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The Cartan–Petrov–Pirani–Penrose classification [52, 53, 50] describes the different degeneracies:
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In NP language, if the tetrad vector is a principle null vector, i.e.,
, then
automatically,
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For the algebraically-special metrics, the special cases are
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An outstanding feature of the algebraically-special metrics is contained in the beautiful Goldberg–Sachs theorem [18].
Theorem. For a nonflat vacuum spacetime, if there is an NGC that is shear-free, i.e., there is a
null vector field with (), then the spacetime is algebraically special and, conversely, if
a vacuum spacetime is algebraically special, there is an NGC with (
).
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