Finally, we remark that scalars naturally arising from contractions of tensors in the
spacetime with various combinations of la and if have no components in S and are thus
all of even-parity. Note that such objects necessarily have zero spin-weight. This provides
enough information to characterize the parity of arbitrary objects.
In practice, we are generally given some spin- and boost-weighted scalar, q' (and/or
its complex conjugate), and we merely want to identify the even- and odd-parity pieces
without explicitly decomposing it according to Equation :37. In this case Equation :37
allows us to do so by simply writing
In the context of a spacetime where 1" and if are fixed by considerations other than being
orthogonal to of and of (e.g. Petrov type D, where we would like them aligned with
the principal null directions), but of and of fail to form a closed 2-surface (the K~err
spacetime provides one such example; this can he seen by noting that B and a' don't
commute), the question arises of whether or not something like Equation :38 is still useful
to consider. It appears so. In such a case the decomposition theorems (the first lines of
Equations :35 and :34) fail to be true, but this isn't a serious issue. Because a~b and Feb
still allow us to decompose tensors into their "proper" and pI-, ud.I" pieces, in place of
Equation :37 we have
70..0, = (-1)nn![n s .. z ,,3 tee, i .,,)+ i ,. .ni,,3 sve + ir ,,) (:39)
where i..~ 1 and "odd" are written in quotes to emphasize the fact that they really
refer to real and imaginary in this context and the bar over tau indicates the proper spin-
and boost-weight. Clearly, Equation :39, lacks the advantage present in Equation :37
of being able to put all of the angular dependence into B and a' and regard the entire
tensor as arising from the two real scalars -rev,, and -r,., Nevertheless it provides a useful
decomposition of spin- and boost-weighted scalars, without separation of variables, that