APPENDIX C:
INTEGRATION A LA HELD
We provide details of the integration that lead to Equation 4-17 and 4-25. As it
turns out, the type II calculation is actually much simpler than the the type D calculation
because it uses a tetrad in which -r = -r' = 0. Therefore we will work out the type D
calculation in detail and the type II result mostly follows by setting certain quantities to
zero, as indicated below.
We will need some results (and their complex conjugates) from the integration of the
type D background:
8 p -o_ o 2 C1
i~p
1 1
o 2 -0 2_
2 2
1 1 1 11
+Tto "~2 2iT 0T~ 0t 2 22p +ToO, (C-2)
-r = -Wo" cto o rpp, (C-3)
92 0 3. (C-5)
As noted in the text, xo" / 0 leads to the accelerating C-metrics, which we include for full
generality. Henceforth the corresponding quantities in type II spacetimes can be obtained
by setting -ro a r = cto 4 0 and Wo 420~1 in the type D result. Thus, in type II
1 This arises from the fact that in type D spacetimes there is only one non-vanishing
Weyl scalar, 92. Ill type II spacetimes, however, both 93 and 94 arT in genOTra alSO
nonzero. Though we do not refer to any of the other Weyl scalars in this work, we would
like maintain agreement with the standard conventions.