(A-10)
APPENDIX A
THE GHP RELATIONS
In this appendix, we give the GHP commutators, field equations and Bianchi
identities, as well as the derivatives of the tetrad vectors. The full set of equations is
obtained by applying to those listed prime, complex conjugation or both. When acting on
a quantity of type {p, q}, the commutators are:
[P, P']
(7 T')8 + (7' f')' p(s'~ Tr'a + 2 11
(A-1)
-q(a'R pfr' + @ol),
(P' p')P + (p p)P' + p(pp'
-q pp'- aa' + ', 1 -I n.
r- 8' = (p p)a ( -r p')
- p's =p2 _/
- P'a -p'a a'p + 'r + k
-- a'-r = pp' + aa' -- -rf -- KK
(A-2)
[a, a']
- ee' + 011 II)
(A-3
The GHP equations are:
Bp
Pp
-r
87
D'p
~ 1 + @ol,
CK 02,
;' I' 2II.
The Bianchi identities are given by:
P 1 a'l',, p@ol + 8 oo
-r'l',, + 4pl 1,' l~_ + -r'@oo 2p~ol
-2a410 + 2n~11 + s@02,