P9a 2 3p2 (2-52)
where we have omitted those equations that can he obtained directly by utilizing the
operations of prime and complex conjugation. By applying the commutators to ('2 and
making use of the equations above, we learn that
pp' = P'p (2-54)
8-r' = 8'-r (2-55)
Pr' = B'p. (2-56)
Note that the preceding equations hold for all type D spacetimes. Next we specialize to
non-accelerating spacetimes by making use of Equation 2-36 in the form Tr' = -~ in
Equation 2-56 to obtain
P-r' = 8'p = 2p-r'.(57
Now we compute the commutator [P, D']p and use the GHP equations and the appropriate
version of Equation 2-57 until we arrive at an expression in which the only derivatives are
8'-r and P'p. This expression can then he used with Equations 2-51 and 2-36 to find the
followingf two relations:
P'p =P pp ('+ (T ') -(2(2-58)
a'T =,, T'+p(p' p') + 2 (2-59)
(2-60)
and our task is complete. It is worth pointing out that due to Equations 2-36, these
expressions are not unique. This is a sign that there is some redundancy in the GHP
equations, which is to be expected when we consider such a special class of spacetimes. We
also point out that having expressions for every derivative on every quantity of interest is
sufficient (but not necessary) to completely integrate the background GHP equations. This