Causal hierarchy in modified gravity

TL;DR

This paper explores how the causal hierarchy in general relativity extends to modified gravity theories with additional structures, analyzing the conditions under which causality can be preserved despite superluminal or infinite signal velocities.

Contribution

It establishes a framework for causal hierarchies in modified gravity theories, highlighting the role of signal velocity and the conditions for maintaining causality.

Findings

Suitable causal hierarchies can be defined in extended gravity frameworks.

Infinite signal velocity necessitates a unique, hypersurface orthogonal aether field.

Global parabolicity is introduced as a condition for causality preservation.

Abstract

We investigate the causal hierarchy in various modified theories of gravity. In general relativity the standard causal hierarchy, (key elements of which are chronology, causality, strong causality, stable causality, and global hyperbolicity), is well-established. In modified theories of gravity there is typically considerable extra structure, (such as: multiple metrics, aether fields, modified dispersion relations, Horava-like gravity, parabolic propagation, etcetera), requiring a reassessment and rephrasing of the usual causal hierarchy. We shall show that in this extended framework suitable causal hierarchies can indeed be established, and discuss the implications for the interplay between "superluminal" propagation and causality. The key distinguishing feature is whether the signal velocity is finite or infinite. Preserving even minimal notions of causality in the presence of infinite signal velocity requires the aether field to be both unique and hypersurface orthogonal, leading us to introduce the notion of global parabolicity.