where (s,,, and (3,,, are real spin-weight 0 scalars (type {b, b}; b indicating the boost-weight).
Thus, given a quantity with boost-weight b and spin-weight 1, the even parity piece
is simply ifev,, and the odd parity piece is iifor,,/ Similarly the complex conjugate of
such a quantity (same boost weight, but spin-weight -1) has even parity piece a'(ee,,
and odd parity piece -id'(oric. The relative minus sign between an odd-parity object
and its complex conjugate is a possible source of confusion, so we must he careful when
performing parity decompositions.
Symmetric, trace-free two-indexed tensors on S also have a simple parity decomposition.
It is easy to recognize the (two) components of such tensors as spin-weight +2 scalars.
That is, the components are of type {b + 2, b + 2}. We consider the parity decomposition
on S by creating the tensor from a vector on S ta, with boost-weight b and spin-weight 0:
Xab = nc~ckb + bc~cks Jub~cd~ckd, (3-5)
which can in turn he further decomposed into its even and odd parity pieces by applying
Equation 3-4 to yield
Xub = L( (2(cb) 8c~cd Jub~cd~c(~ d)xever, + 2(cib d8c~dhele
which provides us with a means of identifying the even and odd hits of symmetric
trace-free tensors on S. This result generalizes quite easily to n-indexed symmetric
trace-free tensors (with components of spin-weight in and boost-weight b) on S:
1 This agrees with the correspondence between the even and odd parity vector and
tensor spherical harmonics and the spin-weighted spherical harmonics (see Thorne's review
[8] for details) (i.e., the "i" comes along for the ride).