spacetimes we have
p' =po 202 6)
-r = 0, (C-7)
7' = 0, (C-8)
92 20 ao3, _9)
the equation for 5 p not following from the limiting process mentioned above. Note
that the quantity B p is never used in any of the integration we perform in the type II
background spacetime. We will also need the definitions of the new operators:
~; 1pW 4 2
P = i-T r( (C-10)
p p 2 p p
+ (C-11)
B (C-12)
where p and q label the GHP type of the quantity being acted on. Additionally, in
Sections 4.2 and 4.4 we make use of the commutator
-; -p' / 1 11 ; p' 1 1 1 -(
[8, 8 ] P' + -i -/ + +~ p +
pp p p 2 p p(C-13)
VCp' 1 1 1/ tr
which is valid in type D and (with -r = 0) type II spacetimes.
We now begin with
DO = 0, (C-14)
which integrates trivially to give
II = Clo. (C-15)
With this information in hand, we can now integrate the equation governing (m:
(P + p)(m + (B + rl)(1 = 0. (C-16)