coefficients in Equation 2-6, while the spin coefficients in Equation 2-7 have no well
defined spin or boost weight since, under Equation 2-17, they pick up terms involving
derivatives of (. When acting on a quantity of well defined spin and boost weight,
the directional derivatives of Equation 2-8 by themselves also fail to create another
quantity of well defined weight. However, it is possible to combine the spin coefficients
in Equation 2-7 with the action of derivative operators in Equation 2-8 to construct
derivative operators that do produce new quantities with well defined spin and boost
weights. With x taken to be of type {p, q}, we can define these operators as follows:
(2-18)
where P and a are Icelandic characters named "thorn" and "edth''",epciey aho
these derivatives has some well defined type {r, s} in the sense that when they act on a
quantity of type {p, q}, a quantity of type {r + p, a + q} is produced. These new derivative
operators inherit their type from their corresponding tetrad vectors:
(2-19)
It is quite often useful to think of a (P) and 8' (P') as spin (boost) weight raising and
lowering operators, respectively. The derivatives in Equation 2-18 can be combined to
form a covariant derivative operator:
1 1 ao
= V, ( q)nb alb (p q mb amb.
2 2
We note in passing that this definition defines the "GHP connection." Our primary use for
Equation 2-20 will be to express things in GHP language via the replacement V, i 0,
With these definitions, all equations in the NP formalism can be translated into GHP