A vector field $X$is called a Killing vector for the metric $g$ if$\mathit{ }{\mathrm{\ℒ}}_{X}g \= 0$, where${\mathrm{\ℒ}}_X$ denotes Lie differentiation with respect to $X\.$ If $X_j \= g_{\mathrm{jk}} X^k$ and $▿$denotes covariant differentiation with respect to the given metric, then this equation can be written as${▿}_{\(i}{X}_{j\)}\= 0.$The set of all Killing vectors forms a finite dimensional Lie algebra of vector fields.

The program KillingVectors generates the defining system of 1st order PDE for a Killing vector and uses pdsolve to find the solutions to these PDE.

The keyword argument coefficientvariables $\= \left[x^1 \, x^2\, ..\. \, x^k\right]$ allows the user to specify the coefficient functions in the Killing vector $X$as functions of the variables $\left[x^1 \, x^2\, ..\. \, x^k\right]$.

The exact form of the Killing vector $X$can be specified with the keyword argument ansatz $\= X$ . For example, if the coordinates on the underlying manifold are $\left[x\, y\, z\right]$and $X_1\, X_2$are defined vectors, then one may solve for Killing vectors of the form $X \= f\left(y\,z\right)X_1 \+ g\left(y\,z\right)X_2$. In this situation the unknown functions must be explicitly specified with the keyword argument unknowns, for example, unknowns $\= \left[f\left(y\,z\right)\, g\left(y\,z\right)\right]\.$

When using the keyword argument ansatz, additional algebraic or differential conditions may be imposed upon the unknowns using the keyword argument auxiliaryequations $\= \mathrm{EqList}\.$Here $\mathrm{EqList}$is a list of the auxiliary equations to be added to the Killing equations.

If the metric $g$depends upon a number of unspecified parameters (either constants or functions), then the keyword argument parameters$\= \mathrm{ParList}\,$where $\mathrm{ParList}$is the list of parameters, will invoke case-splitting with respect to these parameters. Special values of the parameters, where either the number or the explicit form of the Killing vectors changes, are calculated. The keyword argument auxiliaryequations can be used to specify additional algebraic or differential conditions on the parameters.

With keyword argument output = $''pde''\,$the defining partial differential equations for the Killing vectors are returned. The option output = $''general''$returns the general Killing vector in terms of a number of arbitrary constants ${\mathrm{\_C}}_1$, ${\mathrm{\_C}}_{2 }\,$... . The option output = $''list''$returns a list of vectors which form a basis for the solution space. The default value of this keyword argument is output = $''list''\.$

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form KillingVectors(...) only after executing the commands with(DifferentialGeometry), with(Tensor), in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-KillingVectors(...)

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,P\right)\:$

$\mathrm{g1}≔\mathrm{evalDG}\left(\frac{1}{y^2}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}\right)\right)$

$\mathrm{g1}:=\frac{1}{y^2}\mathrm{dx}\mathrm{dx}+\frac{1}{y^2}\mathrm{dy}\mathrm{dy}$

$\mathrm{K1}≔\mathrm{KillingVectors}\left(\mathrm{g1}\right)$

$\mathrm{K1}:=\left[\left(\frac{x^2}{2}-\frac{y^2}{2}\right)\mathrm{D\_x}+yx\mathrm{D\_y}\,x\mathrm{D\_x}+y\mathrm{D\_y}\,\mathrm{D\_x}\right]$

$\mathrm{map}\left(\mathrm{LieDerivative}\,\mathrm{K1}\,\mathrm{g1}\right)$

$\left[0\mathrm{dx}\mathrm{dx}\,0\mathrm{dx}\mathrm{dx}\,0\mathrm{dx}\mathrm{dx}\right]$

$\mathrm{KillingVectors}\left(\mathrm{g1}\,\mathrm{output}=''general''\right)$

$\left(\frac{1}{2}{x}^2\mathrm{\_C1}+\mathrm{\_C2}x-\frac{1}{2}\mathrm{\_C1}{y}^2+\mathrm{\_C3}\right)\mathrm{D\_x}+y\left(\mathrm{\_C1}x+\mathrm{\_C2}\right)\mathrm{D\_y}$

$L≔\mathrm{LieAlgebraData}\left(\mathrm{K1}\,\mathrm{Sym}\right)$

$L:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=-\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=-\mathrm{e3}\right]$

$\mathrm{DGsetup}\left(L\right)$

$\mathrm{Lie\; algebra:\; Sym}$

$\mathrm{Query}\left(''Semisimple''\right)$

$\mathrm{true}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{g2}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\left(ax+bz+1\right)\mathrm{dy}\&t\mathrm{dy}+\frac{1}{ax+by+1}\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g2}:=\mathrm{dx}\mathrm{dx}+\left(ax+bz+1\right)\mathrm{dy}\mathrm{dy}+\frac{1}{ax+by+1}\mathrm{dz}\mathrm{dz}$

