Periodic standing-wave approximation: Eigenspectral computations for linear gravity and nonlinear toy models

The periodic standing-wave approach to binary inspiral assumes rigid rotation of gravitational fields and hence helically symmetric solutions. To exploit the symmetry, numerical computations must solve for ``helical scalars,'' fields that are functions only of corotating coordinates, the labels on the helical Killing trajectories. Here we present the formalism for describing linearized general relativity in terms of helical scalars and we present solutions to the mixed partial differential equations of the linearized gravity problem (and to a toy nonlinear problem) using the adapted coordinates and numerical techniques previously developed for scalar periodic standing-wave computations. We argue that the formalism developed may suffice for periodic standing-wave computations for post-Minkowskian computations and for full general relativity.