On turbulence for spacetimes with stable trapping

TL;DR

The paper investigates nonlinear scalar waves on a fixed, four-dimensional, static spacetime with a stable photon sphere to model gravitational perturbations in ultracompact objects and black strings. It combines rigorous linear theory, showing uniform energy boundedness with slow, logarithmic decay due to stable trapping, with detailed nonlinear simulations of the cubic defocusing wave equation that reveal turbulent behavior in the trapped region via a direct angular-mode cascade to higher $\ell$-modes and growth of higher-order energies. The key finding is that slow linear decay can sustain nonlinear interactions, producing a robust cascade and high-order derivative growth without necessarily triggering collapse, suggesting that turbulence could persist in gravitational perturbations of such spacetimes. This work provides a controlled setting to study turbulence arising from stable trapping, offering insights into the dynamics of ultracompact objects and black strings, and outlining paths for extending the analysis to gravitational waves and more general trapping geometries.

Abstract

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.

Figures (22)

• Figure 1: We choose the scalar function $f(r)= 1-r^2(0.026+11.21\, r^4)^{-1}$ (solid green line). The choice of $f(r)$ is such that the local minimum of the geodesic potential $V(r)=f(r)r^{-2}$ (dashed orange line) is located sufficiently close to the origin (i.e., $r_0\sim 0.22$) to guarantee the necessary numerical resolution within the stable trapping region (see Section \ref{['sec_numerical_implementation']}).
• Figure 2: The function $u(r)$ as defined for the initial data in \ref{['Eq:piecewise_s']}. We recall that the minimum of the radial geodesic potential $V(r)$ is located at $r_0\sim 0.22$ (cf. Figure \ref{['fig:finfo']}).
• Figure 3: The evolution of the quantity \ref{['sup_2nd_deriv_theta']} over the time interval $t\in [0,3]$, with a logarithmic scale on the vertical axis. Six different values of $\epsilon$ are examined, each corresponding to a different color in the plot. For $\epsilon=0.75$ or larger, the solution already exhibits a markedly nonlinear behaviour within the time interval considered. The convergence of these simulations is shown in Figure \ref{['fig:ConvergenceLinToNonLin']}.
• Figure 4: Time snapshots of the evolution of the solution $\phi$ on the spatial domain $\mathcal{P}$ (recall \ref{['compactified_spatial_domain']}) over the time interval $t\in[0,150]$. For different snapshots, time increases from the left to the right (top to bottom). The vertical axis and color correspond to the amplitude of the solution, with orange depicting positive values of the solution and green negative values. The range of the vertical axis varies from one panel to another.
• Figure 5: The evolution of the quantity \ref{['sup_pi_dot']} over the time interval $t\in [0,150]$, with a logarithmic scale on the vertical axis. The point-wise relative numerical error in this quantity is generally sub-percent level (see Appendix \ref{['app:convergence']} for more details).

Theorems & Definitions (2)

• Theorem 1
• Theorem 2