Black Holes at Exp-time

TL;DR

The paper investigates how classical GR descriptions of black hole interiors fail at exponentially long times (Exp-time) and proposes that complexity geometry, via the cut locus, explains the transition from linear complexity growth (ramp) to a plateau and eventual quantum recurrences (Expexp-time). It links wormhole volume to quantum computational complexity (CV duality) and analyzes three post-Exp-time interpretations—continued growth, superpositions of geometries, and minimal-geodesic histories—showing they converge before Exp-time but diverge thereafter. A simple toy model and discussions of firewall scenarios in exact and perturbed TFD states illustrate how interior geometry, boundary observables, and tensor-network descriptions can remain consistent despite underlying non-classical dynamics. The framework highlights how complexity-driven transitions and recurrences shape our understanding of quantum gravity in highly entangled black-hole systems and informs firewall debates and interior geometries at extreme times.

Abstract

Classical GR governs the evolution of black holes for a long time, but at some exponentially large time it must break down. The breakdown, and what comes after it, is not well understood. In this paper I'll discuss the problem using concepts drawn from complexity geometry. In particular the geometric concept of cut locus plays a key role.

Figures (13)

• Figure 1: Time-dependence of unitary operator complexity for a chaotic system.
• Figure 2: Penrose diagram for two-sided eternal black hole. The diagram has been foliated by maximum volume slices.
• Figure 3: A geodesic $\bf{a}\it(t)$ originating at $\bf p$ parameterized by $t.$
• Figure 4: For $t<t_c$ the shortest geodesic connecting $p$ with $\bf{a}\it(t)$ is shown in black. After the geodesic passes the cut locus at $t_c$ the shortest geodesic at any given $t$ is the red curve $\gamma(t).$
• Figure 5: As the black geodesic continues on its way, the shortest geodesic connecting $p$ with $\bf{a}\it(t)$ may pass through a number of cut points at which new families of geodesics come into play.