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arXiv · 1809.06031

Subregion Action and Complexity

Mohsen Alishahiha·Komeil Babaei Velni·M. Reza Mohammadi Mozaffar
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Abstract

We evaluate finite part of the on-shell action for black brane solutions of Einstein gravity on different subregions of spacetime enclosed by null boundaries. These subregions include the intersection of WDW patch with past/future interior and left/right exterior for a two sided black brane. Identifying the on-shell action on the exterior regions with subregion complexity one finds that it obeys subadditivity condition. This gives an insight to define a new quantity named mutual complexity. We will also consider certain subregion that is a part of spacetime which could be causally connected to an operator localized behind/outside the horizon. Taking into account all terms needed to have a diffeomorphism invariant action with a well-defined variational principle, one observes that the main contribution that results to a nontrivial behavior of the on-shell action comes from joint points where two lightlike boundaries (including horizon) intersect. A spacelike boundary gives rise to a linear time growth, while we have a classical contribution due to a timelike boundary that is given by the free energy.

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Mohsen Alishahiha, Komeil Babaei Velni, M. Reza Mohammadi Mozaffar. 2018-09-17. Subregion Action and Complexity. https://doi.org/10.1103/physrevd.99.126016

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