 DifferentialGeometry/Tensor/NPDirectionalDerivatives - Maple Help

Tensor[NPDirectionalDerivatives] - create procedures for the Newman-Penrose directional derivatives

Calling Sequences

NPDirectionalDerivatives(Fr, output)

Parameters

Fr      - the name of an initialized anholonomic frame, created from a null tetrad

output  - (optional) keyword argument output = "sequence" Description

 • Let $g$ be a metric with signature $\left(1,-1,-1,-1\right)$ and be a null tetrad for $g$. The Newman-Penrose directional derivative operators take derivatives of functions in the directions of the 4 vectors of the tetrad. See Newman and Penrose, Stewart.
 • The NPDirectionalDerivatives command returns a table with 4 entries "D", "Delta", "delta", "barDelta" corresponding to the customary labels assigned to the NP directional derivative operators $(D,\mathrm{Δ},$δ, $\stackrel{‾}{\mathrm{δ}}$). With optional keyword argument output = "sequence", the directional derivative operators are returned as a sequence of 4 Maple procedures.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form NPDirectionalDerivatives(...) only after executing the commands with(DifferentialGeometry); with(Tensor);  in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-NPDirectionalDerivatives. Examples

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

Example 1.

Define a manifold $S$ with coordinates $\left(t,x,y,z\right)$.

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],S\right)$
 ${\mathrm{frame name: S}}$ (2.1)

Define a metric of signature (1,-1,-1,-1).

 S > $g≔\mathrm{evalDG}\left({x}^{2}\mathrm{dt}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dt}-{y}^{2}\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}-{z}^{2}\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}-{t}^{2}\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{S}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{x}}^{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}{{y}}^{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}{{z}}^{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-}{{t}}^{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{S}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{x}}^{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}{{y}}^{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}{{z}}^{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-}{{t}}^{{2}}\right]\right]\right]\right)$ (2.2)

Define a null tetrad NTetrad for the metric $g$.  Use GRQuery to verify that NTetrad is a null tetrad.

 S > $\mathrm{NTetrad}≔\mathrm{evalDG}\left(\left[\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{x}\mathrm{D_t}+\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{t}\mathrm{D_z},\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{x}\mathrm{D_t}-\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{t}\mathrm{D_z},\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{y}\mathrm{D_x}+\frac{\frac{1}{2}I{2}^{\frac{1}{2}}}{z}\mathrm{D_y},\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{y}\mathrm{D_x}-\frac{\frac{1}{2}I{2}^{\frac{1}{2}}}{z}\mathrm{D_y}\right]\right)$
 ${\mathrm{NTetrad}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}\frac{\sqrt{{2}}}{{2}{}{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}\frac{\sqrt{{2}}}{{2}{}{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}{-}\frac{\sqrt{{2}}}{{2}{}{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}{-}\frac{\sqrt{{2}}}{{2}{}{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{y}}\right]{,}\left[\left[{3}\right]{,}\frac{\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{y}}\right]{,}\left[\left[{3}\right]{,}\frac{\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{y}}\right]{,}\left[\left[{3}\right]{,}\frac{{-}\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{y}}\right]{,}\left[\left[{3}\right]{,}\frac{{-}\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}\right]\right]\right]\right)\right]$ (2.3)
 S > $\mathrm{GRQuery}\left(\mathrm{NTetrad},g,"NullTetrad"\right)$
 ${\mathrm{true}}$ (2.4)

Calculate the NP directional derivative operators defined by the null tetrad NTetrad.  The result is a table with 4 indices.

 S > $\mathrm{NPDirDer}≔\mathrm{NPDirectionalDerivatives}\left(\mathrm{NTetrad}\right)$
 ${\mathrm{NPDirDer}}{:=}{\mathrm{table}}\left(\left[{"Delta"}{=}\left({f}{→}{\mathrm{LieDerivative}}{}\left({{E}}_{{2}}{,}{f}\right)\right){,}{"delta"}{=}\left({f}{→}{\mathrm{LieDerivative}}{}\left({{E}}_{{3}}{,}{f}\right)\right){,}{"D"}{=}\left({f}{→}{\mathrm{LieDerivative}}{}\left({{E}}_{{1}}{,}{f}\right)\right){,}{"bardelta"}{=}\left({f}{→}{\mathrm{LieDerivative}}{}\left({{E}}_{{4}}{,}{f}\right)\right)\right]\right)$ (2.5)
 S > $\mathrm{indices}\left(\mathrm{NPDirDer}\right)$
 $\left[{"Delta"}\right]{,}\left[{"delta"}\right]{,}\left[{"D"}\right]{,}\left[{"bardelta"}\right]$ (2.6)

The individual directional derivatives operators can be extracted from the table NPDirDer.

 S > $X≔\mathrm{NPDirDer}\left["D"\right]$
 ${X}{:=}{{\mathrm{NPDirDer}}}_{{"D"}}$ (2.7)
 S > $X\left(f\left(t,x,y,z\right)\right)$
 $\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}\left(\frac{{\partial }}{{\partial }{t}}{}{f}{}\left({t}{,}{x}{,}{y}{,}{z}\right)\right)}{{x}}{+}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}\left(\frac{{\partial }}{{\partial }{z}}{}{f}{}\left({t}{,}{x}{,}{y}{,}{z}\right)\right)}{{t}}$ (2.8)

Note that this coincides with the Lie derivative of $f\left(t,x,y,z\right)$ with respect to the first vector in NTetrad.