$\mathrm{K2}≔\mathrm{KillingVectors}\left(\mathrm{g2}\,\mathrm{parameters}=\left[a\,b\right]\right)$

$\mathrm{K2}:=\left[-z\mathrm{D\_x}+x\mathrm{D\_z}\,-z\mathrm{D\_y}+y\mathrm{D\_z}\,\mathrm{D\_z}\,-y\mathrm{D\_x}+x\mathrm{D\_y}\,\mathrm{D\_y}\,\mathrm{D\_x}\right]\,\left[\mathrm{D\_x}\right]\,\left[\mathrm{D\_z}\,\mathrm{D\_y}\right]\,\left[-\frac{b}{a}\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_z}\right]\,\left[\left\{a\=0\,b\=0\right\}\,\left\{a\=0\,b\=b\right\}\,\left\{a\=a\,b\=0\right\}\,\left\{a\=a\,b\=b\right\}\right]$

$\mathrm{Cases}≔\mathrm{K2}\left[-1\right]$

$\mathrm{Cases}:=\left[\left\{a\=0\,b\=0\right\}\,\left\{a\=0\,b\=b\right\}\,\left\{a\=a\,b\=0\right\}\,\left\{a\=a\,b\=b\right\}\right]$

$\mathrm{nops}\left(\mathrm{Cases}\right)$

$4$

$\mathrm{K2}\left[1\right]$

$\left[-z\mathrm{D\_x}+x\mathrm{D\_z}\,-z\mathrm{D\_y}+y\mathrm{D\_z}\,\mathrm{D\_z}\,-y\mathrm{D\_x}+x\mathrm{D\_y}\,\mathrm{D\_y}\,\mathrm{D\_x}\right]$

$\mathrm{K2}\left[2\right]$

$\left[\mathrm{D\_x}\right]$

$\mathrm{K2}\left[3\right]$

$\left[\mathrm{D\_z}\,\mathrm{D\_y}\right]$

$\mathrm{K2}\left[4\right]$

$\left[-\frac{b}{a}\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_z}\right]$

$\mathrm{KillingVectors}\left(\mathrm{g2}\,\mathrm{parameters}=\left[a\,b\right]\,\mathrm{auxiliaryequations}=\left[a^2+b^2\ne 0\right]\right)$

$\left[\mathrm{D\_x}\right]\,\left[\mathrm{D\_z}\,\mathrm{D\_y}\right]\,\left[-\frac{b}{a}\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_z}\right]\,\left[\left\{a\=0\,b\=b\right\}\,\left\{a\=a\,b\=0\right\}\,\left\{a\=a\,b\=b\right\}\right]$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{g3}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\left(f\left(x\right)+y^2\right)\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g3}:=\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}+\left(f\left(x\right)+y^2\right)\mathrm{dz}\mathrm{dz}$

$K≔\mathrm{KillingVectors}\left(\mathrm{g3}\,\mathrm{parameters}=\left\{f\left(x\right)\right\}\,\mathrm{auxiliaryequations}=\left\{f\left(x\right)\ne 0\right\}\right)$

$K:=\left[\mathrm{D\_z}\,\mathrm{D\_x}\right]\,\left[\mathrm{D\_z}\right]\,\left[\mathrm{D\_z}\,-yz\mathrm{D\_x}-\left(\mathrm{\_C1}-x\right)z\mathrm{D\_y}+\arctan \left(\frac{y}{\mathrm{\_C1}-x}\right)\mathrm{D\_z}\,y\mathrm{D\_x}+\left(\mathrm{\_C1}-x\right)\mathrm{D\_y}\right]\,\left[\mathrm{D\_z}\,-2y\mathrm{D\_x}+\left(\mathrm{\_C1}+2x\right)\mathrm{D\_y}\right]\,\left[\left\{f\left(x\right)\=\mathrm{\_C1}\right\}\,\left\{f\left(x\right)\=f\left(x\right)\right\}\,\left\{f\left(x\right)\={\mathrm{\_C1}}^2-2\mathrm{\_C1}x\+x^2\right\}\,\left\{f\left(x\right)\=\mathrm{\_C1}x\+x^2\+\mathrm{\_C2}\right\}\right]$

$\mathrm{g3a}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\left(x^2+y^2\right)\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g3a}:=\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}+\left(x^2+y^2\right)\mathrm{dz}\mathrm{dz}$

