Complexity and shock wave geometries

Abstract

In this paper we refine a conjecture relating the time-dependent size of an Einstein-Rosen bridge (ERB) to the computational complexity of the dual quantum state. Our refinement states that the complexity is proportional to the spatial volume of the ERB. More precisely, up to an ambiguous numerical coefficient, we propose that the complexity is the regularized volume of the largest codimension one surface crossing the bridge, divided by ${G}_{N}{l}_{\mathrm{AdS}}$. We test this conjecture against a wide variety of spherically symmetric shock wave geometries in different dimensions. We find detailed agreement.