De Sitter Horizons & Holographic Liquids
DDe Sitter Horizons & Holographic Liquids
Dionysios Anninos , Dami´an A. Galante , and Diego M. Hofman Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK Institute for Theoretical Physics Amsterdam & ∆ Institute for Theoretical Physics, University of Amsterdam,Science Park 904, 1090 GL Amsterdam, The Netherlands
Abstract
We explore asymptotically AdS solutions of a particular two-dimensional dilaton-gravitytheory. In the deep interior, these solutions flow to the cosmological horizon of dS . Wecalculate various matter perturbations at the linearised and non-linear level. We considerboth Euclidean and Lorentzian perturbations. The results can be used to characterise thefeatures of a putative dual quantum mechanics. The chaotic nature of the de Sitter horizonis assessed through the soft mode action at the AdS boundary, as well as the behaviour ofshockwave type solutions. a r X i v : . [ h e p - t h ] A ug ontents κ <
29B Shockwaves in dS
30C Some details for out-of-time-ordered correlators 31D The γ -theory and γ -Schwarzian 32 D.1 The γ -Schwarzian theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 The AdS/CFT correspondence has provided important evidence in support of the idea that thehorizon of a black hole, at the microscopic level, comprises a large number of strongly interactingdegrees of freedom in a liquid-like dissipative state. The role of AdS is to provide the horizon witha non-gravitating boundary from which the characteristic features of the holographic liquid canbe probed. The simplest way to do so is through the evaluation of correlations of low energy localoperators. The goal of this paper is to make progress toward the hypothesis that the cosmologicalhorizon of an asymptotically de Sitter universe is itself, microscopically, a holographic liquid.1everal thermodynamic features of the de Sitter horizon have been known since the classic workof Gibbons and Hawking [1]. What is missing so far, is a framework to bridge the gap betweenthe thermodynamic and microscopic, in the same spirit that AdS/CFT bridges the gap betweenthe old literature on black hole thermodynamics and the modern perspective of a holographicliquid.What makes the de Sitter problem challenging is the absence of a spatial AdS boundary,or more generally some non-gravitating region of spacetime, from which to probe the de Sitterhorizon. To address this, we construct a phenomenological gravitational theory which containsasymptotically AdS solutions with a region of de Sitter in the deep interior. Our approachis inspired, to an extent, by analogous approaches applying the framework of AdS/CFT toproblems in condensed matter [2]. Though this approach is incomplete, we believe that bringingthe question of the de Sitter horizon to the standards of AdS/CMT is a step in the forwarddirection. Ultimately, a successful approach will require a microscopic completion.Concretely, we construct a class of two-dimensional gravitational theories admitting solutionswhich interpolate between an AdS boundary with a dS horizon in the deep interior [3]. Havingdone so, we probe the de Sitter horizon using the available tools of AdS/CFT. There are severalreasons to work in two-dimensions. Though simpler, the dS horizon shares many features withits higher dimensional cousin, including the characteristics of its quasinormal modes [4] and thefiniteness of the geometry. Also, dS × S appears as solution of Einstein gravity with Λ > is directly relevant to four-dimensional de Sitter. Furthermore, recent progressin our microscopic understanding of AdS holography [6, 7, 8, 9, 10, 11, 12, 13] may guide us inconstructing a microscopic model dual to the interpolating geometry.Embedding an inflating universe in an AdS d +1 spacetime with d > boundary. The reason wecan do this is crucially related to the dimensionality of the AdS boundary, and is precisely theone case not considered in previous literature. It is closer in spirit to a ‘holographic worldline’perspective of the static de Sitter region [16, 17]. From this perspective, it is assumed that thedual description of the static de Sitter region is captured by a large N quantum mechanicalmodel, rather