Self-accelerating massive gravity: Hidden constraints and characteristics

Self-accelerating backgrounds in massive gravity provide an arena to explore the Cauchy problem for derivatively coupled fields that obey complex constraints which reduce the phase space degrees of freedom. We present here an algorithm based on the Kronecker form of a matrix pencil that finds all hidden constraints, for example those associated with derivatives of the equations of motion, and characteristic curves for any 1 + 1 dimensional system of linear partial differential equations. With the Regge-Wheeler-Zerilli decomposition of metric perturbations into angular momentum and parity states, this technique applies to fully 3 + 1 dimensional perturbations of massive gravity around any spherically symmetric self-accelerating background. Five spin modes of the massive graviton propagate once the constraints are imposed: two spin-2 modes with luminal characteristics present in the massless theory as well as two spin-1 modes and one spin-0 mode. Although the new modes all possess the same-typically spacelike-characteristic curves, the spin-1 modes are parabolic while the spin-0 modes are hyperbolic. The joint system, which remains coupled by nonderivative terms, cannot be solved as a simple Cauchy problem from a single noncharacteristic surface. We also illustrate the generality of the algorithm with other cases where derivative constraints reduce the number of propagating degrees of freedom or order of the equations.