempts solving them in one step. When successful, KillingVectors returns a list of Killing vectors, obtained after specializing the integration constants that appear when solving the system of differential equations (that you can see using the option output = equations). When no solution is found, KillingVectors returns NULL, the same way the PDE solver pdsolve does.

A more compact solution can be obtained by passing the optional argument specializeconstants = false, in which case KillingVectors returns an equation with the Killing vector on the left-hand side and a list with its components, depending on integration constants, on the right-hand side.

When passing the optional argument output = solutions, instead of returning a vector equation, the output is a set of solution equations, with each Killing vector component on the left-hand side and the corresponding solution on the right-hand side. This is useful, for instance, for verification purposes: you can test the solutions computed against the output of KillingVectors using the option output = equations by using the pdetest command. -When passing the optional argument output = equations, the Killing equations themselves are returned without attempting solving them. This is useful when KillingVectors fails in finding a solution, or when particular solutions of different forms may be of interest; these solutions can frequently be computed using the symmetry commands of PDEtools like FunctionFieldSolutions and PolynomialSolutions, or mainly InvariantSolutions used together with its various options for particularizing solutions.

The PDE system returned by KillingVectors with the option output = equations is by default returned as a list of equations and automatically reduced taking into account its integrability conditions. The order of the equations in the list corresponds to a total degree, obtained with an orderly ranking (see details in casesplit). Simplifying the PDE system taking into account integrability conditions is a frequently desired but in some cases expensive mathematical computation. To avoid it, pass the optional argument integrabilityconditions = false, in which case KillingVectors will return faster, a set of equations, not a list, and with no differential elimination simplifications.

When the metric g_ depends on parameters, either symbols or functions of spacetime variables, the Killing vectors computed using KillingVectors are expected to be valid for arbitrary values of these parameters. It is sometimes of interest, however, to investigate the different kinds of solutions that may exist for different particular values of these parameters. To perform such an investigation, use the optional argument parameters = ... where the right-hand-side is a set or a list with the parameters, and the system of equations will be split into cases with respect to these parameters using differential algebra techniques, and the Killing vectors will be expressed as a piecewise function according to the different cases for these parameters. As frequently happens when splitting into cases, the resulting lists of equations may involve inequations as well. Note: passing parameters automatically forces reducing the PDE system using integrability conditions, regardless of the value of the keyword integrabilityconditions.

with(Physics):

Setup(mathematicalnotation = true);

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

Setup(coordinates = X);

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

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

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

Define(V);

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

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

KillingVectors(V);

$\left[V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\mathrm{x2}\,-\mathrm{x1}\,0\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,0\,1\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\mathrm{x3}\,0\,-\mathrm{x1}\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\mathrm{x4}\,0\,0\,\mathrm{x1}\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[1\,0\,0\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,\mathrm{x4}\,0\,\mathrm{x2}\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,0\,\mathrm{x4}\,\mathrm{x3}\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,0\,0\,1\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,\mathrm{x3}\,-\mathrm{x2}\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,1\,0\,0\right]\right]$

KillingVectors(V[mu], specialize = false);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{specialize}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{specializeconstants}\text{'}$

$V_{\mathrm{\mu }}=\left[\mathrm{c\_\_1}\mathrm{x2}+\mathrm{c\_\_2}\mathrm{x3}+\mathrm{c\_\_3}\mathrm{x4}+\mathrm{c\_\_4}\,-\mathrm{c\_\_1}\mathrm{x1}+\mathrm{c\_\_5}\mathrm{x3}+\mathrm{c\_\_6}\mathrm{x4}+\mathrm{c\_\_7}\,-\mathrm{c\_\_2}\mathrm{x1}-\mathrm{c\_\_5}\mathrm{x2}+\mathrm{c\_\_8}\mathrm{x4}+\mathrm{c\_\_9}\,-\mathrm{c\_\_3}\mathrm{x1}-\mathrm{c\_\_6}\mathrm{x2}-\mathrm{c\_\_8}\mathrm{x3}+\mathrm{c\_\_10}\right]$

g_[];

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\mathrm{-1}& 0& 0& 0\\ 0& \mathrm{-1}& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& 0& 0& 1\end{array}\right]$

g_[sc];

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

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

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

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

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

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

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

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

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

PDEtools:-declare(V(X));

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

KillingVectors(V[alpha], output = equations);

$\left[{\left(V_3\right)}_r=\frac{2V_3}{r}\,{\left(V_3\right)}_{\mathrm{\theta }}=2\cot \left(\mathrm{\theta }\right)V_3-{\left(V_2\right)}_{\mathrm{\phi }}\,{\left(V_3\right)}_{\mathrm{\phi }}=-\cot \left(\mathrm{\theta }\right){\sin \left(\mathrm{\theta }\right)}^2{V}_2\,{\stackrel{\mathbf{.}}{V}}_3=0\,{\left(V_2\right)}_{\mathrm{\phi },\mathrm{\phi }}=-{\sin \left(\mathrm{\theta }\right)}^2{V}_2\left({\cot \left(\mathrm{\theta }\right)}^2+1\right)\,{\left(V_2\right)}_r=\frac{2V_2}{r}\,{\left(V_2\right)}_{\mathrm{\theta }}=0\,{\stackrel{\mathbf{.}}{V}}_2=0\,{\left(V_4\right)}_r=-\frac{2m{V}_4}{r\left(2m-r\right)}\,{\left(V_4\right)}_{\mathrm{\theta }}=0\,{\left(V_4\right)}_{\mathrm{\phi }}=0\,{\stackrel{\mathbf{.}}{V}}_4=0\,V_1=0\right]$

