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The Cartan–Petrov–Pirani–Penrose classification [68, 69, 66] describes the different degeneracies (i.e., the number of coinciding PNDs):
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In NP language, if the tetrad vector is a principal null direction, 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 [29].
Theorem (Goldberg–Sachs). 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 (
).
In particular, this means that for all algebraically special metrics there is an everywhere shear-free NGC,
and a null tetrad exists such that . The main idea of this review is an asymptotic
generalization of this statement: for all asymptotically flat metrics, there exists a null tetrad such that the
and
harmonic coefficients of the asymptotic Weyl tensor components
and
(namely,
and
) vanish. Note that this is in reality a nontrivial condition only on
, since the
other three components vanish automatically when we recall that
and
are spin-weight two and
one respectively.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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