Abstract
We establish a version of the Momentum/Complexity (PC) duality between the rate of operator complexity growth and an appropriately defined 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.
Highlights
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
By examining the VC complexity of thin spherical shells impinging on double-sided AdS black holes, we can explicitly identify the relevant momentum component
The key to the construction is to measure this momentum with respect to a bulk time foliation by the same maximal surfaces that one uses to compute VC complexity
Summary
For a holographic CFT defined on a spherical spatial manifold Sd−1 of radius L, we consider its gravity dual on AdSd+1, taken to have curvature radius L. Spherical symmetry holds globally in the full spacetime, whereas stationarity is broken at W Both V ± have smooth Killing vectors which are timelike in the asymptotic regions and spacelike inside event horizons. In terms of the shell’s proper time τ and its radius R(τ ), continuity of the spacetime metric across W implies the first junction condition, f±(R). The second junction condition establishes the jump of the extrinsic curvature across W as proportional to the stress-energy on the shell’s world-volume. The particular conditions of spherical symmetry and stationarity along V ± allow us to write the junction conditions in terms of the Killing vectors ξ±, an expression that will be useful later.
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