Degenerate Kundt Spacetimes and the Equivalence Problem

Abstract

This thesis is mainly focused on the equivalence problem for a subclass of Lorentzian manifolds: the degenerate Kundt spacetimes. These spacetimes are not defined uniquely by their scalar curvature invariants. To prove two metrics are diffeomorphic, one must apply Cartan's equivalence algorithm, which is a non-trivial task: in four dimensions Karlhede has adapted the algorithm to the formalism of General Relativity and significant effort has been spent applying this algorithm to particular subcases. No work has been done on the higher dimensional case. First, we study the existence of a non-spacelike symmetry in two well-known subclasses of the N dimensional degenerate Kundt spacetimes: those spacetimes with constant scalar curvature invariants (CSI) and those admitting a covariant constant null vector (CCNV). We classify the CSI and CCNV spacetimes in terms of the form of the Killing vector giving constraints for the metric functions in each case. For the rest of the thesis we fix N=4 and study a subclass of the CSI spacetimes: the CSI-? spacetimes, in which all scalar curvature invariants vanish except those constructed from the cosmological constant. We produce an invariant characterization of all CSI-? spacetimes. The Petrov type N solutions have been classified using two scalar invariants. However, this classification is incomplete: given two plane-fronted gravitational waves in which both pairs of invariants are similar, one cannot prove the two metrics are equivalent. Even in this relatively simple subclass, the Karlhede algorithm is non-trivial to implement. We apply the Karlhede algorithm to the collection of vacuum Type N VSI (CSI-?, ? = 0) spacetimes consisting of the vacuum PP-wave and vacuum Kundt wave spacetimes. We show that the upper-bound needed to classify any Type N vacuum VSI metric is four. In the case of the vacuum PP-waves we have proven that the upper-bound is sharp, while in the case of the Kundt waves we have lowered the upper-bound from five to four. We also produce a suite of invariants that characterize each set of non-equivalent metrics in this collection. As an application we show how these invariants may be related to the physical interpretation of the vacuum plane wave spacetimes.