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Paper   IPM / P / 15357
School of Physics
  Title:   Subregion Action and Complexity
1.  M. Alishahiha
2.  K. Babaei Velni
3.  M.R. Mohammadi Mozaffar
  Status:   Published
  Journal: Phys. Rev. D
  Vol.:  99
  Year:  2019
  Pages:   126016
  Supported by:  IPM
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 these regions with subregion complexity one finds that it obeys subadditivity condition. 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 boundaries give 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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