than some local quantum field theory in d > U (1) isometry as well as an AdS boundary. Of these, one branch isdescribed by a metric which interpolates between an AdS boundary and the static patch of dS .2n section 3 we consider matter perturbations at the linearised level. In section 4 we considerthe dynamics of the soft mode residing at the boundary of the interpolating solutions. In section5 we calculate out-of-time-ordered four-point functions stemming from the exchange of the softmode. We note the absence of exponential Lyapunov behaviour. In section 6 the backreactionof a shockwave pulse on the interpolating geometry is explored. It is noted that in certaincircumstances, the horizon can retreat rather than advance toward the AdS boundary. Insection 7, we conclude with a discussion of the results, their potential holographic implications,and some speculative remarks. Finally, further details for certain calculations can be found inthe appendices. The theory we consider is described by the following dilaton-gravity Euclidean action: S E = − κ (cid:90) d x √ g ( φR + V ( φ )) − κ (cid:90) ∂ M du √ h φ K + S m , (2.1)where S m is the action for some matter theory. One can also add to this action a topologicalterm: S top = − φ (cid:90) d x √ gR − φ (cid:90) ∂ M du √ h K , (2.2)where φ is a positive constant which we consider to be large. The Newton constant is givenby κ = 8 πG . The scalar φ in (2.1), or equivalently κ , may be positive or negative – whatis important is that φ tot = ( φ + φ/κ ) remain everywhere positive. Hence, throughout thediscussion we will assume that φ (cid:29) | φ/κ | (cid:29) κ . The Euclidean geometry lives on a disk topology M with a circular boundary ∂ M .The equations of motion for the two-dimensional metric and dilaton read: ∇ a ∇ b φ − g ab ∇ φ + g ab V ( φ ) = − κ T mab , (2.3) R = − V (cid:48) ( φ ) , (2.4)where T mab is the stress tensor for the matter fields. There are also the matter equations ofmotion. We assume further, that the matter theory interacts only with the two-dimensionalmetric at the classical level. Taking the divergence of equation (2.3) leads to:[ ∇ , ∇ a ] φ + 12 ∇ a V ( φ ) = 0 . (2.5)Some algebra reveals that [ ∇ , ∇ a ] φ = R ∂ a φ/
2. Using this, the divergence equation reduces to:
R ∂ a φ = − ∂ a V ( φ ) . (2.6)3n other words, one finds that equation (2.4) is redundant whenever ∂ a φ (cid:54) = 0. Another usefulequation is obtained by taking the trace of (2.3) that leads to: − ∇ φ + V ( φ ) = − κ T mab g ab . (2.7)Finally, in the absence of matter it is straightforward to check that the equations of motionimply ξ a = (cid:15) ab ∂ b φ is a Killing vector [18]. We will be considering tree level features of the theory (2.1). For this purpose, a useful gaugeis the conformal gauge: ds = e ω ( ρ,τ ) ( dρ + dτ ) , (2.8)where τ is periodic with period 2 π and the origin of the disk lives at ρ → −∞ .Using (2.7), the dilaton equations become: ∂ ρ ∂ τ φ − ∂ τ ω ∂ ρ φ − ∂ ρ ω ∂ τ φ = − κT mρτ , (2.9)12 ( ∂ ρ ∂ ρ − ∂ τ ∂ τ ) φ − ∂ ρ ω ∂ ρ φ + ∂ τ ω ∂ τ φ = − κ (cid:0) T mρρ − T mττ (cid:1) . (2.10)Another gauge which is often convenient is the Schwarzschild gauge: ds = N ( r, T ) dT + dr N ( r, T ) . (2.11)In the Schwarzschild gauge, the Euclidean solution is [19]: N ( r ) = 1 | φ h | (cid:90) r sign φ h dz V ( φ ( z )) , φ ( r ) = | φ h | r . (2.12)The periodic condition T ∼ T + 2 π is fixed by requiring a regular Euclidean geometry. Thepoint r = sign φ h is the location of the Euclidean horizon.Given the Killing vector ξ a = (cid:15) ab ∂ b φ , we will (locally) choose ω to be solely a function ofthe ρ -coordinate. This dramatically simplifies the matter-less equations, which can be now besolved explicitly. We will be interested in the class of potentials introduced in [3]. We take V ( φ ) to be a non-negative function. Outside some transition region | φ | (cid:38) (cid:15) with (cid:15) a small positive number,the potential behaves as V ( φ ) ≈ | φ | . This transition region is not very important, but we4ssume that V ( φ ) is continuous and vanishing at φ = 0. Two simple examples are V ( φ ) = 2 | φ | and V ( φ ) = 2 φ tanh( φ/(cid:15) ). For positive/negative φ the metric has constant negative/positivecurvature. Depending on the value of the dilaton at the origin of the disk, which we denote by φ h , the metric may have both negative and positive curvature, or purely negative curvature.For most of the paper we will be interested in the sharp gluing limit with (cid:15) →
