Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity

TL;DR

This work develops a Lorentzian-flow reformulation of holographic complexity, recasting the complexity=volume proposal in terms of min flow-max cut for divergenceless, timelike flows whose flux through a boundary region equals the homologous maximal bulk Cauchy-slice volume. It provides explicit geometric constructions of Lorentzian threads in AdS and BTZ spacetimes, interprets these threads as gatelines for state preparation, and shows the bulk linearized Einstein equations arise from a holographic first law of complexity via the bulk–boundary symplectic structure. Perturbative Lorentzian threads are mapped to closed differential forms, with the canonical perturbative form given by the bulk symplectic current; the closedness of this form encodes linearized gravity. A key conceptual advance is the interpretation of spacetime dynamics as emergent from boundary complexity and an ensemble-average refinement of complexity that accounts for suboptimal tensor networks, connecting to tensor-network models of AdS and suggesting a broader, time-dependent holographic dictionary. The results point toward a principle of spacetime complexity and open avenues for extending to higher-derivative gravities and non-static states via an ensemble framework, with potential implications for bulk reconstruction and quantum gravity phenomenology.

Abstract

Holographic entanglement entropy was recently recast in terms of Riemannian flows or 'bit threads'. We consider the Lorentzian analog to reformulate the 'complexity=volume' conjecture using Lorentzian flows -- timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. By the nesting of Lorentzian flows, holographic complexity is shown to obey a number of properties. Particularly, the rate of complexity is bounded below by conditional complexity, describing a multi-step optimization with intermediate and final target states. We provide multiple explicit geometric realizations of Lorentzian flows in AdS backgrounds, including their time-dependence and behavior near the singularity in a black hole interior. Conceptually, discretized flows are interpreted as Lorentzian threads or 'gatelines'. Upon selecting a reference state, complexity thence counts the minimum number of gatelines needed to prepare a target state described by a tensor network discretizing the maximal volume slice, matching its quantum information theoretic definition. We point out that suboptimal tensor networks are important to fully characterize the state, leading us to propose a refined notion of complexity as an ensemble average. The bulk symplectic potential provides a specific 'canonical' thread configuration characterizing perturbations around arbitrary CFT states. Consistency of this solution requires the bulk satisfy the linearized Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating for a principle of 'spacetime complexity'. Lastly, we argue Lorentzian threads provide a notion of emergent time. This article is an expanded and detailed version of [arXiv:2105.12735], including several new results.

Figures (25)

• Figure 1: WDW patch in double sided AdS-Schwarzschild black hole. The shaded region (green) is the domain of dependence of the bulk Cauchy slice (purple) asymptoting to the $(r=\infty$) boundary Cauchy surfaces defined at left and right boundary times $t_{L}$ and $t_{R}$. Here we have considered the case when $t_{L}=t_{R}$. More precisely, the left and right corners and upper and lower tips of the causal diamond hit UV regulating surfaces just before reaching the timelike boundaries $r=\infty$ or past and future singularities $r=0$, respectively. CA duality is defined by the gravitational action evaluated on the WDW patch, while CV duality is given by the spatial volume of the maximal hypersurface extending to $t_{L}$ and $t_{R}$.
• Figure 2: Asymptotically AdS $M$ in Lorentzian signature foliated by maximal bulk slices $\Sigma$. Here the boundary subregion $A$ (gold) has boundary $\partial A=\sigma_{A}$ on both timelike boundaries of $M$, anchoring the maximal volume slice $\Sigma(A)$. Disconnected boundary subregion $B$ (light blue) is disjoint from $A$ and has boundary $\sigma_{AB}$ anchor $\Sigma(AB)$. The surfaces $A$ and $AB\supset A$ are nested boundary regions. The upper shaded region (light green) is the bulk region $r(A)$, while the middle shaded region (blue) is the bulk region $r(B)$. Bulk regions $r(A)$ and $r(B)$ are nested inside the bulk region of $AB$, $r(AB)=r(A)\cup r(B)$.
• Figure 3: Examples of flows $v(A,AB)$ for two nested regions. Left: $\mathcal{C}(AB)=\mathcal{C}(A)$, occurs when there is no flux through $B$; all flux passing through $\Sigma(AB)$ also passes $\Sigma(A)$. Middle: $\mathcal{C}(AB)>\mathcal{C}(A)$, where flux crosses $\Sigma(AB)$ but exits through $B$. Right: $\mathcal{C}(AB)<\mathcal{C}(A)$, when flux enters through $B$ and exits through $A$.
• Figure 4: Nested cuts associated with the three disjoint boundary regions $A,B,C$.
• Figure 5: A partition of a boundary region $A\subset \partial M$ into $A=A_{X}\cup A_{Y}$, where $A_{X}\cap A_{Y}=\emptyset$, and with associated non-intersecting boundary Cauchy slices $\sigma_{A_{X}}\cup\sigma_{A_{Y}}=\sigma_{A}$. The red line connecting $\sigma_{X}$ and $\sigma_{Y}$ denotes the maximal Cauchy slice $\Sigma$ containing the HRT surface $R$. The minimal flux of a Lorentzian flow $v$ through the boundary $A^{X,Y}$ computes the subregion complexity of the boundary regions $\sigma_{X,Y}$, given by the maximum volumes of $V_{A_{X}}$ and $V_{A_{Y}}$, respectively. The flow which computes the volume of the entire boundary slice will always have more flux through $A^{X}$ and $A^{Y}$ than the minimal flux, leading to superadditivity \ref{['eq: superAdditivityProof']}.