Holography beyond AdS/CFT by Nicholas S Salzetta Adissertationsubmittedinpartialsatisfactionofthe requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Yasunori Nomura, Chair Professor Ori Ganor Assistant Professor Jessica Lu Fall 2019 Holography beyond AdS/CFT Copyright 2019 by Nicholas S Salzetta 1 Abstract Holography beyond AdS/CFT by Nicholas S Salzetta Doctor of Philosophy in Physics University of California, Berkeley Professor Yasunori Nomura, Chair Physicists have long sought to fully understand how gravity can be fully formulated within aquantummechanicalframework.Apromisingavenueofresearchinthisdirectionwas born from the idea of holography - that gravitational physics can be recast as a di↵erent theory living in fewer dimensions. Evidence of this phenomena was first observed in the arena of black hole physics, where the entropy of a black hole was calculated to scale with its area, not its volume. The advent of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence provided an explicit realization of holography for gravitational theories in AdS space. This brought a flurry of activity dedicated to dissecting this correspondence. Ultimately, however, it will be necessary to move beyond the confines of AdS/CFT in order to understand our universe, as we live in a de Sitter type universe. The research presented in this dissertation attempts to broaden the scope of holgraphic theories to include more phenomenologically relevant universes. To do so, we utilize a top-down approach and take results from AdS/CFT that appear to be general holographic results and see how they can be applied in spacetimes other than AdS. In particular, we take the Ryu-Takayanagi (RT) formula, along with its related results, and investigate what we can learn by applying it to general spacetimes. Doing so naturally forces us to utilize holographic screens, as these are the largest such surfaces where the RT formula can be self-consistently applied. This approach allows us to examine properties of the purported boundary theory for general spacetimes, including the entanglement structure and propagation speeds of excitations dual to bulk excitations. This is done in chapters 2 and 3. In chapters 4 and 5, we use this lens of generalized holography to elucidate the nature of the relationship between entanglement and emergent geometry. Finally, in chapter 6, we revisit one the important underlying assumption that the RT formula is general and demonstrate the validity of this assumption. i To my parents, Meg and Paul ii Contents Contents ii List of Figures iv 1 Introduction 1 2 Toward holography for general spacetimes 6 2.1 Introduction . 6 2.2 HolographyandQuantumGravity. 10 2.3 HolographicDescriptionofFRWUniverses . 14 2.4 Interpretation:BeyondAdS/CFT . 27 2.5 Discussion . 42 2.6 Appendix ..................................... 48 3 Butterfly Velocities for Holographic Theories of General Spacetimes 55 3.1 Introduction . 55 3.2 Definition of the Butterfly Velocity in Holographic Theories of General Space- times . 56 3.3 Butterfly Velocities for the Holographic Theory of FRW Universes . 60 3.4 Discussion . 65 4 Spacetime from Unentanglement 67 4.1 Introduction . 67 4.2 MaximallyEntropicStatesHaveNoSpacetime. 69 4.3 Spacetime Emerges through Deviations from Maximal Entropy . 84 4.4 HolographicHilbertSpaces. 90 4.5 Conclusion . 93 4.6 Appendix ..................................... 99 5 Pulling the Boundary into the Bulk 113 5.1 Introduction . 113 5.2 Motivation . 116 5.3 HolographicSlice ................................. 117 iii 5.4 Examples . 122 5.5 InterpretationandApplications . 130 5.6 Relationship to Renormalization . 137 5.7 Discussion . 139 5.8 Appendix ..................................... 141 6 HolographicEntanglementEntropyinTTDeformedCFTs 148 6.1 Introduction . 148 6.2 FieldTheoryCalculation. 149 6.3 Bulk Calculation . 153 6.4 Discussion . 154 Bibliography 157 iv List of Figures 2.1 For a fixed semiclassical spacetime, the holographic screen is a hypersurface ob- tained as the collection of codimension-2 surfaces (labeled by ⌧)onwhichthe expansion of the light rays emanating from a timelike curve p(⌧)vanishes,✓ =0. This way of erecting the holographic screen automatically deals with the redun- dancy associated with complementarity. The ambiguity of choosing p(⌧)reflects alargefreedominfixingtheredundancyassociatedwithholography. 11 2.2 The congruence of past-directed light rays emanating from p0 (the origin of the reference frame) has the largest cross sectional area on a leaf σ,wheretheholo- graphic theory lives. At any point on σ,therearetwofuture-directednullvectors orthogonal to the leaf: ka and la.ForagivenregionΓoftheleaf,wecanfinda codimension-2 extremal surface E(Γ) anchored to the boundary @ΓofΓ,which is fully contained in the causal region Dσ associated with σ............ 13 2.3 Various FRW universes, I, II, III, ,havethesameboundaryarea at di↵erent ⇤ times, t (I),t (II),t (III), .Quantumstatesrepresentinguniversesatthese··· A ⇤ ⇤ ⇤ moments belong to Hilbert··· space specified by the value of the boundary area. 16 ⇤ 2.4 A region L(γ)oftheleafσ is parameterizedH by an angle γ :[0,⇡]. The extremal ⇤ surface E(γ)anchoredtoitsboundary,@L(γ), is also depicted schematically. (In fact, E(γ)bulgesintothetimedirection.) . 