The late-time attractor structure of dynamical black branes

Abstract

The ringdown phase of a perturbed black hole is conventionally described by a linear superposition of quasinormal modes. However, as the AdS black brane approaches its final global equilibrium, this linear quasinormal mode description becomes inadequate, and nonlinear dynamics play a significant role in the late-stage evolution. We show that the interplay between nonlinear evolution and horizon dissipation in general relativity drives dynamical AdS black branes towards the final state along a unique path, independent of their initial perturbations. Through numerical simulations, we identify this late-time attractor and uncover the associated universal nonlinear behavior, characterized by a simple dimensionless relative amplitude in the dual hydrodynamic variables.

Figures (8)

• Figure 1: This figure shows the typical spectrum of fluctuations for the system. In a compact system, the wavenumber $k$ can only take on discrete values, as labeled by dots. The dispersion relation for the hydro-modes \ref{['eq:disp']} is plotted as red dots in the blue curve, which might play important role in early-time evolution. Late-time evolution is dominated by the fundamental mode and its excitation, shown as purple dots in the yellow dashed line.
• Figure 2: This figure shows the amplitude of first three Fourier modes (FM) for energy current as a function of $\bar{t}=t/L$. Here $E_0=1$, $L=400$.
• Figure 3: The time evolution of the relative amplitude for the energy current. The dashed lines ( $L=800$, $E_0=1$) and solid lines ($L=400$, $E_0=1$) correspond to different initial conditions and different size. For all initial conditions, ${\bar{\alpha}}_n$ approaches the same constant value, indicating that the final state is independent of the initial configuration.
• Figure 4: This figure shows that the final $\bar{\alpha}^S$ is a monotonic increasing function of radial position.
• Figure S1: The time evolution of the amplitude for energy current $j_n(t)$ in a log plot from initial condition ${j_1=10^{-2},j_{n>1}=0}$. Here $E_0=1$, $L=800$. Due to nonlinear effects, Fourier modes $j_2$ and $j_3$ are excited during the time evolution. Their late time decay is also governed by the driven mode, scaling as $e^{2\text{Im}[\omega_1]t}$ and $e^{3\text{Im}[\omega_1]t}$ respectively, as shown in the black line.