Black hole subregion action and complexity

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 the Wheeler-DeWitt 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 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 diffeomorphism-invariant action with a well-defined variational principle, one observes that the main contribution that results in a nontrivial behavior of the on-shell 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.