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Paper IPM / P / 15975  


Abstract:  
We evaluate finite part of the onshell action for black brane solutions of Einstein gravity on different subregions of spacetime enclosed by null boundaries. These subregions include the intersection of the WheelerDeWitt patch with past/future interior and left/right exterior for a twosided black brane. Identifying the onshell action on the exterior regions with subregion complexity, one finds that it obeys the subadditivity condition. This gives an insight to define a new quantity named mutual complexity. We will also consider a 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 diffeomorphisminvariant action with a welldefined variational principle, one observes that the main contribution that results in a nontrivial behavior of the onshell action comes from joint points where two lightlike boundaries (including the 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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