0. In theconformal gauge with φ h <
0, the solution for the metric is given by: ds = cos − ρ (cid:0) dρ + dτ (cid:1) , ρ ∈ (0 , π/ , cosh − ρ (cid:0) dρ + dτ (cid:1) , ρ ∈ ( −∞ , . (2.13)Smoothness requires τ ∼ τ + 2 π . The dilaton equation is solved by: φ = − φ h tan ρ , ρ ∈ (0 , π/ , − φ h tanh ρ , ρ ∈ ( −∞ , . (2.14)We will refer to expressions (2.13) and (2.14) as the interpolating solution. The Euclidean AdS boundary lives at ρ = π/
2, whereas for negative values of ρ the metric is the standard metric ona half-sphere. The Euclidean AdS piece of the geometry (2.13), between ρ ∈ (0 , π/ geometry. It can be viewed as a pieceof the hyperbolic cylinder.For φ h >
0, the solution has negative curvature everywhere and is given by: ds = sinh − ρ (cid:0) dρ + dτ (cid:1) , ρ ∈ ( −∞ , , (2.15) φ = − φ h coth ρ . (2.16)Again, smoothness requires τ ∼ τ + 2 π . The geometry (2.15) is the hyperbolic disk. It is theEuclidean continuation of the AdS black hole geometry.As we mentioned earlier, we will allow both signs of κ in (2.1). To give some context to this,consider the spherically symmetric sector of Einstein gravity with a positive cosmological con-stant. In this case, the dimensionally reduced theory in two-dimensions is itself a dilaton-gravitytheory. In a particular limit, known as the Nariai limit, the dilaton potential is approximatelylinear. The Euclidean solution is given by the two-sphere with a running dilaton. The dilatonincreases from one pole to the other. Viewing the two-sphere as two hemispheres joined at theequator, we see that one hemisphere has an increasing dilaton profile towards the pole, whereasthe other has a decreasing dilaton profile. Switching from one sign to the other switches the signof κ in the effective two-dimensional theory. In appendix A we discuss a qualitatively similarphenomenon in the context of pure Jackiw-Teitelboim theory [20]. In appendix D we discussa broader family of dilaton potentials where the potential changes behaviour at some value φ and we relax the assumption V ( φ ) >
0. 5 orentzian continuation
As a final note, we mention that the Euclidean solutions can be continued to Lorentzian solutions.This continuation can be done in several ways. The simplest continuation takes τ → it . Theresulting solution is static and has a horizon at the value where the Euclidean geometry smoothlycapped off, namely ρ → −∞ . The boundary is that of an asymptotically Lorentzian AdS geometry. We can extend the geometry beyond the horizon. The interior of the horizon is eitherlocally dS or the interior of the AdS black hole, depending on the sign of φ h . The Penrosediagram for the interpolating geometry is shown in figure 1. κ = 0 x + x − =0 x + x − = e π x + x − =1 AdS dS dS AdS
𝒞 𝒞
Figure 1:
Penrose diagram for the interpolating solution. The dashed lines interpolate between a negative anda positive curvature region, that are coloured in light and darker blue, respectively. C is the boundary curve closeto the AdS boundaries. Inside the horizons, the geometry is locally dS , but depending on whether κ is positiveor negative, the dilaton behaves as in the interior of a dS black hole or as in the dS cosmological patch. One could also imagine analytically continuing ρ instead. In that case, one obtains a cosmo-logical type solution and the dilaton becomes time-dependent. If τ -remains periodically iden-tified upon continuing ρ , the interpolating cosmological geometry will have a big bang/crunchtype singularity and asymptote to the future/past boundary of global dS . Though interestingin their own right, these solutions have compact Cauchy surfaces and are thus not asymptoticallyAdS . We leave their consideration for future work. We now discuss several boundary conditions for the Euclidean dilaton-gravity theory (2.1).The Dirichlet problem is given by specifying the value of the dilaton φ b ( u ) and inducedmetric h ( u ) at ∂ M . Here, u is a compact coordinate that parameterises points on ∂ M . In the6eyl gauge: h ( u ) = e ω ( ρ ( u ) ,τ ( u )) (cid:0) ( ∂ u τ ( u )) + ( ∂ u ρ ( u )) (cid:1) , φ b ( u ) = φ ( ρ ( u ) , τ ( u )) , (2.17)where C = { τ ( u ) , ρ ( u ) } is the curve circumscribing ∂ M . The curve C is not independent data.Rather, it is fixed by the particular solution to the Dirichlet problem. Requiring the variation ofthe action to vanish under these conditions leads to the addition of the usual Gibbons-Hawkingboundary term. If the boundary value φ b ( u ) is positive everywhere, then C circumscribes part ofthe negatively curved geometry. If, moreover, h ( u ) and φ b ( u ) scale with some large parameter Λ,such that {√ h, φ b } = Λ { (cid:112) ˜ h, ˜ φ b } in the limit Λ → ∞ , we can approximate the Gibbons-Hawkingboundary term by: S bdy [ τ ( u )] ≈ − κ (cid:90) du (cid:112) ˜ h ˜ φ b (cid:18) Λ + sign φ h ( ∂ u τ ) h ( u ) + Sch ˜ h [ τ ( u ) , u ] (cid:19) . (2.18)The above action and its properties will be developed further in section 4. The covariantSchwarzian action Sch ˜ h [ τ ( u ) , u ] is obtained by taking the standard Schwarzian Sch [ τ ( u ) , u ] andcoupling it to a non-trivial metric ˜ h ( u ). We will write the more general expression later on in(4.14). For now, we fix ˜ h ( u ) = 1 and state the familiar expression:Sch [ τ ( u ) , u ] = ∂ u τ∂ u τ − (cid:18) ∂ u τ∂ u τ (cid:19) . (2.19)The reason for the sign φ h in (2.18) is that the background solution has a different radial slicingdepending on whether or not there is a two-sphere in the interior. In particular, for φ h > clock is that of the Euclidean AdS black hole, i.e. the isometric direction ofthe hyperbolic disk. When φ h <
0, the asymptotic AdS clock is that of Euclidean global AdS with periodically identified time, i.e. the hyperbolic cylinder. Consequently the function τ ( u )must be a map from S to S .Using the equations of motion, we can obtain an expression for the on-shell Euclidean actionfor the Dirichlet problem: − S cl = 1 κ (cid:90) du (cid:112) h ( u ) φ b ( u ) K ( u ) + 12 κ (cid:90) d x √ g ( V ( φ ) − φ ∂ φ V ( φ )) . (2.20)It is straightforward to evaluate the above action in the Schwarzschild gauge (2.11). The secondterm in (2.20) is non-vanishing only near the region where V ( φ ) is non-linear. This region canbe made parametrically small, such that the dominant contribution to the on-shell action isfrom the boundary term. For instance, take V ( φ ) ≈ ζ − φ near the transition region around φ = 0, and let the width of the region be 2 (cid:15) . Then, it can be shown, using the general solution More generally, one could also consider adding additional local boundary terms [21]. − S int = 12 κ (cid:90) d x √ g ( V ( φ ) − φ∂ φ V ( φ )) = α(cid:15) κζ , φ h (cid:28) − (cid:15) , (2.21)where α = 4 π/V ( φ h ). For small enough (cid:15) , the dominant contribution to the on-shell action comesfrom the boundary term (2.18). As a concrete example, one can consider V ( φ ) = 2 φ tanh φ/(cid:15) ,for which an exact calculation can be performed. In this case, ζ = (cid:15) , and one can indeed checkthat S int ∼ (cid:15) . Note that this contribution is not present if φ h (cid:29) (cid:15) .The Neumann problem is given by specifying the value of the conjugate momenta π φ b ( u )and π h ( u ) along a prescribed closed boundary curve C = { τ ( u ) , ρ ( u ) } parameterized by somecoordinate u . Explicitly, the conjugate (Euclidean) momenta to the boundary metric and dilatonare: κ π φ b = − K , κ π h = − n a ∂ a φ h . (2.22)Here n a is a unit vector that is normal to C . Since now the dilaton and induced metric areallowed to vary along C , we may find saddle point solutions for which the metric is nowhereAdS . As a simple example, if we fix { π h , π φ b } = {− / κ , } as our boundary condition, thesolution is given by the half-sphere.In addition to the Dirichlet and Neumann boundary conditions, we may also consider mixedDirichlet-Neumann boundary conditions or even conditions on linear combinations of the mo-menta and boundary values of the fields. The choice depends on the physics of interest. Notall boundary data { φ b ( u ) , h ( u ) } admit a Euclidean solution which is both smooth and real. Forinstance, if φ b is u -independent and negative, and h is u -independent and sufficiently large, onecannot realise a smooth and real Euclidean saddle. There may be complex saddles that areallowed however. In recent literature relating two-dimensional gravity to the SYK model (seefor example [22]), the Dirichlet problem is considered for the two-dimensional bulk theory, andwe will mostly focus on this choice.As a concrete example, we consider the Dirichlet problem with u -independent boundaryvalues φ b (cid:29) h (cid:29)
