Turbulence for spacetimes with stable trapping

Motivated by understanding the nonlinear gravitational dynamics of spacetimes admitting stably trapped null geodesics, such as ultracompact objects and black string solutions to general relativity, we explore the dynamics of nonlinear scalar waves on a simple (fixed) model geometry with stable trapping. More specifically, we consider the time evolution of solutions to the cubic (defocusing) wave equation on a four-dimensional static, spherically symmetric, and asymptotically flat (horizonless) spacetime admitting a stable photon sphere. Unlike the known results for scalar waves on spacetimes with unstable trapping, our study shows fundamental differences between linear and nonlinear scalar dynamics. The local energy, as well as all local higher-order energies, of solutions to the linear wave equation on our model spacetime can be rigorously proven to remain uniformly bounded and to decay uniformly in time. However, due to the presence of stable trapping, the uniform decay rate is slow. To help elucidate how the slow linear decay affects solutions to the nonlinear wave equation considered, we examine numerical solutions of the latter, restricting to axisymmetric initial data in this work. In contrast to the linear dynamics, we exhibit a family of nonlinear solutions with turbulent behavior. Within the region of stable trapping, the slow linear decay allows local higher-order energies of the nonlinear solution to grow over the time interval that we numerically evolve. The growth is induced by a direct energy cascade: Beginning with initial data containing a small number of low-order multipole modes, a spectrum of high-order multipole modes are populated in time by the nonlinear interactions and eventually dominate over the low-modes in the evolution. That the system exhibits a direct cascade limits the time over which our numerical scheme can provide convergent solutions for the short-wavelength structure that develops, and hence we can only speculate what this intermediate-time dynamics implies for the nonlinear stability of the motivating spacetimes in general relativity. Nevertheless, we provide a heuristic argument suggesting that, if a similar behavior occurs for gravitational wave perturbations of these spacetimes, it would likely not generically lead to black hole or singularity formation.