Putting these simplifications into the (odd-parity) Einstein equations, we see that
is satisfied identically by virtue of the gauge conditions. Furthermore, we have that
SIm = -Stra (and likewise for the nm and uti components), which is no surprise because,
as a tensor, Sub respects the parity decomposition. This leaves us with Sinz, S,m and Enzo.
Starting with the last piece, we can commute the derivatives to write
Enzm = B((p' p')hnz + (P p)hnm>z = 0,
which we can !~Ita.,i Il~e" by peeling off the B to give us
(P' -p')Anz + (P p)h,mz = 0 (:317)
(P' -p')Anz + (P p)h,z, = 0, (:318)
where the second relation follows from complex conjugation of the first (or integration of
Enzo), and we have set the !.Il, I s!i .11.1 <..0!-I .11' to zero for convenience (it would cancel
helow). We now turn our attention to Sim. By successive applications of Equation :317 we
can eliminate all terms involving Ph,tm, arriving at
Sim= (S' 29' +4pP' pp' -4t/'2 Im 2p2 ims
Taking the prime of this (which introduces an overall minus sign because of the parity
decomposition) leads to a similar expression for S,m. Next we take the (sourcefree)
combination
(P' 2p')B'Sim + (P 2p)BS,z~ = 0. (:320)
We can remove from this expression all references to al;,.. and a'h,m: using the gauge
conditions in Equations :316, which, after some serious commuting leads to the quite