The Ricci tensor is represented by the following ten scalars:
1 1
oo0 = R1, 21 -24 ,
2 2
= -(R1 R34), ~02 2 R33,
1 1
oi 2R13, 22 2 -R22, (2-12)
1 1
~12 2 R23, 20 2 -R44 >
to =R14, 1 '
2 24
The field equations then follow from Equations 2-9 and 2-10. A full set of equations for
the NP formalism is composed of the commutators, the equations involving dependence on
matter, and the Bianchi identities. This is given in Appendix A.
2.2 GHP
In 1973 Geroch, Held and Penrose (GHP) [9] introduced some convenient modifications
of the NP formalism. Specifically, they identified the notions of spin and boost weight and
make explicit use of an inherent discrete symmetry of the NP equations.
In the NP formalism, there is an implicit invariance under a certain interchange
of the basis vectors which GHP have built on through the introduction of the prime (')
operation, defined by its action on the tetrad vectors:
(2-13)
A glance at Equations 2-6 and 2-7 -11- -- -; the adoption of a change in notation:
and similarly for the directional derivatives of Equation 2-8
D' = a and 6' = 6. (2-15)