1. There are two-solutions satisfying these boundary conditions. For φ h < − φ h tan ρ c = φ b and cos − ρ c = h . For φ h > − φ h coth ρ c = φ b and sinh − ρ c = h . The boundary lies at a constant ρ = ρ c surface nearthe AdS boundary, with τ = u . The configuration with least action depends on the sign of κ .For κ > κ <
0) the dominant configuration has φ h > φ h < Perturbative analysis
In this section we consider the effect of a small matter perturbation. The goal is to gain someunderstanding as to how the geometry responds to matter perturbations when the backgroundhas an interpolating region. Unlike the case of the Jackiw-Teitelboim model [20] which has alinear dilaton potential which fixes the internal geometry entirely, our perturbations will alsoinvolve the metric.We consider turning on some matter content with non-trivial boundary profile. This will in-teract with the graviton and dilaton fields with strength κ , such that the perturbative expansionis one in small κ . The geometry communicates with the matter through the effect of the matteron the dilaton. We choose our background solution to be the interpolating geometry (2.13) withdilaton profile (2.14), which assumes the sharp-gluing limit where (cid:15) = 0, i.e. V ( φ ) = 2 | φ | . Giventhe background U (1) isometry in the τ -direction, it is convenient to consider the Fourier modesof the linearised fields. As matter, we consider a complex, massless, free scalar χ with action: S χ = (cid:90) d x √ g g ab ∂ a ¯ χ ∂ b χ . (3.1)In the conformal gauge, the general solution that is well behaved in the interior is: χ ( ρ, τ ) = (cid:88) m ∈ Z h m e | m | ( ρ − π/ imτ , h m ∈ C . (3.2)At the AdS boundary, where ρ = π/
2, the Fourier modes of the boundary profile for thematter field are h m . The ρτ -component of the matter stress-tensor will be most relevant forour calculations. We first solve the general linearised equations, and then consider particularboundary conditions. To leading order in a small matter field expansion, we must solve for the fluctuation δφ ( ρ, τ )of the background dilaton ¯ φ ( ρ ) and the fluctuation δω ( ρ, τ ) of the background conformal factor¯ ω ( ρ ). The perturbed geometry can be expressed as: ds = e ω ( ρ )+ δω ( ρ,τ )) (cid:0) dρ + dτ (cid:1) . (3.3)We obtain the following linearised equations: ∂ ρ ∂ τ δφ − ∂ ρ ¯ ω∂ τ δφ − ∂ τ δω∂ ρ ¯ φ = − κ T χτρ , (3.4) − e − ω (cid:0) ∂ ρ + ∂ τ (cid:1) δω − R δω = − ∂ φ V ( ¯ φ ) δφ , (3.5)9here now T χτρ is the stress tensor corresponding to the scalar χ . The inhomogeneous solutionto (3.4) can be expressed as an integral: δφ inh ( ρ, τ ) = e ¯ ω ( ρ ) (cid:90) ρ −∞ dρ (cid:48) e − ¯ ω ( ρ (cid:48) ) (cid:18) − κ (cid:90) τ dτ (cid:48) T χρτ ( ρ (cid:48) , τ (cid:48) ) + ∂ ρ (cid:48) ¯ φ ( ρ (cid:48) ) δω ( ρ (cid:48) , τ ) (cid:19) . (3.6)Substituting (3.6) into (3.5) we find: (cid:16) − e − ω ( ρ ) (cid:0) ∂ ρ + ∂ τ (cid:1) − R (cid:17) δω ( τ, ρ ) + ∂ φ V ( ¯ φ ) δφ inh ( ρ, τ ) = 0 . (3.7)For V ( φ ) = 2 | φ | we have ∂ φ V ( φ ) = 4 δ ( φ ). Away from the ρ = 0 interpolating region, we mustsolve the free wave-equation: (cid:16) − e − ω ( ρ ) (cid:0) ∂ ρ + ∂ τ (cid:1) − ¯ R (cid:17) δω ( ρ, τ ) = 0 . (3.8)The above equation states that the linearised correction to the Ricci scalar must vanish. Itssolution corresponds to a linearised diffeomorphism that preserves the conformal gauge. Foreither ρ > ρ <
0, (3.8) has two solutions. In the ρ < (cid:0) cosh ρ (cid:0) ∂ ρ + ∂ τ (cid:1) + 2 (cid:1) δω − ( ρ, τ ) = 0 . (3.9)We keep the solution that is non-singular at ρ = −∞ . The Fourier modes are then found to be: δω − ( ρ, q ) = α q e | q | ρ ( −| q | + tanh ρ ) , q ∈ Z , (3.10)with α q = α ∗− q . For ρ >
