CHAPTER 2
NEW TOOLS FOR PERTURBATION THEORY
In this chapter we develop the basic formalism we will be working within for the
remainder of this work. We begin with a description of the spin coefficient formalism
of N. i.--us! Ia and Penrose [22] and introduce the modifications of it due to Geroch, Held
and Penrose [9]. Within the latter formalism, we develop the properties of the general
class of spacetimes with which we will be working. Included is a discussion of gauge and
the general framework of relativistic perturbation theory. The chapter ends with the
introduction to the framework we will exploit in subsequent chapters.
2.1 NP
The ?-. i.--us! lIs-Penrose (henceforth NP) formalism has its roots in the spinor
formulation of General Relativity. Despite the great beauty and generality of the spinor
approach, we will approach the subject as a special case of the tetrad formalism. In
this view, the NP formalism is developed by (1) introducing a basis of null vectors for
the spacetime and (2) contracting everything in sight with unique combinations of the
aforementioned basis vectors.
We begin by introducing an orthogonal tetrad of null vectors, 16, n", m" and m", with
la and n" being real and m" and m" being complex conjugates. We will impose a relative
normalization
lnan = -mema = 1, (2-1)
with all other inner products vanishing. As an example to keep in mind, consider an
orthonormal tetrad on Minkowski space, (t", x", y, za), such that t"t, = -x"x, = -y"y,
-zaza = 1. Since the vectors are properly normalized, it is easy to verify that
1 1
la (t" + z), na (t" z")
(2-2)
1 1
m" = (xa iya), m" (Xa iya),