Weyl scalar

In the Newman–Penrose (NP) formalism of general relativity, Weyl scalars refer to a set of five complex scalars formula_1 which encode the ten independent components of the Weyl tensor of a four-dimensional spacetime.
Definitions.
Given a complex null tetrad formula_2 and with the convention formula_3, the Weyl-NP scalars are defined by
Note: If one adopts the convention formula_9, the definitions of formula_10 should take the opposite values; that is to say, formula_11 after the signature transition.
Alternative derivations.
According to the definitions above, one should find out the Weyl tensors before calculating the Weyl-NP scalars via contractions with relevant tetrad vectors. This method, however, does not fully reflect the spirit of Newman–Penrose formalism. As an alternative, one could firstly compute the spin coefficients and then use the NP field equations to derive the five Weyl-NP scalars
where formula_17 (used for formula_18) refers to the NP curvature scalar formula_19 which could be calculated directly from the spacetime metric formula_20.
Physical interpretation.
Szekeres (1965) gave an interpretation of the different Weyl scalars at large distances:
For a general asymptotically flat spacetime containing radiation (Petrov Type I), formula_22 & formula_23 can be transformed to zero by an appropriate choice of null tetrad. Thus these can be viewed as gauge quantities.
A particularly important case is the Weyl scalar formula_25.
It can be shown to describe outgoing gravitational radiation (in an asymptotically flat spacetime) as
Here, formula_30 and formula_31 are the "plus" and "cross" polarizations of gravitational radiation, and the double dots represent double time-differentiation.
There are, however, certain examples in which the interpretation listed above fails. These are exact vacuum solutions of the Einstein field equations with cylindrical symmetry. For instance, a static (infinitely long) cylinder can produce a gravitational field which has not only the expected "Coulomb"-like Weyl component formula_18, but also non-vanishing "transverse wave"-components formula_24 and formula_25. Furthermore, purely outgoing Einstein-Rosen waves have a non-zero "incoming transverse wave"-component formula_24.