17 2.5 The value of Q(γ)asafunctionofγ (0 γ ⇡/2) for w = 1(vacuumenergy), 0.98, 0.8, 0 (matter), 1/3 (radiation), and 1. The dotted− line indicates the lower− bound− given by the flat space geometry, which can be realized in a curvature dominatedopenFRWuniverse. 20 2.6 The value of Q(⇡/2) as a function of w.......................21 2.7 The shape of the extremal surfaces E(⇡/2) for w = 1, 0.98, 0.8, 0, 1/3, and 1. The horizontal axis is the cylindrical radial coordinate− − normalized− by the apparent horizon radius, ⇠/⇠AH, and the vertical axis is the Hubble time, tH... 22 2.8 An FRW universe whose dominant component changes from w to w0 at time t0. Two surfaces depicted by orange lines are the latest extremal surface fully con- tained in the w region (bottom) and the earliest extremal surface fully contained in the w0 region (top), each anchored to the leaves at t and t0.. 25 ⇤ v 2.9 The ratio of the screen entanglement entropies, Rw = R1w(⇡/2), before and after the transition from a universe with the equation of state parameter w to that with w0 = 1, obtained from Figs. 2.6 and 2.7 using Eq. (2.58). The dot at w = 1 − represents R 1 = R1 1(⇡/2) obtained in Eq. (2.59). 26 − − 2.10 A steep potential (a) leading to the time evolution of the scalar field (b), the area of a leaf hemisphere (c), and the screen entanglement entropy (d). The same for abroadpotential(e)–(h). 28 2.11 Possible structures of the Hilbert space for a fixed boundary space B.In H⇤ the direct sum structure (left), each semiclassical spacetime in Dσ has its own ⇤ Hilbert space ,w.TheRussiandollstructure(right)correspondstothesce- ⇤ nario of “spacetimeH equals entanglement,” i.e. the entanglement entropies of the holographic degrees of freedom determine spacetime in Dσ .Thisimpliesthata ⇤ superposition of exponentially many semiclassical spacetimes can lead to a dif- ferent semiclassical spacetime. 31 2.12 If a black hole forms inside the holographic screen, future-directed ingoing light rays emanating orthogonally from the leaf σ at an intermediate time may hit the ⇤ singularity before reaching a caustic. While the diagram here assumes spherical symmetry for simplicity, the phenomenon can occur more generally. 36 2.13 To determine a state in the future, we need information on the “exterior” light sheet, the light sheet generated by light rays emanating from σ in the ka ⇤ directions, in addition to that on the “interior” light sheet, i.e. the one generated− by light rays emanating in the +ka directions. 37 2.14 In a universe beginning with a big bang, obtaining a future state requires a spec- ification of signals from the big bang singularity, in addition to the information contained in the original state. In an FRW universe this is done by imposing spa- tial homogeneity and isotropy, which corresponds to selecting a fine-tuned state from the viewpoint of the big bang universe. 43 3.1 A local operator at p, represented by the dot, is on the HRT surface anchored to the boundary of a spherical cap region, 0 γ,ontheleafattime⌘ ,located ⇤ at r = r . Note that the figure suppresses the time direction; for example, the ⇤ operator is not at the same time as the leaf. 61 3 3.2 Butterfly velocity vB multiplied by γ as a function of f. Here, γ is the angular size of the leaf region, and f is the fractional displacement of the bulk local operatorfromthetipoftheHRTsurface;seeEq.(3.15). 63 3.3 Butterfly velocity vB of a bulk local operator at the tip of the HRT surface, f =0,asafunctionoftheangularsizeγ of the leaf region for w =1,1/3, 0, 1/3, and 2/3 (solid curves, from top to bottom). The dashed curve represents − 3− vB =4/3γ ,theanalyticresultobtainedforγ 1inEq.(3.31).Thehorizontal dashed line represents the speed of light. .⌧ . 65 vi 4.1 The volume V (r+,R) of the Schwarzschild-AdS spacetime that can be recon- structed from the boundary theory, normalized by the corresponding volume V (R) in empty AdS space: f = V (r+,R)/V (R). Here, R is the infrared cut- o↵of (d + 1)-dimensional AdS space, and r+ is the horizon radius of the black hole. 73 4.2 The Penrose diagram of de Sitter space. The spacetime region covered by the flat- slicing coordinates is shaded, and constant time slices in this coordinate system are drawn. The codimension-1 null hypersurface ⌃0 is the cosmological horizon for an observer at r = 0, to which the holographic screen of the FRW universe asymptotes in the future. 75 4.3 Constant time slices and the spacetime region covered by the coordinates in static slicing of de Sitter space.