DifferentialGeometry/Tensor/SegreTypeDetails - Maple Help

Details for SegreType

Description

 • The command SegreType uses the algorithm of E. Zakhary and J. Carminati, A New Algorithm for the Segre Classification of the Trace-Free Ricci Tensor, General Relativity and Gravitation,Vol 36, (2004), 1015-1038 to determine the Segre type. The algorithm first calculates the Plebanski-Petrov type and then the Segre type.
 • If the Plebanski-Petrov type of the Ricci tensor is O, then the Segre type of $R$ is [(1,111)], [1,(111)], [(1,11),1], or [(2,11)].
 • If the Plebanski-Petrov type of the Ricci tensor $R$ is N, then the Segre type of $R$ is [(2,1)1] or [(3,1)].
 • If the Plebanski-Petrov type of the Ricci tensor $R$ is $D$, then the Segre type of $R$ is [(1,1)(11)], [1,1(11)], [(1,1)11], [2,(11)], or [$\stackrel{‾}{Z}Z$,(11)].
 • If the Plebanski-Petrov type of the Ricci tensor $R$ is III, then the Segre type of $R$ is [3,1].
 • If the Plebanski-Petrov type of the Ricci tensor $R$ is II, then the Segre type of $R$ is [2,11].
 • If the Plebanski-Petrov type of the Ricci tensor $R$ is I, then the Segre type of $R$ is [1,111] or [$\stackrel{‾}{Z}\mathrm{Z11}$].
 • The algorithm depends upon certain invariants calculated from the Newman-Penrose Ricci scalars ${\mathrm{Φ}}_{00},{\mathrm{Φ}}_{01},{\mathrm{Φ}}_{02},{\mathrm{Φ}}_{10},{\mathrm{Φ}}_{11},{\mathrm{Φ}}_{20},{\mathrm{Φ}}_{21},{\mathrm{Φ}}_{22}$. These invariants are:

 no summing

 • Here are the details of the algorithm.

A. The Plebanski-Petrov type of the Ricci tensor $R$ is O.

Step A1. If all the Ricci scalars , then the Segre type $S$ = [(1,111)].

Step A2. Otherwise, if ${\mathrm{Δ}}_{\mathrm{Φ}}=0$, then S = [(2,11)].

Step A3. Otherwise, if ${E}_{00}>0$, then S = [1,(111)].

Step A4. Otherwise, if ${E}_{00}\le 0$, then S = [(1,11)1].

B. The Plebanski-Petrov type of the Ricci tensor $R$ is N.

Step B1. If ${r}_{1}=0$, then $S$ = [(3,1)], otherwise $S$ = [(2,1)1].

C. The Plebanski-Petrov type of the Ricci tensor $R$ is D.

Step C1. If ${r}_{1}=0$, then $S=\left[Z\stackrel{&conjugate0;}{Z}\left(11\right)\right]$.

Step C2. If ${r}_{1}\ne 0$ and all the , then $S$ = [(1,1)(11)].

Step C3. If ${r}_{1}\ne 0$, $\mathrm{χ__0}\ne 0$ and $\mathrm{χ}{'}_{0}=0$, then $S$ = [(2,11)].

Step C4. If ${r}_{1}\ne 0$, and $\mathrm{χ}{'}_{2}=0$, then $S$ = [2,(11)].

Step C5. If  $H=0$, then $S$ = [2,(11)], while if $H<0$, then $S=\left[Z\stackrel{&conjugate0;}{Z}\left(11\right)\right]$.

D. The Plebanski-Petrov type of the Ricci tensor $R$ is $\mathrm{III}$.

Step D1. The Segre type of $R$ is [3,1].

E. The Plebanski-Petrov type of the Ricci tensor $R$ tensor is II.

Step E1. Then Segre type of $R$ is [2, 11].

F. The Plebanski-Petrov type of the Ricci tensor $R$ tensor is I.

Step F1. If ${\mathrm{D}}_{p}<0$, then $S$ =[1,111] while if ${\mathrm{D}}_{p}>0$, then $S$ = [$\stackrel{‾}{Z}\mathrm{Z11}$].