Black holes and the butterfly effect

TL;DR

Explores how tiny early-time perturbations lead to rapid scrambling of entanglement between two holographic CFTs in the thermofield double state. Uses both a simple qubit model and a holographic AdS3 BTZ setup with a shock wave to quantify the destruction of L–R correlations via mutual information and geodesic probes, showing t_* ≈ (β/2π) log S. Analyzes string/Planck-scale corrections and discusses potential implications for the firewall debate and horizon smoothness. Concludes that black holes exhibit fast scrambling with a geometric mechanism tied to exponentially blue-shifted perturbations.

Abstract

We use holography to study sensitive dependence on initial conditions in strongly coupled field theories. Specifically, we mildly perturb a thermofield double state by adding a small number of quanta on one side. If these quanta are released a scrambling time in the past, they destroy the local two-sided correlations present in the unperturbed state. The corresponding bulk geometry is a two-sided AdS black hole, and the key effect is the blueshift of the early infalling quanta relative to the $t = 0$ slice, creating a shock wave. We comment on string- and Planck-scale corrections to this setup, and discuss points that may be relevant to the firewall controversy.

Figures (4)

• Figure 1: Mutual information (upper, blue) and spin-spin correlation function (lower, red) in the perturbed state $|\Psi'\rangle$, as a function of the time of the perturbation $t_w$. The delay is a propagation effect; if the perturbation at site five is sufficiently recent, sites one and two are unaffected.
• Figure 2: The Kruskal diagram (center) and Penrose diagram (right) for the BTZ geometry.
• Figure 3: The Kruskal and Penrose diagrams for the geometry with a shock wave from the left, represented by the double line. The dashed $v=0$ and $\tilde{v}=0$ horizons miss by an amount $\alpha$.
• Figure 4: In the unperturbed BTZ geometry (left), a smooth horizon requires the black mode on the left to be highly entangled with the blue mode on the right. By contrast, in the shock wave geometry (right) the black and blue modes are far apart and unentangled. Instead, the black mode is entangled with the green mode coming out of the white hole. The arguments of Leichenauer:2013kaa suggest that the green mode may be complicated in the CFT.