CHAPTER 6
THE NON-R ADIATED 1\ULTIPOLES
In this chapter we will address the issue of the non-radiated multipoles alluded to in
C'!s Ilter 1. The issue is that the metric constructed from a Hertz potential is incomplete
in the sense that its multiple decomposition necessarily begins at -e = 2 because the
angular dependence of the potential is that of a spin-weight +2 angular function. To see
this explicitly, we focus our attention on the IR G metric perturbation (Equation 3-35) in
the Schwarzschild spacetime, where the potential, 9, can he decomposed into some radial
function, R(r), with exponential time dependence, e-i", and a spin-weight 2 spherical
harmonic, -2 Loz(0, 4) (see Appendix D, for details about the spin-weighted spherical
harmonics). Ignoring the radial and time dependence, we see that the components of the
metric perturbation have angular dependence given by
hit ~ 82-2 at = [(e 1) ( + 1)(e + 2)]1/20Ym ine)
him ~ -2 Bat = [(-e 1)(e + 2)]1/2-1 z,, (6-2)
and similarly for him and hmm. Because the spin-weighted spherical harmonics are
undefined for |8| > -e, the above expressions make it clear that the metric perturbation
in this gauge has no -e = 0, 1 pieces and therefore provides an incomplete description of
the physical spacetime. By continuity, the situation persists in the K~err spacetime. How
incomplete is this description?
For the n, I iR~~ly of this work, we have focused our attention on gravitational
radiation in type D spacetimes. This information is contained in the perturbation of either
I',, or ('4, a Tesult established by Wald [16]. In particular, Wald was able to show that
well-behaved perturbations of I',. and #'4 determine each other and furthermore that either
one characterizes the entire perturbation of the spacetime up to It l i .! perturbations
in mass and angular momentum. With I',, and #'4 determined hv the Hertz potential