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Six-point functions and collisions in the black hole interior

Felix M. Haehl, Alexandre Streicher, Ying Zhao

TL;DR

This work computes and resums Lorentzian six-point out-of-time-order correlators ${\cal F}_6$ in the eternal AdS black hole to diagnose collisions of signals behind the horizon, quantify two-sided operator growth, and extract collision information via traversable-wormhole couplings. Using the eikonal gravity framework, the authors achieve all-orders in $G_N$ and reveal a factorization ${\cal F}_6 \approx \frac{\langle W_1 W_1 {\cal O}_j {\cal O}_j \rangle}{\langle W_1W_1 \rangle \langle {\cal O}_j {\cal O}_j \rangle} \times \frac{\langle {\cal O}_j {\cal O}_j W_2 W_2 \rangle}{\langle {\cal O}_j {\cal O}_j \rangle \langle W_2 W_2 \rangle} \times \frac{\langle W_1 W_1 W_2 W_2 \rangle}{\langle W_1 W_1 \rangle \langle W_2 W_2 \rangle} \times {\cal F}_{6,\text{conn}}$, with the connected piece ${\cal F}_{6,\text{conn}}$ of order unity. The heavy-probe limit yields explicit expressions showing a scrambling-time scale $2t_*$ and a six-point decay form, while geodesic calculations in JT gravity and AdS$_3$ gravity corroborate the results. The traversable-wormhole analysis demonstrates how the collision product can be inferred when the interior interaction is mild and a left–right coupling is introduced. Overall, the paper provides a nonperturbative gravity framework for fine-grained chaotic dynamics and interior-collision physics in holographic systems, with connections to Schwarzian reparametrization dynamics and Virasoro blocks.

Abstract

In the eternal AdS black hole geometry, we consider two signals sent from the boundaries into the black hole interior shared between the two asymptotic regions. We compute three different out-of-time-order six-point functions to quantify various properties of the collision of these signals behind the horizons: (i) We diagnose the strength of the collision by probing the two-signal state on a late time slice with boundary operators. (ii) We quantify two-sided operator growth, which provides a dual description of the signals meeting in the black hole interior, in terms of the quantum butterfly effect and quantum circuits. (iii) We consider an explicit coupling between the left and right CFTs to make the wormhole traversable and extract information about the collision product from behind the horizon. At a technical level, our results rely on the method of eikonal resummation to obtain the relevant gravitational contributions to Lorentzian six-point functions at all orders in the $G_N$-expansion. We observe that such correlation functions display an intriguing factorization property. We corroborate these results with geodesic computations of six-point functions in two- and three-dimensional gravity.

Six-point functions and collisions in the black hole interior

TL;DR

This work computes and resums Lorentzian six-point out-of-time-order correlators in the eternal AdS black hole to diagnose collisions of signals behind the horizon, quantify two-sided operator growth, and extract collision information via traversable-wormhole couplings. Using the eikonal gravity framework, the authors achieve all-orders in and reveal a factorization , with the connected piece of order unity. The heavy-probe limit yields explicit expressions showing a scrambling-time scale and a six-point decay form, while geodesic calculations in JT gravity and AdS gravity corroborate the results. The traversable-wormhole analysis demonstrates how the collision product can be inferred when the interior interaction is mild and a left–right coupling is introduced. Overall, the paper provides a nonperturbative gravity framework for fine-grained chaotic dynamics and interior-collision physics in holographic systems, with connections to Schwarzian reparametrization dynamics and Virasoro blocks.

Abstract

In the eternal AdS black hole geometry, we consider two signals sent from the boundaries into the black hole interior shared between the two asymptotic regions. We compute three different out-of-time-order six-point functions to quantify various properties of the collision of these signals behind the horizons: (i) We diagnose the strength of the collision by probing the two-signal state on a late time slice with boundary operators. (ii) We quantify two-sided operator growth, which provides a dual description of the signals meeting in the black hole interior, in terms of the quantum butterfly effect and quantum circuits. (iii) We consider an explicit coupling between the left and right CFTs to make the wormhole traversable and extract information about the collision product from behind the horizon. At a technical level, our results rely on the method of eikonal resummation to obtain the relevant gravitational contributions to Lorentzian six-point functions at all orders in the -expansion. We observe that such correlation functions display an intriguing factorization property. We corroborate these results with geodesic computations of six-point functions in two- and three-dimensional gravity.

Paper Structure

This paper contains 5 sections, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview and summary of results
  3. The structure of gravitational interactions
  4. Diagnosing collisions in the wormhole interior
  5. Setup and general properties

Figures (2)

  • Figure 1: The three setups we consider in this paper. $(a)$ Scattering experiment in the eternal AdS black hole geometry dual to the thermofield double state. Two perturbations are inserted, one on the left and one on the right, and we probe the state using a two-point function of ${\cal O}_j$ on a later time slice. $(b)$ The same setup but with ${\cal O}_j$ operators inserted at $a=b=0$ serves to quantify the two-sided scrambling process and the 'overlap' of the two growing perturbations. $(c)$ A coupling between the left and right theories modifies the time evolution in such a way that the wormhole becomes traversable and we can extract information about the scattering process behind the horizon.
  • Figure 2: Illustration of the 'irreducibly' out-of-time-order configuration \ref{['eq:F6def']}. We show the complex time contours where the three perturbations have support, as well as the left-right configuration, which is inspired by the bulk picture. The complex contour time coordinate is $t_R$ and $-t_L - i\pi$ for right and left boundary times, respectively.