Locally Boost Isotropic Spacetimes and the Type ${\bf D}^k$ Condition

TL;DR

This paper classifies a special class of spacetimes called type D^k, characterized by boost isotropy and alignment properties, and identifies the conditions on their metrics within degenerate Kundt spacetimes.

Contribution

It provides a complete characterization of type D^k spacetimes by identifying degenerate Kundt metrics and their metric function conditions.

Findings

All type D^k spacetimes are identified within degenerate Kundt metrics.

Conditions on metric functions for type D^k classification are explicitly determined.

Type D^k spacetimes can be distinguished by scalar polynomial curvature invariants.

Abstract

We consider the class of locally boost isotropic spacetimes in arbitrary dimension. For any spacetime with boost isotropy, the corresponding curvature tensor and all of its covariant derivatives must be simultaneously of alignment type ${\bf D}$ relative to some common null frame. Such spacetimes are known as type ${\bf D}^k$ spacetimes and are contained within the subclass of degenerate Kundt spacetimes. Although, these spacetimes are $\mathcal{I}$-degenerate, it is possible to distinguish any two type ${\bf D}^k$ spacetimes, as the curvature tensor and its covariant derivatives can be characterized by the set of scalar polynomial curvature invariants for any type ${\bf D}^k$ spacetime. In this paper we find all type ${\bf D}^k$ spacetimes by identifying degenerate Kundt metrics that are of type ${\bf D}^k$ and determining the precise conditions on the metric functions.