Type III and N universal spacetimes

Abstract

Universal spacetimes are spacetimes for which all conserved symmetric rank-2 tensors, constructed as contractions of polynomials from the metric, the Riemann tensor and its covariant derivatives of arbitrary order, are multiples of the metric. Consequently, metrics of universal spacetimes solve vacuum equations of all gravitational theories with Lagrangian being a polynomial curvature invariant constructed from the metric, the Riemann tensor and its derivatives of arbitrary order. In the literature, universal metrics are also discussed as metrics with vanishing quantum corrections and as classical solutions to string theory. Widely known examples of universal metrics are certain Ricci-flat pp waves. In this paper, we start a general study of geometric properties of universal metrics in arbitrary dimension and we arrive at a broader class of such metrics. In contrast with pp waves, these universal metrics also admit non-vanishing cosmological constant and in general do not have to possess a covariantly constant or recurrent null vector field. First, we show that a universal spacetime is necessarily a CSI spacetime, i.e. all curvature invariants constructed from the Riemann tensor and its derivatives are constant. Then we focus on type N spacetimes, where we arrive at a simple necessary and sufficient condition: a type N spacetime is universal if and only if it is an Einstein Kundt spacetime. A class of type III Kundt universal metrics is also found. Several explicit examples of universal metrics are presented.