The Typical-State Paradox: Diagnosing Horizons with Complexity

TL;DR

The paper tackles how horizons can be opaque for typical black-hole states yet transparent for black holes formed naturally. It proposes an Increasing Complexity Criterion, tied to a complexity/volume duality, as the unifying diagnostic for horizon transparency. The analysis shows that complexity equilibrium lasts exponentially long, explaining natural transparency, while opacity is shown to be fragile and easily destroyed by small perturbations, such as a single thermal photon. Through geometric and information-theoretic arguments, including AdS black-hole models and shock-wave diagnostics, the work links horizon behavior to the growth of quantum state complexity and delineates regimes of black, grey, and white hole-like dynamics.

Abstract

The concept of transparent and opaque horizons is defined. One example of opaqueness is the presence of a firewall. Two apparently contradictory statements are reconciled: The overwhelming number of black hole states have opaque horizons; and: All black holes formed by natural processes have transparent horizons. A diagnostic is proposed for transparency, namely that the computational complexity of the state be increasing with time. It is shown that opaque horizons are extremely unstable and that the slightest perturbation will make them transparent within a scrambling time.

Figures (7)

• Figure 1: The left panel shows a one-sided ADS black hole. The pink region contains the bridge-to-nowhere. The geometry can be foliated by maximal volume slices anchored at the boundary. The right panel shows a white hole similarly foliated. In the black hole case the volume of the bridge increases with $t$ and in the white hole the volume decreases.
• Figure 2: Eternal black/white hole foliated by maximal volume slices anchored on the boundary. The dark blue lines represent the ERB behind the horizon. The volume of the ERB decreases in the white hole region and increases in the black hole region.
• Figure 3: To diagnose the right horizon we anchor the maximal slices at $t_L = \infty$ and vary $t_R.$
• Figure 4: A shock wave geomety with a shock injected on the left at time $t_w.$
• Figure 5: Eddington-Finkelstein diagram for a shock wave initially within a Planck distance of the horizon. At the indicated time the shock separates from the horizon and falls into the singularity.