Self-accelerating Massive Gravity: Superluminality, Cauchy Surfaces and Strong Coupling

TL;DR

This paper explores the causal structure, superluminality, and strong coupling issues in self-accelerating solutions of massive gravity, revealing spacetime pathologies and the impact of two-metric frameworks on global spacetime properties.

Contribution

It constructs conformal diagrams for characteristic surfaces in new and existing solutions, analyzing superluminality and pathologies in massive gravity with two metrics.

Findings

Most solutions exhibit spacelike characteristics indicating superluminality.

Kinetic terms can vanish in certain frames, complicating degrees of freedom counting.

Spacetime singularities require ad hoc rules for continuity in two-metric theories.

Abstract

Self-accelerating solutions in massive gravity provide explicit, calculable examples that exhibit the general interplay between superluminality, the well-posedness of the Cauchy problem, and strong coupling. For three particular classes of vacuum solutions, one of which is new to this work, we construct the conformal diagram for the characteristic surfaces on which isotropic stress-energy perturbations propagate. With one exception, all solutions necessarily possess spacelike characteristics, indicating perturbative superluminality. Foliating the spacetime with these surfaces gives a pathological frame where kinetic terms of the perturbations vanish, confusing the Hamiltonian counting of degrees of freedom. This frame dependence distinguishes the vanishing of kinetic terms from strong coupling of perturbations or an ill-posed Cauchy problem. We give examples where spacelike characteristics do and do not originate from a point where perturbation theory breaks down and where spacelike surfaces do or do not intersect all characteristics in the past light cone of a given observer. The global structure of spacetime also reveals issues that are unique to theories with two metrics: in all three classes of solutions, the Minkowski fiducial space fails to cover the entire de Sitter spacetime allowing worldlines of observers to end in finite proper time at determinant singularities. Characteristics run tangent to these surfaces requiring {\it ad hoc} rules to establish continuity across singularities.