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.
Highlights
Based on the earlier works of Refs. [1,2], it was conjectured that computational complexity associated with a boundary state may be identified with the on-shell action evaluated on a certain subregion of the bulk spacetime [3,4]
Even for a static geometry, such as an eternal black hole, the interior of black hole grows with time, indicating that there could be a quantity in the dual field theory that grows with time far after the system reaches the thermal equilibrium
In this paper, motivated by the “complexity equals action” proposal, we have evaluated the on-shell action on certain spacetime subregions enclosed by null boundaries that, includes the WDW patch itself, too
Summary
Based on the earlier works of Refs. [1,2], it was conjectured that computational complexity associated with a boundary state may be identified with the on-shell action evaluated on a certain subregion of the bulk spacetime [3,4]. ; ð1:2Þ where Lis an undetermined length scale and γ is the determinant of the induced metric on the joint point where two null segments intersect, and Even with this extra term the length scale Lremains undetermined, adding this term to action (1.1) would define a consistent theory. One may want to get the finite on-shell action (as we do for gravitational free energy) that requires adding certain counterterms These terms are required from holographic renormalization The aim of this article is to compute on-shell action on certain subregions behind and outside the horizon enclosed by null boundaries. We will consider an eternal black brane that provides a gravitational dual for a thermofield double state Those subregions that are behind the horizon are UV finite and time dependent, though those outside the horizon are typically UV divergent and time independent. The last section is devoted to a discussion and conclusion, in which we present the interpretation of our results
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