0, we must solve: (cid:0) cos ρ (cid:0) ∂ ρ + ∂ τ (cid:1) − (cid:1) δω + ( ρ, τ ) = 0 . (3.11)Interestingly, the above equation is that of a free scalar in AdS with ∆ = 2. The two solutionsare given by: δω + ( ρ, q ) = β q e | q | ρ ( | q | + tan ρ ) + γ q e −| q | ρ ( | q | − tan ρ ) , q ∈ Z , (3.12)with β q = β ∗− q and γ q = γ ∗− q . We are now in a position to compute the integral (3.6). For thejumping condition, we need the integral at ρ = 0. In order to fix some of the coefficients of thesolution, we must consider matching δω − ( ρ, τ ) and δω + ( ρ, τ ) across ρ = 0. The first matchingcondition is continuity of the metric at ρ = 0, so that δω − (0 , τ ) = δω + (0 , τ ) . (3.13)10rom this, it is straightforward to get α q = − ( γ q + β q ) . (3.14)The second matching condition relates the radial derivatives of δω ( ρ, τ ) across ρ = 0. From ∂ φ V ( φ ) = 4 δ ( φ ), one obtains the following jump condition:( ∂ ρ δω + ( ρ, τ ) − ∂ ρ δω − ( ρ, τ )) (cid:12)(cid:12)(cid:12) ρ =0 = − φ h δφ inh (0 , τ ) . (3.15)The continuity and jump condition allow us to fix two of the three integration constants { α q , β q , γ q } . Finally, requiring that δω + ( ρ, τ ) is fast-falling as one approaches ρ = π/ γ q = e π | q | β q , which in turn allows us to completely determine the linearisedprofile δω ( ρ, τ ). One finds: β q = i κφ h sgn q e − π | q | (1 + q )(1 − q ) T ρτ (0 , q ) . (3.16)Given δω ( ρ, τ ) we can obtain δφ ( ρ, τ ) by evaluating the integral in (3.6). Notice that δφ ( ρ, τ )has no jump in its first derivative. It is also interesting to obtain an equation for the curve S separating the region of positive and negative φ . To linear order, this reads¯ φ ( ρ ) + δφ inh ( ρ, τ ) = 0 . (3.17)Solutions to the above equation ρ S ( τ ) will produce a curve S separating positive and negativevalues of φ , and hence regions of positive and negative curvature. In the absence of matter, thiscurve is a circle. Given that there are no positivity constraints on δφ inh ( ρ, τ ) this original circlecan become deformed in both directions, as we shall shortly explore in a concrete example.Once a linearised solution is found, one must impose the appropriate boundary conditions.For instance, let us consider the Dirichlet problem with boundary values { h, φ b } which wetake to be u -independent with φ b large and positive. In the absence of matter, the boundaryvalue problem is solved by finding a ρ c such that h = e ω ( ρ c ) and φ b = − φ h tan ρ c . Thus,in the absence of matter the boundary curve is fixed to by the circle C = { u, ρ c } . Uponturning on a linear perturbation, C will be slightly deformed to a new curve pataremeterised by C p = { u + δτ ( u ) , ρ c + δρ ( u ) } , where δτ ( u ) and δρ ( u ) are fixed by solving: h = e ω ( ρ c ) (1 + 2 ∂ u δτ ( u ) + 2 δω ( ρ c , u ) + 2 ∂ ρ ¯ ω ( ρ c ) δρ ( u )) , (3.18) φ b = ¯ φ ( ρ c ) + ∂ ρ ¯ φ ( ρ c ) δρ ( u ) + δφ inh ( ρ c , u ) . (3.19)11he above equations are solved by: δρ ( u ) = − δφ inh ( ρ c , u ) ∂ ρ ¯ φ ( ρ c ) , ∂ u δτ ( u ) = ∂ ρ ¯ ω ( ρ c ) ∂ ρ ¯ φ ( ρ c ) δφ inh ( ρ c , u ) . (3.20)From the above we see that δτ ( u ) and δρ ( u ) are indeed O ( κ ) quantities, justifying our assumptionthat the resulting curve lies near the unperturbed curve C = { u, ρ c } . As a concrete example, suppose the matter field is characterised by a single harmonic m ≥ χ ( ρ, τ ) is simply: χ ( ρ, τ ) = 2 h m e m ( ρ − π/ cos( mτ ) , (3.21)from which it immediately follows that: T χρτ ( ρ, τ ) = − h m m e m ( ρ − π/ sin(2 mτ ) . (3.22)It is then straightforward to obtain the perturbative solution. We first evaluate the integral inequation (3.6) for the given stress tensor. The next step is to find the curves where the dilatonvanishes. This can be done numerically and shown for an example with m = 1 in figure 2. We (a) (b) Figure 2:
The perturbative solution for m = − φ h = 1. (a) shows the full perturbative solution on the unit disk,while (b) zooms in the interpolating region. The unperturbed interpolating geometry is given by a separationbetween negative (blue) and positive (yellow) curvature at ρ = 0 (dashed, red circle). The perturbation for κh = 2 .