$\mathrm{KillingVectors}\left(\mathrm{g3a}\right)$

$\left[yz\mathrm{D\_x}-zx\mathrm{D\_y}\+\arctan \left(\frac{y}{x}\right)\mathrm{D\_z}\,\mathrm{D\_z}\,-y\mathrm{D\_x}\+x\mathrm{D\_y}\right]$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,t\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{\Omega }≔\mathrm{evalDG}\left(\left[\mathrm{dx}\,\mathrm{dy}\,\frac{1}{\mathrm{sqrt}\left(2\right)}\exp \left(x\right)\mathrm{dz}\,\mathrm{dt}+\exp \left(x\right)\mathrm{dz}\right]\right)$

$\mathrm{\Ω}:=\left[\mathrm{dx}\,\mathrm{dy}\,\frac{\sqrt{2}{\ⅇ}^x}{2}\mathrm{dz}\,{\ⅇ}^x\mathrm{dz}+\mathrm{dt}\right]$

$\mathrm{FD}≔\mathrm{FrameData}\left(\mathrm{\Omega }\,''Godel''\right)$

$\mathrm{FD}:=\left[d\mathrm{\Θ1}\=0\,d\mathrm{\Θ2}\=0\,d\mathrm{\Θ3}\=\mathrm{\Θ1}\bigwedge \mathrm{\Θ3}\,d\mathrm{\Θ4}\=\sqrt{2}\mathrm{\Θ1}\bigwedge \mathrm{\Θ3}\right]$

$\mathrm{DGsetup}\left(\mathrm{FD}\,\mathrm{verbose}\right)$

$\mathrm{The\; following\; coordinates\; have\; been\; protected:}$

$\left[x\,y\,z\,t\right]$

$\mathrm{The\; following\; vector\; fields\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{E1}\,\mathrm{E2}\,\mathrm{E3}\,\mathrm{E4}\right]$

$\mathrm{The\; following\; differential\; 1-forms\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\Θ1}\,\mathrm{\Θ2}\,\mathrm{\Θ3}\,\mathrm{\Θ4}\right]$

$\mathrm{frame\; name:\; Godel}$

$\mathrm{g4}≔\mathrm{evalDG}\left(\mathrm{\Θ1}\&t\mathrm{\Θ1}+\mathrm{\Θ2}\&t\mathrm{\Θ2}+\mathrm{\Θ3}\&t\mathrm{\Θ3}-\mathrm{\Θ4}\&t\mathrm{\Θ4}\right)$

$\mathrm{g4}:=\mathrm{\Θ1}\mathrm{\Θ1}+\mathrm{\Θ2}\mathrm{\Θ2}+\mathrm{\Θ3}\mathrm{\Θ3}-\mathrm{\Θ4}\mathrm{\Θ4}$

$\mathrm{\Phi }≔\mathrm{Transformation}\left(M\,''Godel''\,\left[x=x\,y=y\,z=z\,t=t\right]\right)$

$\mathrm{\Φ}:=\left[x=x\,y=y\,z=z\,t=t\right]$

$\mathrm{PushPullTensor}\left(\mathrm{\Phi }\,\mathrm{g4}\right)$

$\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}-\frac{{\ⅇ}^{2x}}{2}\mathrm{dz}\mathrm{dz}-{\ⅇ}^x\mathrm{dz}\mathrm{dt}-{\ⅇ}^x\mathrm{dt}\mathrm{dz}-\mathrm{dt}\mathrm{dt}$

$\mathrm{K4}≔\mathrm{KillingVectors}\left(\mathrm{g4}\right)$

$\mathrm{K4}:=\left[z\mathrm{E1}-\frac{\left({\ⅇ}^x{z}^2-2{\ⅇ}^{-x}\right)\sqrt{2}}{4}\mathrm{E3}-\left(\frac{{\ⅇ}^x{z}^2}{2}+{\ⅇ}^{-x}\right)\mathrm{E4}\,\mathrm{E1}-\frac{{\ⅇ}^{x}z\sqrt{2}}{2}\mathrm{E3}-{\ⅇ}^{x}z\mathrm{E4}\,-\frac{\sqrt{2}{\ⅇ}^x}{2}\mathrm{E3}-{\ⅇ}^x\mathrm{E4}\,-\mathrm{E4}\,\mathrm{E2}\right]$

$\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{K4}\,\mathrm{Sym}\right)$

$\mathrm{L2}:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e3}\right]$

$\mathrm{DGsetup}\left(\mathrm{L2}\right)$

$\mathrm{Lie\; algebra:\; Sym}$

$\mathrm{DA}≔\mathrm{Decompose}\left(\mathrm{factoralgebras}=\mathrm{true}\right)$

$\mathrm{L3}≔\mathrm{LieAlgebraData}\left(\mathrm{DA}\left[2\right]\right)$

$\mathrm{L3}:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e3}\right]$

$\mathrm{DGsetup}\left(\mathrm{L3}\,\left[f\right]\,\left[\mathrm{\alpha }\right]\right)$

$\mathrm{Lie\; algebra:\; L1}$

$\mathrm{MultiplicationTable}\left(''LieTable''\right)$