  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.
