Locally homogeneous Kundt triples and CSI metrics

Abstract

A pseudo-Riemannian manifold is called CSI if all scalar polynomial invariants constructed from the curvature tensor and its covariant derivatives are constant. In the Lorentzian case, the CSI spacetimes have been studied extensively due to their application to gravity theories. It is conjectured that a CSI spacetime is either locally homogeneous or belongs to the subclass of degenerate Kundt metrics. Independent of this conjecture, any CSI spacetime can be related to a particular locally homogeneous degenerate Kundt metric sharing the same scalar polynomial curvature invariants. In this paper we will invariantly classify the entire subclass of locally homogeneous CSI Kundt spacetimes which are of alignment type D to all orders and show that any other CSI Kundt metric can be constructed from them.

I -preserving diffeomorphisms of Lorentzian manifolds

“…Thus, a given pseudo-Riemannian space is I-degenerate if the curvature tensor and its covariant derivatives satisfy the S G 1 property relative to a common null coframe. As in the Lorentzian case [16], the proof of this result relies on the limit of a diffeomorphism associated with an appropriately chosen boost in order to generate a non-diffeomorphic space with the same set I. Motivated by this result, and theorem 3.1 we can state a simple existence theorem for IP D vector fields in pseudo-Riemannian spaces: Theorem 7.3.…”

“…In fact, the locally homogeneous Kundt ∞ triples are of alignment type D k , i.e., the curvature tensor and its covariant derivatives are of type D to all orders [16]. As the SP Is fully determine the Cartan invariants of a type D k spacetime [7,16,8], the Cartan invariants must be constant, ensuring the existence of a fully transitive set of Killing vector fields. Thus, for any Kundt-CSI spacetime, corollary 5.3 and proposition 4.3 give the following proposition: Proposition 6.2.…”

“…The subset of CSI spacetimes belonging to the Kundt class are called Kundt-CSI. For Kundt-CSI metrics, the transverse space is a locally homogeneous Riemannian manifold and the metric functions must satisfy particular differential equations [12,13,15,16].…”

“…In the Lorentzian case there are Kundt-CSI spacetimes that do not have enough Killing vector fields to determine the CSI property. However, any Kundt-CSI spacetime can be mapped to a related locally homogeneous Kundt-CSI spacetime with the same set I which provides an explanation for the CSI property [2,16]. Such a metric is known as a Kundt ∞ triple and will be defined in section 6.…”