Momentum/Complexity Duality and the Black Hole Interior

TL;DR

The paper establishes a bulk momentum/complexity (PC) duality within the Volume-Complexity framework by analyzing a spherically symmetric thin-shell collapse in AdS, relating the rate of VC complexity growth to a specific radial component of bulk momentum measured on extremal-volume foliations. The main result expresses the complexity rate as $\dot{\cal C}[O_{shell}] = {d-1\over 8\pi G}(\Pi_+ - \Pi_-)$, which is recast as the projection of the shell momentum along a defined complexity field $\mathcal{C}_\Sigma^\mu$, i.e. $P_C = \int_{S_W} {\cal P}_\mu {\cal C}_\Sigma^\mu$, and equivalently as $-\int_\Sigma N_\Sigma^\mu T_{\mu\nu} {\cal C}_\Sigma^\nu$. At late times, extremal surfaces saturate inside the black hole, producing a constant interior momentum and a linear growth of complexity with slope equal to the shell's energy, highlighting a deep link between interior geometry and operator complexity. The analysis suggests the PC duality generalizes beyond the thin-shell setup and motivates further ab initio derivations and extensions to less symmetric configurations.

Abstract

We establish a version of the Momentum/Complexity (PC) duality between the rate of operator complexity growth and a radial component of bulk momentum for a test system falling into a black hole. In systems of finite entropy, our map remains valid for arbitrarily late times after scrambling. The asymptotic regime of linear complexity growth is associated to a frozen momentum in the interior of the black hole, measured with respect to a time foliation by extremal codimension-one surfaces which saturate without reaching the singularity. The detailed analysis in this paper uses the Volume-Complexity (VC) prescription and an infalling system consisting of a thin shell of dust, but the final PC duality formula should have a much wider degree of generality.

Figures (6)

• Figure 1: Standard notions of PC duality are defined in terms of near-horizon dynamics, using radial and time coordinates which remain outside the horizon.
• Figure 2: Penrose diagram of the collapsing shell geometry. The shell is injected in the bulk at late times compared with $T^{-1}$, causing the initial black hole of mass $M_-$ to grow up to the bigger mass $M_+$. The worldvolume of the matter shell is labelled $\mathcal{W}$ and sets the boundary between the two black hole spacetimes $\mathscr{V}^\pm$.
• Figure 3: Extremal codimension-one surface $\Sigma$ of interest. Its boundary $\partial \Sigma$ consists of two spheres at infinity, located at times $t_L = t_R = t$.
• Figure 4: Configuration of relevant vectors at the intersection sphere $\mathcal{S}_{\mathcal{W}} = \Sigma \cap {\cal W}$.
• Figure 5: The saturation slice $\Sigma_{\infty}$ interpolates between the extremal surface barrier inside $\tilde{r}_{-}$ and outside $\tilde{r}_{+}$.