  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.
