that the even-parity (real) parts of the source vanish identically and thus the angular
momentum perturbation is completely odd-parity (imaginary). However, we still have
pieces in the source (such as 1) that contribute to the even parity perturbation. We will
treat each parity individually.
6.1.2.1 Odd-parity angular momentum perturbations
Because our gauge vector only has one nonzero component, our task is greatly
simplified. The contribution of the odd-parity gauge vector to the metric perturbation
takes the form
0 0 -(8tS)(sin 8)-lY- (8tS) sin 0Y+
hO 0 -(8,S -2r-1S)(sin 8)-lY- (8,S -2r-1S) sin 0Y+
sym syvm 0 -S [(sin B)l18Y- + cos 0Yf sinl 08oY ]
sym sym sym 0
(6-56)
where the "-" on hab, referring to the interior spacetime, is to be distinguished from the
-"on Ye which refers to a combination of spin-weight 1 spherical harmonics. In this
situation, we must modify our requirement of form invariance (which is already broken by
the perturbation) to the requirement that only ht- remains nonzero, which preserves the
minimum freedom to match to the exterior. First we set has = 0, which implies me = 0
or 1Yem = -1 m,. This can only hold if m = 0, which means the perturbation is axially
symmetric. Moving on, we turn our attention to eliminating 604. This entails
cos 8Y sin 0804eY =
which has the solution
Ye = 1 og + -1 Yon ~sin 0.
This is just the statement that -e = 1, which is to be expected from the fact that the
perturbation appears in the spin-1 part of the metric. The angular dependence of the