Terminal Holographic Complexity

TL;DR

The paper develops a quasilocal version of holographic complexity tailored to terminal spacelike singularities by restricting the action-complexity prescription to the past domain of dependence of a terminal set. It introduces a nesting construction that combines the on-shell action with a codimension-two area term to define a quasilocal complexity, and it analyzes the local YGH contribution to extract a comoving complexity density via coarse-graining. Through analytic studies of AdS black holes, topological crunches, and Kasner interiors, the authors demonstrate monotonic growth of the nesting complexity under suitable counterterms and reveal how FRW and generic BKL singularities exhibit vanishing or evanescent local density, respectively. The work draws parallels with Penrose's Weyl criterion, highlights the special role of the YGH term, and outlines directions for extending the framework to de Sitter and other terminal boundaries.

Abstract

We introduce a quasilocal version of holographic complexity adapted to `terminal states' such as spacelike singularities. We use a modification of the action-complexity ansatz, restricted to the past domain of dependence of the terminal set, and study a number of examples whose symmetry permits explicit evaluation, to conclude that this quantity enjoys monotonicity properties after the addition of appropriate counterterms. A notion of `complexity density' can be defined for singularities by a coarse-graining procedure. This definition assigns finite complexity density to black hole singularities but vanishing complexity density to either generic FRW singularities or chaotic BKL singularities. We comment on the similarities and differences with Penrose's Weyl curvature criterion.

Figures (14)

• Figure 1: On the left, the codimension-one asymptotic surface ${\cal S}_\infty$, accounting for the total complexity 'flowing' into the black-hole singularity $\cal S^*$. On the right, the subtracted codimension-one surface ${\cal S}'_t$ which accounts for the time-dependence of VC complexity in the eternal black hole geometry.
• Figure 2: Generic terminal set $\cal S^*$ (in red) and its past domain of dependence $D^-(\cal S^*)$ and causal past $J^-(\cal S^*)$ .
• Figure 3: The total VC complexity flowing into the singular set ${\cal S}^*$ is the volume of the asymptotic surface ${\cal S}_\infty$. Its AC analog is the on-shell action integrated over the past domain of dependence $D^- ({\cal S}^*)$.
• Figure 4: The WdW patch ${\cal W}_t$, associated to the cut-off surface ${\cal S}'_t$, intersects the singularity at ${\cal S}^*_t$.
• Figure 5: The WdW patch ${\cal W}_u$ (in yellow), associated to a given ${\cal S}^*_u$ subset (in black) of the full terminal set $\cal S^*$ (in red). The codimension-two set ${\cal V}_u$ is the (possibly disconnected) boundary of Cauchy surfaces $\Sigma_u$ for ${\cal W}_u$.