 KillingSpinors - Maple Help

Tensor[KillingSpinors] - calculate the Killing spinors for a given spacetime

Calling Sequences

KillingSpinors( ${\mathbf{σ}}$$,$pqoptions)

Parameters

$\mathrm{σ}$        - a solder form on a 4-dimensional space-time

p, q     - non-negative integers which specify the number of un-primed and primed indices for the Killing spinor

options  - any of the following keyword arguments: ansatz, unknowns, auxiliaryequations, coefficientvariables, parameters, output Description

 • Let $▿$denote covariant differentiation with respect to the given solder form $\mathrm{σ}.$ A type  symmetric spinor with components ${S}_{B\cdot \cdot \cdot C}^{B'\cdot \cdot \cdot C'}$ ( $p$ lower unprimed indices and upper primed indices) is a Killing spinor if${▿}_{(A}^{(A'}{S}_{B\cdot \cdot \cdot C)}^{B'\cdot \cdot \cdot C')}$ = 0.
 • The command KillingSpinor generates the defining system of 1st order PDE for a Killing spinor and uses pdsolve to find the solutions to these PDE.
 • The keyword argument coefficientvariables  allows the user to specify the coefficient functions in the Killing spinor $S$as functions of the variables  .
 • The exact form of the spinor can be specified with the keyword argument ansatz  For example, if the coordinates on the underlying manifold are and are defined type spinors, then one may solve for Killing spinors tensors of the form . In this situation the unknown functions must be explicitly specified with the keyword argument unknowns, for example, unknowns
 • When using the keyword argument ansatz, additional algebraic or differential conditions may be imposed upon the unknowns using the keyword argument auxiliaryequations Here is a list of the auxiliary equations to be added to the Killing spinor equations.
 • If the solder form depends upon a number of unspecified parameters (either constants or functions), then the keyword argument parameterswhere 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 spinors changes, are calculated. Additional algebraic or differential conditions may be imposed upon the parameters using the keyword argument auxiliaryequations
 • With keyword argument output = $"pde",$the defining partial differential equations for the Killing spinor are returned. The option output = returns the general solution in terms of a number of arbitrary constants ${\mathrm{_C}}_{1}$, ${\mathrm{_C}}_{2}$, ... while the option output = returns a list of tensors 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 KillingSpinor(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-KillingSpinor(...). Examples

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

Example 1.

We find the Killing spinors of type (1, 0), (0, 1) and (1, 1) on the spacetime with metric $g$$.$

 > $\mathrm{DGsetup}\left(\left[u,v,x,y\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 M > $gâ‰”\mathrm{evalDG}\left(2{ⅇ}^{2\mathrm{ρ}x}\mathrm{du}&t\mathrm{du}-\mathrm{du}&t\mathrm{dv}-\mathrm{dv}&t\mathrm{du}-\mathrm{dx}&t\mathrm{dx}-\mathrm{dy}&t\mathrm{dy}\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{2}{}{{ⅇ}}^{{2}{}{\mathrm{\rho }}{}{x}}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{2}{}{{ⅇ}}^{{2}{}{\mathrm{\rho }}{}{x}}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{2}{}{{ⅇ}}^{{2}{}{\mathrm{\rho }}{}{x}}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{2}{}{{ⅇ}}^{{2}{}{\mathrm{\rho }}{}{x}}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right)$ (2.2)

Define an orthonormal tetrad for this metric.

 M > $\mathrm{OT}â‰”\mathrm{evalDG}\left(\left[-\frac{1{ⅇ}^{-\mathrm{ρ}x}\mathrm{D_u}}{2\mathrm{ρ}}+\frac{1{ⅇ}^{\mathrm{ρ}x}\left(2{\mathrm{ρ}}^{2}-1\right)\mathrm{D_v}}{2\mathrm{ρ}},\mathrm{D_x},\mathrm{D_y},\frac{1{ⅇ}^{-\mathrm{ρ}x}\mathrm{D_u}}{2\mathrm{ρ}}+\frac{1{ⅇ}^{\mathrm{ρ}x}\left(1+2{\mathrm{ρ}}^{2}\right)\mathrm{D_v}}{2\mathrm{ρ}}\right]\right)$
 ${\mathrm{OT}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{2}\right]{,}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\left({2}{}{{\mathrm{\rho }}}^{{2}}{-}{1}\right)}{{2}{}{\mathrm{\rho }}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{2}\right]{,}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\left({2}{}{{\mathrm{\rho }}}^{{2}}{-}{1}\right)}{{2}{}{\mathrm{\rho }}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{2}\right]{,}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\left({2}{}{{\mathrm{\rho }}}^{{2}}{-}{1}\right)}{{2}{}{\mathrm{\rho }}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{2}\right]{,}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\left({2}{}{{\mathrm{\rho }}}^{{2}}{-}{1}\right)}{{2}{}{\mathrm{\rho }}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{2}\right]{,}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\left({2}{}{{\mathrm{\rho }}}^{{2}}{+}{1}\right)}{{2}{}{\mathrm{\rho }}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{2}\right]{,}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\left({2}{}{{\mathrm{\rho }}}^{{2}}{+}{1}\right)}{{2}{}{\mathrm{\rho }}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{2}\right]{,}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\left({2}{}{{\mathrm{\rho }}}^{{2}}{+}{1}\right)}{{2}{}{\mathrm{\rho }}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{2}\right]{,}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\left({2}{}{{\mathrm{\rho }}}^{{2}}{+}{1}\right)}{{2}{}{\mathrm{\rho }}}\right]\right]\right]\right)\right]$ (2.3)

Use the command SolderForm to find the solder form defined by this orthonormal tetrad.

