APPENDIX D
SPIN-WEIGHTED SPHERICAL HARMONICS
In this appendix, we present the basics of the theory of spin-weighted spherical
harmonics [15, 80]. These functions have a natural place in the GHP formalism and
provide a simple alternative to the more complicated tensor spherical harmonics. The
discussion in this section takes place on the round 2-sphere. In that case, the action of a
on some quantity, X, of spin-weight s is given by
EX = (sin 8)" +i csc 0 (sin 8)-"X, (D-1)
and the action of 8' is
B'X = (sin 8)-" i csc 0 (sin 8)"y. (D-2)
The spin-weighted spherical harmonics, shm(0, ~), are then defined in terms of the
ordinary spherical harmonics by
s em(e, #)= (,)0~~ -e. The basic properties of the s m, are easily seen to be
shm, = (-1)m+s-syem, (D-4)
ishmn = (- 8)(c+ s+) s+1Im,, (D-5)
alsT m = (+ sl) (- + 1) s-ibm (D-6)
a'ashem = (e- s)(e+ s + 1) sYem. (D-7)
For each value of s, the spin-weighted spherical harmonics are complete:
s~e(0 4)hm0',#' =6(# #')6(cos 0 cos 0'), (D-8)
= 0 m= -
~~