Since the usual scalar harmonics, }*1,>, define a complete set of functions on the two-sphere,
we can use them to construct two types of vectors. The first is the so-called even parity
vector defined (up to a constant) byl
where VA is the derivative compatible with Y4s (Equation 1-5). The other vector is the
odd-parity (pseudo-) vector
YBceAB A em, (17)
wher FABis just the standardu Levi-Civita symbol. To define tensor harmonics, we
essentially just take one more derivative of Equations 1-6 and 1-7. The even parity
tensors are given by
VA Blinz, and 34slm,, (1-8)
and the odd-parity (pseudo-) tensor by
714CFCD D B~e (1m
Even parity objects pick up minus signs under a parity transformation (8 i x 8,~
xr + ~) according to (-1) and odd parity objects pick up minus signs according to
(-1) Bl. For this reason the even parity parts are sometimes referred to as "electric" and
the odd parity parts n,! I,e!tic" in the older literature. Because parity is an inherent
syninetry of spherically syninetric backgrounds, it provides a natural way of decoupling
the two degrees of freedom of the gravitational field. Note, however, that parity is not a
good syninetry in even slightly less syninetric spacetintes (e.g. K~err). We will return to
1 The tensor harmonics defined in this chapter are not those generally used, but have
been chosen for their heuristic value. See Thorne's review [8] for the standard tensor
harmonics and their relation to various other representations of the sphere, or Appendix
D for the spin-weighted spherical harmonics which provide another alternative for the
angular decomposition.