3, generates the new curve with φ = 0 that is given by the black line in the plot. The black dashed lineclose to the boundary in Fig. (a) shows boundary conditions given by φ b = 30, h = 900. still need to fix the boundary conditions. Given h and φ b , then δρ ( u ) and δτ ( u ) are determinedby (3.20). In figure 2 we display the solution with h = 900, φ b = 30.More generally, if we take ∂ M to lie at ρ c ≈ π/ τ = u , the condition that φ b remains12ositive and large along ∂ M enforces: | φ h | (cid:38) κ h m m (4 m + 1) (cid:18) m e − πm (4 m − (cid:19) . (3.23)Relaxing the above condition leads to solutions where regions of negative and positive curvaturemight separate, as shown in figure 3. If we impose a Dirichlet boundary condition for which φ b is everywhere positive along ∂ M , solutions such as the one in figure 3 (b) are no longer allowed.Perhaps this suggests for the Dirichlet problem with AdS asymptotia, one can entirely removethe dS region in the interior by turning on strong enough matter sources at ∂ M . It would beinteresting to futher explore this at the non-linearised level. (a) κh = 8 . κh = 8 . Figure 3:
Perturbative solution with m = − φ h = 1 on the unit disk. Yellow regions indicate positive curvaturewhile blue ones, negative. The interpolating region with φ = 0 is the solid curve between them. The red dashedcircle shows the interpolating region of the unperturbed geometry. As the strength of the perturbation increases,the positive curvature regions grow until negative curvature regions become disconnected. We would like to end this section with some remarks on the general non-linear problem. Again,we will consider the limit where there is a sharp transition from positive to negative curvature.The geometric problem consists of gluing a constant negative curvature geometry to a constantpositive curvature geometry across some curve S . In the conformal gauge, constant curvaturemetrics in two-dimensions are given by solutions to the Liouville equation. To express the spaceof solutions, it is convenient to introduce a complex coordinate z = e ρ + iτ . A large class ofpositive constant curvature geometries are: ds = 4 f (cid:48) + ( z ) g (cid:48) + (¯ z )(1 + f + ( z ) g + (¯ z )) dz d ¯ z , (3.24)13here f + ( z ) and g + (¯ z ) are meromorphic and anti-meromorphic functions giving rise to a realgeometry. For example the round metric on the two-sphere has f + ( z ) = z and g + (¯ z ) = ¯ z .Negative constant curvature geometries are: ds = 4 f (cid:48)− ( z ) g (cid:48)− (¯ z )(1 − f − ( z ) g − (¯ z )) dz d ¯ z . (3.25)The standard hyperbolic geometry has f − ( z ) = z and g − (¯ z ) = ¯ z . The hyperbolic cylinder isgiven by f − ( z ) = e i log z and g − (¯ z ) = e i log ¯ z . Poles in f − ( z ) and g − (¯ z ) translate to conical defectsin the two-dimensional geometry.We wish to glue the two geometries together along an arbitrary closed curve S , such thatboth the metric and extrinsic curvature matches along the curve. The equation for S will besome non-holomorphic function s ( z, ¯ z ) = 0. Thus, f + ( z ) and g + (¯ z ) should be analytic in U S .Throughout the region U S within S we require a smooth constant positive curvature geometry.By the Riemann mapping theorem, we can map U S to the open disk with circular boundary.We can then fill the interior of the disk with the standard metric on the two-sphere and glueit to the appropriate hyperbolic geometry. Finding the explicit form of the map is generallycomplicated. Upon performing the map, the boundary values of the fields will transform. It isimportant to understand how the boundary values of the fields transform since they are to beinterpreted as sources for the operators in a putative dual quantum mechanics. In this section we consider the effective theory of the soft mode arising close to the AdS boundary. We emphasise that the AdS clock near the boundary of the unperturbed, or slightlyperturbed, interpolating geometry is the isometric direction of the hyperbolic cylinder. Asmentioned earlier, this results in a sign difference for the boundary action (2.18) as comparedto that of the hyperbolic disk. Let us take h and φ b to be u -independent, with φ b positive and the two scaling as {√ h, φ b } =Λ { , ˜ φ b } in the large-Λ limit. We take the proper size of the boundary circle to be Λ ˜ β such that u ∼ u + ˜ β . The boundary action for the interpolating geometry (2.13) is found by calculatingthe extrinsic curvature for the curve C = { τ ( u ) , ρ ( u ) } .If we remain off-shell by leaving τ ( u ) unfixed, we can calculate the boundary action for theinterpolating geometry: S bdy = ˜ φ b κ (cid:90) du (cid:18)
