Size and momentum of an infalling particle in the black hole interior

TL;DR

The paper proposes treating the black hole interior in AdS/CFT as a quantum circuit and develops boundary observables, notably the two-sided quantity $\mathcal{F}(a,t_R)$, to diagnose interior properties and the growth of interior operator size. By modeling perturbations as an epidemic of sick gates in the interior circuit, it links the progression of interior size to the time dependence of complexity and to firewall-like phenomena, providing a boundary-operator framework for interior dynamics. It also connects operator size to momentum, showing that interior size tracks interior momentum via a complementarity relation with exterior notions, and demonstrates this in the low-energy SYK/Schwarzian (JT gravity) setup. The work points to generalizations to higher dimensions and pure states, offering a pathway to relate circuit growth, complexity, and horizon smoothness in a boundary-theoretic description of black hole interiors.

Abstract

The future interior of black holes in AdS/CFT can be described in terms of a quantum circuit. We investigate boundary quantities detecting properties of this quantum circuit. We discuss relations between operator size, quantum complexity, and the momentum of an infalling particle in the black hole interior. We argue that the trajectory of the infalling particle in the interior close to the horizon is related to the growth of operator size. The notion of size here differs slightly from the size which has previously been related to momentum of exterior particles and provides an interesting generalization. The fact that both exterior and interior momentum are related to operator size growth is a manifestation of complementarity.

Figures (11)

• Figure 1: Illustration of the quantum circuit representing the future interior of the black hole dual to thermofield double perturbed from the left side.
• Figure 2: The size of the wormhole grows as the circuit grows.
• Figure 3: Illustration of the evolution of healthy and infected gates in (a) compared to the size of the associated bulk regions on time slices in (b).
• Figure 4: When Bob decreases the right time, he is scanning through the quantum circuit from the right to the left.
• Figure 5: The number of healthy gates inside the blue dashed squares at time $t_{Ri}$ is given by ${\cal F}$.