 M > $\mathrm{σ}â‰”\mathrm{SolderForm}\left(\mathrm{OT}\right)$
 ${\mathrm{σ}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{5}{,}{7}\right]{,}{-}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{1}{,}{6}{,}{8}\right]{,}{-}{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}{}{\mathrm{\rho }}\right]{,}\left[\left[{2}{,}{5}{,}{7}\right]{,}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{3}{,}{5}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{8}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{5}{,}{7}\right]{,}{-}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{1}{,}{6}{,}{8}\right]{,}{-}{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}{}{\mathrm{\rho }}\right]{,}\left[\left[{2}{,}{5}{,}{7}\right]{,}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{3}{,}{5}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{8}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{5}{,}{7}\right]{,}{-}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{1}{,}{6}{,}{8}\right]{,}{-}{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}{}{\mathrm{\rho }}\right]{,}\left[\left[{2}{,}{5}{,}{7}\right]{,}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{3}{,}{5}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{8}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{5}{,}{7}\right]{,}{-}\frac{{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{1}{,}{6}{,}{8}\right]{,}{-}{{ⅇ}}^{{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}{}{\mathrm{\rho }}\right]{,}\left[\left[{2}{,}{5}{,}{7}\right]{,}\frac{{{ⅇ}}^{{-}{\mathrm{\rho }}{}{x}}{}\sqrt{{2}}}{{2}{}{\mathrm{\rho }}}\right]{,}\left[\left[{3}{,}{5}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{8}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right)$ (2.4)

We can check this solder form is compatible with the metric using the SpinorInnerProduct command.

 M > $\mathrm{SpinorInnerProduct}\left(\mathrm{σ},\mathrm{σ}\right)$
 ${2}{}{{ⅇ}}^{{2}{}{\mathrm{ρ}}{}{x}}{}{\mathrm{du}}{}{\mathrm{du}}{-}{\mathrm{du}}{}{\mathrm{dv}}{-}{\mathrm{dv}}{}{\mathrm{du}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.5)

There is one Killing spinor of type (1, 0) , one of type (0, 1) , and 5 of type (1, 1).

 M > $\mathrm{KS1}â‰”\mathrm{KillingSpinors}\left(\mathrm{σ},1,0\right)$
 ${\mathrm{KS1}}{:=}\left[{{ⅇ}}^{{-}\frac{{1}}{{2}}{}{\mathrm{ρ}}{}{x}}{}{\mathrm{D_z1}}\right]$ (2.6)
 M > $\mathrm{KS2}â‰”\mathrm{KillingSpinors}\left(\mathrm{σ},0,1\right)$
 ${\mathrm{KS2}}{:=}\left[{{ⅇ}}^{{-}\frac{{1}}{{2}}{}{\mathrm{ρ}}{}{x}}{}{\mathrm{dw2}}\right]$ (2.7)
 M > $\mathrm{KS3}â‰”\mathrm{KillingSpinors}\left(\mathrm{σ},1,1\right)$
 ${\mathrm{KS3}}{:=}\left[{u}{}{\mathrm{D_z1}}{}{\mathrm{dw1}}{+}\frac{{I}{}{{ⅇ}}^{{-}{\mathrm{ρ}}{}{x}}{}{y}{}{\mathrm{D_z1}}{}{\mathrm{dw2}}}{{\mathrm{ρ}}}{+}{u}{}{\mathrm{D_z2}}{}{\mathrm{dw2}}{,}{\mathrm{D_z1}}{}{\mathrm{dw1}}{+}{\mathrm{D_z2}}{}{\mathrm{dw2}}{,}\frac{{\mathrm{D_z1}}{}{\mathrm{dw1}}}{{{\mathrm{ρ}}}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{\left({{ⅇ}}^{{\mathrm{ρ}}{}{x}}{}{u}{+}{{ⅇ}}^{{-}{\mathrm{ρ}}{}{x}}{}{v}\right){}{\mathrm{D_z1}}{}{\mathrm{dw2}}}{{{\mathrm{ρ}}}^{{2}}}{+}{{ⅇ}}^{{\mathrm{ρ}}{}{x}}{}{u}{}{\mathrm{D_z2}}{}{\mathrm{dw1}}{,}{-}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{\mathrm{ρ}}{}{x}}{}{\mathrm{D_z1}}{}{\mathrm{dw2}}}{{{\mathrm{ρ}}}^{{2}}}{+}{{ⅇ}}^{{\mathrm{ρ}}{}{x}}{}{\mathrm{D_z2}}{}{\mathrm{dw1}}{,}{{ⅇ}}^{{-}{\mathrm{ρ}}{}{x}}{}{\mathrm{D_z1}}{}{\mathrm{dw2}}\right]$ (2.8)