CHAPTER 7: GRAVITY

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Remarks on the thermodynamic stability of TT¯ deformations

We point out that negative specific heat at high energies is a characteristic feature of many TT deformations, both in the original d = 2 case and in d = 1 quantum mechanical cousins. This paper is a contribution to the memorial volume in honor of P G O Freund.The work of EB is partially supported by the Israeli Science Foundation Center of Excellence. The work of JLFB is partially supported by the Spanish Research Agency (Agencia Estatal de Investigación) through the Grants IFT Centro de Excelencia Severo Ochoa SEV-2016-0597, FPA2015-65480-P and PGC2018-095976-B-C2

Momentum/Complexity Duality and the Black Hole Interior

We establish a version of the Momentum/Complexity (PC) duality between the rate of operator complexity growth and an appropriately defined radial component of bulk momentum for a test system falling into a black hole. In systems of finite entropy, our map remains valid for arbitrarily late times after scrambling. The asymptotic regime of linear complexity growth is associated to a frozen momentum in the interior of the black hole, measured with respect to a time foliation by extremal codimension-one surfaces which saturate without reaching the singularity. The detailed analysis in this paper uses the Volume-Complexity (VC) prescription and an infalling system consisting of a thin shell of dust, but the final PC duality formula should have a much wider degree of generality.Peer reviewe

Holographic Bulk Reconstruction and Cosmological Singularities

We study the structure of entanglement wedges in the Kasner-AdS geometry, which provides an example of AdS/CFT engineered cosmological singularity. We investigate the specific limitations of causal reconstruction methods, imposed by the presence of the cosmological singularities, and we show the supremacy of modular reconstruction. This model provides an example where modular reconstruction based on a proper operator subalgebra is more powerful than the strongest possible causal reconstruction, based on the complete operator algebra.Peer reviewe

Terminal Holographic Complexity

We introduce a quasilocal version of holographic complexity adapted to ‘terminal states’ such as spacelike singularities. We use a modification of the action-complexity ansatz, restricted to the past domain of dependence of the terminal set, and study a number of examples whose symmetry permits explicit evaluation, to conclude that this quantity enjoys monotonicity properties after the addition of appropriate counterterms. A notion of ‘complexity density’ can be defined for singularities by a coarse-graining procedure. This definition assigns finite complexity density to black hole singularities but vanishing complexity density to either generic FRW singularities or chaotic BKL singularities. We comment on the similarities and differences with Penrose’s Weyl curvature criterion.Peer reviewe

Proof of a Momentum/Complexity Correspondence

We show that the holographic complexity = volume proposal satisfies a very general notion of momentum/complexity correspondence (PC), based on the momentum constraint of general relativity. It relates the rate of complexity variation with an appropriate matter momentum flux through spacelike extremal surfaces. This formalizes the intuitive idea that “gravitational clumping” of matter increases complexity, and the required notion of “infall momentum” is shown to have a Newtonian avatar which expresses this idea. The proposed form of the PC correspondence is found to be exact for any solution of Einstein’s equations in 2 + 1 dimensions, and any spherically symmetric solution in arbitrary dimensions, generalizing all previous calculations using spherical thin shells. Gravitational radiation enters through a correction which does not have a straightforward interpretation as a PC correspondence. Other obstructions to an exact PC duality have a topological origin and arise in the presence of wormholes.Peer reviewe

A Generalized Momentum/Complexity Correspondence

Holographic complexity, in the guise of the Complexity = Volume prescription, comes equipped with a natural correspondence between its rate of growth and the average infall momentum of matter in the bulk. This Momentum/Complexity correspondence can be related to an integrated version of the momentum constraint of general relativity. In this paper we propose a generalization, using the full Codazzi equations as a starting point, which successfully accounts for purely gravitational contributions to infall momentum. The proposed formula is explicitly checked in an exact pp-wave solution of the vacuum Einstein equations.Agencia Estatal de Investigación through the grants IFT Centro de Excelencia Severo Ochoa SEV- 2016-0597, FPA2015-65480-P and PGC2018-095976-B-C21. The work ofFPA2017-84436-P from Ministerio de Economia y CompetitividadFPU Grant FPU16/00639.Peer reviewe

On The Evolution Of Operator Complexity Beyond Scrambling

CC-BY 4.0We study operator complexity on various time scales with emphasis on those much larger than the scrambling period. We use, for systems with a large but finite number of degrees of freedom, the notion of K-complexity employed in [1] for infinite systems. We present evidence that K-complexity of ETH operators has indeed the character associated with the bulk time evolution of extremal volumes and actions. Namely, after a period of exponential growth during the scrambling period the K-complexity increases only linearly with time for exponentially long times in terms of the entropy, and it eventually saturates at a constant value also exponential in terms of the entropy. This constant value depends on the Hamiltonian and the operator but not on any extrinsic tolerance parameter. Thus K-complexity deserves to be an entry in the AdS/CFT dictionary. Invoking a concept of K-entropy and some numerical examples we also discuss the extent to which the long period of linear complexity growth entails an efficient randomization of operators.The work of J.L.F. Barbon and R. Sinha is partially supported by the Spanish Research Agency (Agencia Estatal de Investigaci ́on) through the grants IFT Centro de Excelencia Severo Ochoa SEV-2016-0597, FPA2015-65480-P and PGC2018-095976-B-C21. The work of E. Rabinovici and R. Shir is partially supported by the Israeli Science Foundation Center of Excellence and by the I Core Program “The Quantum Universe”, sponsored by the 25 Planning and Budgeting Committee and the Israeli Science Foundation.Peer reviewe