Black hole subregion action and complexity

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 onshell 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.

On subregion action complexity in AdS3 and in the BTZ black hole

“…Keeping just the terms linear in l in eq. (3.27), we find agreement with the subregion complexity C BTZ,R A computed for one side of the Kruskal diagram, see [46,47]:…”

“…A similar behaviour of subregion CA is found in the thermofield double state where the subsystems are taken as the two disconnected boundaries of spacetime. This case was investigated for asymptotically AdS black holes in D dimensions [46,47], showing that the complexity=action is subadditive when η <η D and superadditive for η >η D . The value ofη D is given by the zero of g D (η) [46]:…”

Complexity of mixed states in QFT and holography

“…(4.6) and with masses 1/δ. Of course, the diagonalization process can also be performed directly for the continuum Hamiltonian and in the infinite volume limit, 50 in which case one obtains the eigenfrequencies ω k = k 2 + m 2 and the sum over the (dimensionless) k i is replaced by the (dimensionful) momentum integral V d−1…”

“…(4.8) 49 The lattice sites are designated with n = n ix i , wherex i are unit normals along the spatial axes. 50 Recall that there are two independent limits here. The continuum limit refers to taking the lattice spacing δ small compared to the other physical parameters in the problem, e.g., δm → 0 and δ/L → 0.…”

Complexity growth of rotating black holes with a probe string

“…The action calculation for some cases includes divergence because of the AdS boundary [142,143]. Its normalization methods are also discussed in some works [77,144,145,146,147,148,149,150].…”