Causal structure and algebraic classification of area metric spacetimes in four dimensions

TL;DR

This paper develops an algebraic classification of four-dimensional area metric manifolds to identify which can serve as physically viable spacetime backgrounds supporting causal matter propagation.

Contribution

It introduces algebraic criteria and a complete classification of area metric tensors, excluding most classes as viable spacetimes and analyzing their causal structures.

Findings

Identified conditions for area metric manifolds to support causal matter.

Developed a complete algebraic classification of area metric tensors.

Proved most algebraic classes are unsuitable as physical spacetimes.

Abstract

Area metric manifolds emerge as a refinement of symplectic and metric geometry in four dimensions, where in numerous situations of physical interest they feature as effective matter backgrounds. In this article, this prompts us to identify those area metric manifolds that qualify as viable spacetime backgrounds in the first place, in so far as they support causally propagating matter. This includes an identification of the timelike future cones and their duals associated to an area metric geometry, and thus paves the ground for a discussion of the related local and global causal structure in standard fashion. In order to provide simple algebraic criteria for an area metric manifold to present a consistent spacetime structure, we develop a complete algebraic classification of area metric tensors up to general transformations of frame. Remarkably, a suitable coarsening of this classification allows to prove a theorem excluding the majority of algebraic classes of area metrics as viable spacetimes.