DifferentialGeometry/Tensor/PetrovTypeDetails - Maple Help

Details for PetrovType

Description

 • The command PetrovType uses the algorithm of M. A. Acvevedo, M. M. Enciso-Aguilar, J. Lopez-Bonilla, M. A. Acvevedo, Petrov classification of the conformal tensor, Electronic Journal of Theoretical Physics, Vol. 9, (2006), 79-82 to determine the Petrov type. The algorithm depends upon certain invariants calculated from the Newman Penrose Weyl scalars ${\mathrm{Ψ}}_{0},{\mathrm{Ψ}}_{1},{\mathrm{Ψ}}_{2},{\mathrm{Ψ}}_{3},{\mathrm{Ψ}}_{4}$. These invariants are:

${G}_{0}=2\left({\mathrm{Ψ}}_{0}{\mathrm{Ψ}}_{2}-{\mathrm{Ψ}}_{1}^{2}\right)$

${G}_{1}=2\left({\mathrm{Ψ}}_{0}{\mathrm{Ψ}}_{3}-{\mathrm{Ψ}}_{1}{\mathrm{Ψ}}_{2}\right)$

${G}_{3}={\mathrm{Ψ}}_{1}{\mathrm{Ψ}}_{4}-{\mathrm{Ψ}}_{2}{\mathrm{Ψ}}_{3}$

${G}_{4}=2\left({\mathrm{Ψ}}_{2}{\mathrm{Ψ}}_{4}-{\mathrm{Ψ}}_{3}^{2}\right)$

${G}_{5}=2\left({\mathrm{Ψ}}_{1}{\mathrm{Ψ}}_{3}-{\mathrm{Ψ}}_{2}^{2}\right)$

$J=-{\mathrm{Ψ}}_{3}{G}_{1}+\frac{1}{2}\left({\mathrm{Ψ}}_{2}{G}_{5}+{\mathrm{Ψ}}_{4}{G}_{0}\right)$

If  then $\mathrm{λ}$ is determined by ${\mathrm{λ}}^{2}=\frac{1}{3}I$ and and

 • The algorithm is as follows:

Step 1. If ${\mathrm{Ψ}}_{0}={\mathrm{Ψ}}_{1}={\mathrm{Ψ}}_{2}={\mathrm{Ψ}}_{3}={\mathrm{Ψ}}_{4}$=0, then the Petrov type is O.

Step 2. Otherwise, if ${G}_{0}={G}_{1}={G}_{2}={G}_{3}={G}_{4}=0$, then the Petrov type is N.

Step 3. Otherwise, if $I=J=0$, then the Petrov type is III.

Step 4. Otherwise, if , then the Petrov type is I.

Step 5. If  and ${M}_{r}=0$ for $r=0,1,2,3,4$ and , then the Petrov type is D.

Step 6. Otherwise, the Petrov type is II.