KillingVectors(V[alpha], output = equations, integrabilityconditions = false);

KillingVectors(V);

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

KillingVectors(V, specializeconstants = false);

$V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,\mathrm{c\_\_2}\sin \left(\mathrm{\phi }\right)+\mathrm{c\_\_3}\cos \left(\mathrm{\phi }\right)\,-\left(\mathrm{c\_\_3}\sin \left(\mathrm{\phi }\right)-\mathrm{c\_\_2}\cos \left(\mathrm{\phi }\right)\right)\cot \left(\mathrm{\theta }\right)+\mathrm{c\_\_4}\,\mathrm{c\_\_1}\right]$

KillingVectors(V, parameters = {m});

$\left\{\begin{array}{cc}\left[V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\cos \left(\mathrm{\theta }\right)t\,-\frac{t\sin \left(\mathrm{\theta }\right)}{r}\,0\,\cos \left(\mathrm{\theta }\right)r\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,0\,1\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\sin \left(\mathrm{\phi }\right)t\sin \left(\mathrm{\theta }\right)\,\frac{t\left(\sin \left(\mathrm{\theta }+\mathrm{\phi }\right)+\sin \left(-\mathrm{\theta }+\mathrm{\phi }\right)\right)}{2r}\,\frac{\cos \left(\mathrm{\phi }\right)t\csc \left(\mathrm{\theta }\right)}{r}\,\sin \left(\mathrm{\theta }\right)r\sin \left(\mathrm{\phi }\right)\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\cos \left(\mathrm{\phi }\right)t\sin \left(\mathrm{\theta }\right)\,\frac{t\left(\cos \left(-\mathrm{\theta }+\mathrm{\phi }\right)+\cos \left(\mathrm{\theta }+\mathrm{\phi }\right)\right)}{2r}\,-\frac{\sin \left(\mathrm{\phi }\right)t\csc \left(\mathrm{\theta }\right)}{r}\,\sin \left(\mathrm{\theta }\right)r\cos \left(\mathrm{\phi }\right)\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\cos \left(\mathrm{\theta }\right)\,-\frac{\sin \left(\mathrm{\theta }\right)}{r}\,0\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\sin \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\,\frac{\cos \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)}{r}\,\frac{\cos \left(\mathrm{\phi }\right)\csc \left(\mathrm{\theta }\right)}{r}\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)\,\frac{\cos \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)}{r}\,-\frac{\sin \left(\mathrm{\phi }\right)\csc \left(\mathrm{\theta }\right)}{r}\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,0\,0\,1\right]\,V_{\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]\,V_{\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]\right]& m=0\\ \left[V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,0\,0\,1\right]\,V_{\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]\,V_{\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]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,0\,1\,0\right]\right]& \mathrm{otherwise}\end{array}\right.$

KillingVectors(V, parameters = {m}, specializeconstants);

$\left\{\begin{array}{cc}\left[V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\cos \left(\mathrm{\theta }\right)t\,-\frac{t\sin \left(\mathrm{\theta }\right)}{r}\,0\,\cos \left(\mathrm{\theta }\right)r\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,0\,1\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\sin \left(\mathrm{\phi }\right)t\sin \left(\mathrm{\theta }\right)\,\frac{t\left(\sin \left(\mathrm{\theta }+\mathrm{\phi }\right)+\sin \left(-\mathrm{\theta }+\mathrm{\phi }\right)\right)}{2r}\,\frac{\cos \left(\mathrm{\phi }\right)t\csc \left(\mathrm{\theta }\right)}{r}\,\sin \left(\mathrm{\theta }\right)r\sin \left(\mathrm{\phi }\right)\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\cos \left(\mathrm{\phi }\right)t\sin \left(\mathrm{\theta }\right)\,\frac{t\left(\cos \left(-\mathrm{\theta }+\mathrm{\phi }\right)+\cos \left(\mathrm{\theta }+\mathrm{\phi }\right)\right)}{2r}\,-\frac{\sin \left(\mathrm{\phi }\right)t\csc \left(\mathrm{\theta }\right)}{r}\,\sin \left(\mathrm{\theta }\right)r\cos \left(\mathrm{\phi }\right)\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\cos \left(\mathrm{\theta }\right)\,-\frac{\sin \left(\mathrm{\theta }\right)}{r}\,0\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\sin \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\,\frac{\cos \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)}{r}\,\frac{\cos \left(\mathrm{\phi }\right)\csc \left(\mathrm{\theta }\right)}{r}\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)\,\frac{\cos \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)}{r}\,-\frac{\sin \left(\mathrm{\phi }\right)\csc \left(\mathrm{\theta }\right)}{r}\,0\right]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,0\,0\,1\right]\,V_{\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]\,V_{\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]\right]& m=0\\ \left[V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,0\,0\,1\right]\,V_{\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]\,V_{\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]\,V_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[0\,0\,1\,0\right]\right]& \mathrm{otherwise}\end{array}\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.