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Details for SegreType

Description

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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.

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If the Plebanski-Petrov type of the Ricci tensor R is O, then the Segre type of R is [(1,111)], [1,(111)], [(1,11),1], or [(2,11)].

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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)].

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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 [ZZ,(11)].

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If the Plebanski-Petrov type of the Ricci tensor R is III, then the Segre type of R is [3,1].

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If the Plebanski-Petrov type of the Ricci tensor R is II, then the Segre type of R is [2,11].

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If the Plebanski-Petrov type of the Ricci tensor R is I, then the Segre type of R is [1,111] or [ZZ11].

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The algorithm depends upon certain invariants calculated from the Newman-Penrose Ricci scalars Φ00,Φ01,Φ02,Φ10,Φ11,Φ20,Φ21,Φ22. These invariants are:

  χ0=2 Φ00Φ02Φ012

  χ1=Φ00Φ12+Φ02Φ102 Φ01Φ11

  χ2=13Φ00Φ222 Φ01Φ21+Φ02Φ20+4 Φ10Φ124 Φ112

  χ3=Φ10Φ222 Φ11Φ21+Φ12Φ20

  χ4=2Φ20Φ22Φ212

  E00=4Φ00Φ11Φ01Φ10

  E01=2Φ00Φ12Φ02Φ10

  E02=4Φ01Φ12Φ02Φ11

  E10=2Φ00Φ21Φ20Φ01

  E11= Φ00Φ22Φ02Φ20

  E12=2Φ01Φ22Φ02Φ21

  E20=4Φ01Φ21Φ20Φ11

  E21=2Φ10Φ22Φ20Φ12

  E22=4Φ11Φ22Φ12Φ21

  ΔΦ=Φ112Φ01Φ21

  r1=2χ22 Φ10Φ12+2 Φ112

  χ'0=2E00E02E012    χ'1=E00E12+E__02E__102 E__01E__11

  χ'2=13E00E222E01E21+E02E20+4E10E124E112  χ'3=E00E222 E__11E__21+ E__12E__20     χ'4=2E20E22E212  

  Ip=13χ0χ44 χ1χ3+3 χ22  r__3=r__12I__p

  Jp=χ0χ2χ4+2 χ1χ2χ3χ0χ32χ12χ4χ23

  H=r1IpJp

  r2=2Φ00Φ11Φ22+Φ01Φ12Φ20+Φ02Φ10Φ21Φ00Φ12Φ21Φ01Φ10Φ22Φ02Φ11Φ20   ki=JpEii2 r2IpΦiiH,ii=0, 1, 2;  no summing

  Dp=Jp2Ip3

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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 Φ00=Φ01= ... = Φ22=0, then the Segre type S = [(1,111)].

Step A2. Otherwise, if ΔΦ=0, then S = [(2,11)].

Step A3. Otherwise, if E00>0, then S = [1,(111)].

Step A4. Otherwise, if E000, then S = [(1,11)1].

 

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

Step B1. If r1=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 r1=0, then S=ZZ&conjugate0;11.  

Step C2. If r10 and all the E00=E01= ... =E22=0, then S = [(1,1)(11)].

Step C3. If r10, χ__00 and χ'0=0, then S = [(2,11)].  

Step C4. If r10, χ0=0 and χ'2=0, then S = [2,(11)].

Step C5. If  H&equals;0, then S = [2,(11)], while if H<0, then S&equals;ZZ&conjugate0;11.

 

D. The Plebanski-Petrov type of the Ricci tensor R is 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 Dp<0, then S =[1,111] while if Dp&gt;0, then S = [ZZ11].