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

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arxiv 2106.12585 v3 pith:UUJ2XWG7 submitted 2021-06-23 hep-th gr-qc

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

classification hep-th gr-qc
keywords complexitylorentzianflowsthreadsbulkholographicstatevolume
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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.

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    hep-th 2026-06 unverdicted novelty 7.0

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    hep-th 2026-07 conditional novelty 5.0

    TTbar-induced corrections to complexity=anything expand into generalized Willmore functionals that admit a local multi-flavor Lorentzian-thread decomposition by curvature sector.

  5. A Timelike Quantum Focusing Conjecture

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