Holographic complexity and spacetime singularities

HJE Holographic complexity and spacetime singularities Jose L.F. Barbon 0 1 2 Eliezer Rabinovici 0 1 2 3 4 Instituto de F sica Teorica IFT UAM/CSIC 0 1 2 C/ Nicolas Cabrera 0 1 2 Campus Universidad Autonoma de Madrid 0 1 2 0 4 Place Jussieu , 75252 Paris Cedex 05 , France 1 Jerusalem 91904 , Israel 2 Madrid 28049 , Spain 3 Racah Institute of Physics, The Hebrew University 4 Laboratoire de Physique Theorique et Hautes Energies, Universite Pierre et Marie Curie We study the evolution of holographic complexity in various AdS/CFT models containing cosmological crunch singularities. We nd that a notion of complexity measured by extremal bulk volumes tends to decrease as the singularity is approached in CFT time, suggesting that the corresponding quantum states have simpler entanglement structure at the singularity. AdS-CFT Correspondence; Black Holes; Spacetime Singularities - 1 Introduction 2 3 2.1 2.2 3.1 3.2 3.3 3.4 state, of order T 1eS, when complexity should level o at a value of order eS, [11]. Further motivation for some sort of complexity/volume relation stems from tensor network representations of various CFT states with holographic interpretation [9, 12{17]. In this set up, there is a natural notion of `complexity' associated to the size of the tensor network. A continuum version of this idea is embodied by the formula (cf. [18{22]) where t is a codimension-one space-like section of the bulk with extremal volume, G is the e ective Newton's constant in AdS and ` its curvature radius. The numerical normalization C(t) / Vol( t) ; G` { 1 { (1.1) (1.2) in (1.2) is a somewhat arbitrary choice, unless we nd some ab initio de nition of complexity in the continuum CFT (see [ 23, 24 ] and [25] for recent discussions in this direction). The extremal codimension-one surfaces leading to the result (1.1) probe the interior of the eternal black hole geometry. However, they turn out to be maximal surfaces, staying far from the black hole singularity as t ! 1, because space-like volumes actually get `crunched' at the singularity. This fact prevents a straightforward association between the large complexity of late-time eternal black holes and the occurrence of a singularity in the bulk. In this paper we examine the behavior of (1.2) in situations which can be interpreted as representing holographic versions of cosmological singularities, and yet one can make a good crunches, and proceed in section 3 to discuss their salient properties from the point of view of a volume/complexity relation. 2 Cosmological frames and bulk singularities There are a number of strategies to engineer bulk cosmological singularities in AdS/CFT models, some of which we brie y review in this section. In general, the models discussed can be described as time-dependent deformations of CFTs. The deformed Hamiltonian is chosen in such a way that it becomes singular in nite time, while we are still able to construct the dual bulk dynamics as explicitly as possible. In order to bene t from the UV completeness of CFTs, we should specify the Hamiltonian deformation in terms of marginal or relevant operators. This leads to two major classes of models. In the rst case, either a dimensionless coupling or the background metric where the CFT is de ned is given a time dependence. In the second case, there is a time-dependent mass scale. 2.1 Models based on singular CFT frames One implementation of these ideas is to work with a CFT on a world-volume frame with a singular time dependence, and study how this singularity is realized by the bulk dynamics. In the previous broad classi cation, this strategy corresponds to adding a time-dependent marginal operator to the CFT. Among the in nite set of models of this type, one looks for those whose dual bulk dynamics is relatively easy to construct. An example of this sort is provided by the Kasner metrics with d > 3 and parameters pi satisfying P pi = Pi pi2 = 1. These metrics have a curvature singularity at t = 0 provided at least one of the coe cients pi is di erent from 0 or 1. The singularity looks like a big crunch in directions with pi > 0 and a big rip in the directions with pi < 0 (of which there is at least one for any d > 3). The Pi pi = 1 relation ensures that the total spatial volume vanishes linearly with t at the t = 0 singularity. Despite its anisotropic character, the Kasner frame has the technical advantage of being Ricci at, which allows us to write one (d + 1)-dimensional bulk solution with no extra work, namely ds2 = dr2 r2 + r2 This solution provides a large-N de nition of a certain CFT state on the Kasner frame, which we refer to as the `Kasner state'. Its global structure resembles an AdS Poincare patch, cut by the singularity t = 0 on the time-re ection spatial surface, with additional null singularities at t = 1 (see Fig 1). The absence of a clear `turn-o ' of the time-dependence makes the Kasner state somewhat di cult to interpret from the point of view of the CFT. In a (...truncated)