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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 is O, then the Segre type of is [(1,111)], [1,(111)], [(1,11),1], or [(2,11)].
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If the Plebanski-Petrov type of the Ricci tensor is N, then the Segre type of is [(2,1)1] or [(3,1)].
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If the Plebanski-Petrov type of the Ricci tensor is , then the Segre type of is [(1,1)(11)], [1,1(11)], [(1,1)11], [2,(11)], or [,(11)].
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If the Plebanski-Petrov type of the Ricci tensor is III, then the Segre type of is [3,1].
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If the Plebanski-Petrov type of the Ricci tensor is II, then the Segre type of is [2,11].
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If the Plebanski-Petrov type of the Ricci tensor is I, then the Segre type of is [1,111] or [].
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The algorithm depends upon certain invariants calculated from the Newman-Penrose Ricci scalars . These invariants are:
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no summing
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Here are the details of the algorithm.
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A. The Plebanski-Petrov type of the Ricci tensor is O.
Step A1. If all the Ricci scalars , then the Segre type = [(1,111)].
Step A2. Otherwise, if , then S = [(2,11)].
Step A3. Otherwise, if , then S = [1,(111)].
Step A4. Otherwise, if , then S = [(1,11)1].
B. The Plebanski-Petrov type of the Ricci tensor is N.
Step B1. If , then = [(3,1)], otherwise = [(2,1)1].
C. The Plebanski-Petrov type of the Ricci tensor is D.
Step C1. If , then .
Step C2. If and all the , then = [(1,1)(11)].
Step C3. If , and , then = [(2,11)].
Step C4. If , and , then = [2,(11)].
Step C5. If , then = [2,(11)], while if , then .
D. The Plebanski-Petrov type of the Ricci tensor is .
Step D1. The Segre type of is [3,1].
E. The Plebanski-Petrov type of the Ricci tensor tensor is II.
Step E1. Then Segre type of is [2, 11].
F. The Plebanski-Petrov type of the Ricci tensor tensor is I.
Step F1. If , then =[1,111] while if , then = [].