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.
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Barbón, J.L., Martín-García, J. & Sasieta, M. Momentum/Complexity duality and the black hole interior. J. High Energ. Phys. 2020, 169 (2020). https://doi.org/10.1007/JHEP07(2020)169
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DOI: https://doi.org/10.1007/JHEP07(2020)169