 S > $\mathrm{LieDerivative}\left(\mathrm{NTetrad}\left[1\right],f\left(t,x,y,z\right)\right)$
 $\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}\left(\frac{{\partial }}{{\partial }{t}}{}{f}{}\left({t}{,}{x}{,}{y}{,}{z}\right)\right)}{{x}}{+}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}\left(\frac{{\partial }}{{\partial }{z}}{}{f}{}\left({t}{,}{x}{,}{y}{,}{z}\right)\right)}{{t}}$ (2.9)

Example 2.

With the keyword argument output = "sequence", the command NPDirectionalDerivatives will return the directional derivatives operators as a sequence. (Note that the name D is protected by Maple.)

 S > $d,\mathrm{\Delta },\mathrm{\delta },\mathrm{bardela}≔\mathrm{NPDirectionalDerivatives}\left(\mathrm{NTetrad},\mathrm{output}="Sequence"\right)$
 ${d}{,}{\mathrm{Δ}}{,}{\mathrm{δ}}{,}{\mathrm{bardela}}{:=}{f}{→}{\mathrm{DifferentialGeometry:-LieDerivative}}{}\left({{E}}_{{1}}{,}{f}\right){,}{f}{→}{\mathrm{DifferentialGeometry:-LieDerivative}}{}\left({{E}}_{{2}}{,}{f}\right){,}{f}{→}{\mathrm{DifferentialGeometry:-LieDerivative}}{}\left({{E}}_{{3}}{,}{f}\right){,}{f}{→}{\mathrm{DifferentialGeometry:-LieDerivative}}{}\left({{E}}_{{4}}{,}{f}\right)$ (2.10)
 S > $\mathrm{\Delta }\left(f\left(t,x,y,z\right)\right)$
 $\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}\left(\frac{{\partial }}{{\partial }{t}}{}{f}{}\left({t}{,}{x}{,}{y}{,}{z}\right)\right)}{{x}}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}\left(\frac{{\partial }}{{\partial }{z}}{}{f}{}\left({t}{,}{x}{,}{y}{,}{z}\right)\right)}{{t}}$ (2.11)

Example 3.

When working with the NP formalism, it is usually advantageous to work with the anholonomic frame defined by the null tetrad.  To create anholonomic frames in DifferentialGeometry, see FrameData.

 S > $\mathrm{FD}≔\mathrm{FrameData}\left(\mathrm{NTetrad},\mathrm{NP}\right)$
 ${\mathrm{FD}}{:=}\left[\left[{\mathrm{E1}}{,}{\mathrm{E2}}\right]{=}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}{\mathrm{E1}}}{{t}{}{x}}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}{\mathrm{E2}}}{{t}{}{x}}{,}\left[{\mathrm{E1}}{,}{\mathrm{E3}}\right]{=}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E1}}}{{x}{}{y}}{+}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E2}}}{{x}{}{y}}{-}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E3}}}{{t}{}{z}}{+}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E4}}}{{t}{}{z}}{,}\left[{\mathrm{E1}}{,}{\mathrm{E4}}\right]{=}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E1}}}{{x}{}{y}}{+}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E2}}}{{x}{}{y}}{+}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E3}}}{{t}{}{z}}{-}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E4}}}{{t}{}{z}}{,}\left[{\mathrm{E2}}{,}{\mathrm{E3}}\right]{=}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E1}}}{{x}{}{y}}{+}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E2}}}{{x}{}{y}}{+}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E3}}}{{t}{}{z}}{-}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E4}}}{{t}{}{z}}{,}\left[{\mathrm{E2}}{,}{\mathrm{E4}}\right]{=}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E1}}}{{x}{}{y}}{+}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E2}}}{{x}{}{y}}{-}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E3}}}{{t}{}{z}}{+}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}{\mathrm{E4}}}{{t}{}{z}}{,}\left[{\mathrm{E3}}{,}{\mathrm{E4}}\right]{=}{-}\frac{\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{E3}}}{{z}{}{y}}{-}\frac{\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{E4}}}{{z}{}{y}}\right]$ (2.12)
 S > $\mathrm{DGsetup}\left(\mathrm{FD}\right)$
 ${\mathrm{frame name: NP}}$ (2.13)

We can now calculate the directional derivatives operators for the null tetrad with the second calling sequence.

 NP > $\mathrm{NPDirectionalDerivatives}\left(\mathrm{NP}\right)$
 ${\mathrm{table}}\left(\left[{"Delta"}{=}\left({f}{→}{\mathrm{DifferentialGeometry:-LieDerivative}}{}\left({{E}}_{{2}}{,}{f}\right)\right){,}{"delta"}{=}\left({f}{→}{\mathrm{DifferentialGeometry:-LieDerivative}}{}\left({{E}}_{{3}}{,}{f}\right)\right){,}{"D"}{=}\left({f}{→}{\mathrm{DifferentialGeometry:-LieDerivative}}{}\left({{E}}_{{1}}{,}{f}\right)\right){,}{"bardelta"}{=}\left({f}{→}{\mathrm{DifferentialGeometry:-LieDerivative}}{}\left({{E}}_{{4}}{,}{f}\right)\right)\right]\right)$ (2.14) See Also