Lorentzian spacetimes with constant curvature invariants in three dimensions

TL;DR

This paper classifies three-dimensional Lorentzian spacetimes with constant scalar invariants, showing they are either locally homogeneous or Kundt spacetimes, and provides explicit constructions for these metrics.

Contribution

It explicitly determines all three-dimensional CSI metrics and proves their classification into locally homogeneous or Kundt spacetimes, with explicit construction methods.

Findings

All three-dimensional CSI spacetimes are either locally homogeneous or Kundt.

Explicit forms of all CSI metrics in three dimensions are provided.

CSI spacetimes can be constructed from locally homogeneous and VSI spacetimes.

Abstract

In this paper we study Lorentzian spacetimes for which all polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant (CSI spacetimes) in three dimensions. We determine all such CSI metrics explicitly, and show that for every CSI with particular constant invariants there is a locally homogeneous spacetime with precisely the same constant invariants. We prove that a three-dimensional CSI spacetime is either (i) locally homogeneous or (ii) it is locally a Kundt spacetime. Moreover, we show that there exists a null frame in which the Riemann (Ricci) tensor and its derivatives are of boost order zero with constant boost weight zero components at each order. Lastly, these spacetimes can be explicitly constructed from locally homogeneous spacetimes and vanishing scalar invariant spacetimes.