On 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. 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 behaviour: 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, with the growth being induced by a direct energy cascade. As a complement to the numerical analysis, we provide a heuristic argument suggesting that, if a similar behaviour occurs for gravitational wave perturbations of the motivating spacetimes in general relativity, it would likely not generically lead to black hole or singularity formation.