I-preserving diffeomorphisms of Lorentzian manifolds

Abstract

We examine the existence of one parameter groups of diffeomorphisms whose infinitesimal generators annihilate all scalar polynomial curvature invariants through the application of the Lie derivative, known as $\mathcal{I}$-preserving diffeomorphisms. Such mappings are a generalization of isometries and appear to be related to nil-Killing vector fields, for which the associated Lie derivative of the metric yields a nilpotent rank two tensor. We show that the set of nil-Killing vector fields contains Lie algebras, although the Lie algebras may be infinite and can contain elements which are not $\mathcal{I}$-preserving diffeomorphisms. We then study the curvature structure of a general Lorenztian manifold, or spacetime, to show that $\mathcal{I}$-preserving diffeomorphism will only exists for the $\mathcal{I}$-degenerate spacetimes and to determine when the $\mathcal{I}$-preserving diffeomorphisms are generated by nil-Killing vector fields. We identify necessary and sufficient conditions for the degenerate Kundt spacetimes to admit an additional $\mathcal{I}$-preserving diffeomorphism and conclude with an application to the class of Kundt spacetimes with constant scalar polynomial curvature invariants to show that a finite transitive Lie algebra of nil-Killing vector fields always exists for these spacetimes.