MiSaTa~uWa equations. In practice, the more widely used prescription for computing
the self force is due to Detweiler and Whiting [3]. In either case, the fundamental
object of interest is the metric perturbation, hub, introduced by the particle on the
large black hole's spacetime. Therefore the EAIRI problem also requires us to compute
the metric perturbation, before we can compute the self-force on the particle. This is
the piece of the problem to which the present work aims to contribute. Determining
the metric perturbation is a task that depends quite sensitively on the spacetime being
perturbed. For spherically symmetric backgrounds, this problem is well understood and
most of the remaining problems are computational in nature. However, for the more
interesting and .I-r lInIlai--; I I11y relevant situation where the larger black hole is rol Iflrin
our understanding is not quite complete. It is on this more general situation that we
focus. Before we continue, we note that all of the .I-r initsli--;I I11y interesting spacetimes,
including the K~err and Schwarzschild metrics, possess curvature tensors with the same
basic algebraic structure. We will elaborate on this more fully in the next chapter, but for
now we merely point out that these spacetimes belong to the larger class of iy ol l.
special spacetimes.
The remainder of this chapter is devoted to providing a review of the literature [4].
Every attempt has been made to phrase the current discussion in generally accessible
language. Alany of these results will be explored in further detail in later chapters, after
the appropriate formalism has been developed.
1.1 Perturbations of Spherically Symmetric Spacetimes
Historically, the subject of black hole perturbation theory got its start with the
pioneering work of Re -~- and Wheeler [5] (henceforth RW), who provided an analysis
of first order perturbations of the Schwarzschild solution (which was later completed by
Zerilli [6, 7]). The fact that the background is spherically symmetric is crucial to their
analysis. The basics will be presented here. A more complete discussion, in a very different
language, is provided in ('! .pter 3.