to Equation 2-45 to obtain a similar operator associated with rl" (Equation 2-43):
P 21/ a -ic -1/3]( l pi -/ 2/
[e-ic -1/3 eic -1/3 2( / /)
+2(pv q)pp' 2-1/3e' 71 --/ -,1/3
(2-46)
2(pT +I q/ '2-136 -13 -/
1pe ,/3i ?-4ic 221/3 1/3 ;2/3 -2/3)
/ 4ic 2 Y/3 1/ -2Y/32/
which also commutes with all four GHP derivations.
On a final note, we remark that in recent work Be o;r [44] obtained an operator
related to K~illing tensor that commutes with the scalar wave equation. The operator
has the feature that it is first order in time. In this context it is tempting to ask if there
exists an operator analogous to those defined for the K~illingf vectors that commutes with
each of the GHP derivatives. The answer is currently unclear and so we leave it for future
investigation.
2.4 The Simplified GHP Equations for Type D Backgrounds
With Equations 2-36 in hand, we are now in a position to completely simplify the
GHP equations for the special case of type D backgrounds. Our starting point is the GHP
equations and Bianchi identities adapted to a Type D background:
Pp = p2 (2-47)
Dr =p~r 7')(2-48)
Sp = -r(p -p) (2-49)
B7 = 72 (2-50)
D'p a'r = pp' -rf 2a (2-51)