An intrinsic characterization of 2+2 warped spacetimes

TL;DR

This paper provides an intrinsic, tensor-based characterization of 2+2 warped spacetimes, identifying conditions involving Killing-Yano tensors and Riemann tensor concomitants that distinguish these geometries.

Contribution

It introduces necessary and sufficient invariant conditions for a metric to be a 2+2 warped product, using explicit algebraic and tensorial criteria.

Findings

Characterization via Killing-Yano tensors and algebraic restrictions

Invariant classification of conformally 2+2 product spacetimes

Explicit conditions involving Riemann tensor concomitants

Abstract

We give several equivalent conditions that characterize the 2+2 warped spacetimes: imposing the existence of a Killing-Yano tensor $A$ subject to complementary algebraic restrictions; in terms of the projector $v$ (or of the canonical 2-form $U$) associated with the 2-planes of the warped product. These planes are principal planes of the Weyl and/or Ricci tensors and can be explicitly obtained from them. Therefore, we obtain the necessary and sufficient (local) conditions for a metric tensor to be a 2+2 warped product. These conditions exclusively involve explicit concomitants of the Riemann tensor. We present a similar analysis for the conformally 2+2 product spacetimes and give an invariant classification of them. The warped products correspond to two of these invariant classes. The more degenerate class is the set of product metrics which are also studied from an invariant point of view.