Petrov classification of perturbed spacetimes: the Kasner example

In this paper we consider vacuum Kasner spacetimes, focusing on those that can be parametrized as linear perturbations of the special Petrov type D case. In particular we analyse in detail the perturbations which map the Petrov type D Kasner spacetime into another Kasner spacetime of Petrov type I. For these 'quasi-D' Kasner models we first investigate the modification to some curvature invariants and the principal null directions, both related to the Petrov classification of the spacetime. This simple Kasner example allows us to clarify the fact that perturbed spacetimes do not retain in general the speciality character of the background. In fact, there are four distinct principal null directions, although they are not necessarily first-order perturbations of the background principal null directions. Then in the Kasner type D background we derive a Teukolsky master equation, a classical tool for studying black-hole perturbations of any spin. This further step allows us to control totally general cosmologies around such a background as well as to show, from a completely new point of view, the well-known absence of gravitational waves in Kasner spacetimes