12 ( ∂ u τ ( u )) − Sch [ τ ( u ) , u ] (cid:19) . (4.1)14n deriving the above action, we have related ρ ( u ) to τ ( u ) via (2.17):Λ = ( ∂ u ρ ( u )) + ( ∂ u τ ( u )) cos ρ ( u ) . (4.2)In the large Λ-limit, we have Λ cos ρ ( u ) ≈ ∂ u τ ( u ) such that ρ ( u ) must be parameterically near ρ = π/
2. This forces the boundary curve C to live parametrically close to the AdS boundary.For τ ( u ) non-compact, the theory (4.1) is invariant under the SL (2 , R ) transformation: σ ( u ) → a σ ( u ) + bc σ ( u ) + d , ad − bc = 1 , (4.3)with a, b, c, d real, and σ ( u ) = tanh τ ( u ) /
2. Due to the fact that τ ( u ) is compact, the SL (2 , R )invariance is broken to a U (1) subgroup corresponding to shifts in τ ( u ).The equations of motion stemming from (4.1) are: ∂ u τ = ∂ u (cid:18) ∂ u τ ∂ u (cid:18) ∂ u τ∂ u τ (cid:19)(cid:19) . (4.4)One solution to the above equations is τ ( u ) = 2 πu/ ˜ β , such that u ∼ u + ˜ β . In [22, 23], ˜ β isinterpreted as the temperature of a putative ultraviolet system. An example where this occursis the SYK model, for which u becomes the clock of the dual quantum mechanical fermions. Wewill keep ˜ β as a parameter, and occasionally interpret it as a temperature. In the case whereour bulk theory has an additional scale, such as the field range of the interpolating region in thedilaton potential, the theory already contains a bulk tunable parameter that can be viewed as atemperature. This is in sharp contrast to linear dilaton potentials for which the bulk geometryis entirely fixed at the classical level.The on-shell boundary action for the τ ( u ) = 2 πu/ ˜ β saddle becomes: S bdy,cl = − sign φ h ˜ φ b ˜ βκ . (4.5)If we interpret ˜ β as an inverse temperature, we see that (4.5) is linear in the temperature. For κ > β is negative for the interpolating solution,which as we recall has φ h < Nevertheless, as we shall soon see, the κ > τ ( u ).For the remainder of the solution space, we can map (4.4) to the equations of motion ofLiouville quantum mechanics [25]: ∂ u ξ ( u ) = e ξ ( u ) , (4.6) Note that for a class of generalised potentials that is analysed in appendix D, it is possible to obtain interpo-lating geometries with κ > ∂ u τ ( u ) = e ξ ( u ) . The solutions to (4.6) are: e ξ ( u ) = a sec au . (4.7)Due to the condition ξ ( u ) = ξ ( u + ˜ β ), we require a = nπ/ ˜ β with n ∈ Z . All these saddles havea divergence at certain values of u . Evaluating the on-shell action for the solutions (4.7) revealsa divergent answer. The action in terms of ξ is given by: S ξ = ˜ φ b κ (cid:90) du (cid:20) ( ∂ u ξ ) + 12 e ξ ( u ) (cid:21) . (4.8)As shown in [25], the path integral measure is flat in ξ ( u ) with the zero mode removed. Case (i): κ > κ > φ h > β = 2 π , and expanding the soft action around the saddle as τ ( u ) = u + δτ ( u )we obtain the following perturbative action for the Schwarzian piece: S fluct = ˜ φ b κ (cid:90) du (cid:16)(cid:0) ∂ u δτ ( u ) (cid:1) + ( ∂ u δτ ( u )) (cid:17) . (4.9)Interestingly, though the interpolating geometry is a sub-dominant saddle, the action for fluc-tuations (4.9) of its soft boundary mode is Gaussian suppressed, allowing for a perturbativeanalysis of δτ ( u ) in the sub-dominant saddle. We will analyse this shortly. Case (ii): κ < κ <
0. In this case, the boundary action for the interpolating geometry(4.1) dominates over that of the hyperbolic disk. This is consistent with the analysis of [3].However, care must be taken with the perturbative analysis of fluctuations. Indeed, for κ < κ <
0. For instance, we could consider a mixed Neumann-Dirichletensemble where π φ = − K and h are fixed, rather than the standard Dirichlet problem we havebeen considering so far. Fixing the extrinsic curvature in Euclidean quantum gravity has beenconsidered in other contexts also (see for example [26, 27]). In going to this ensemble, wemust allow φ b ( u ) to fluctuate, and hence, it is no longer guaranteed that our geometry will beasymptotically AdS . Another possibility is that we must view the κ < κ . One possible lesson is that for κ <
0, we should no longerinterpret ˜ β as a temperature.An example of a well defined theory where one encounters a term in the action corresponding16o a Schwarzian action with negative coefficient arises if one considers coupling the soft modeaction to a non-trivial geometry ˜ h ( u ). In order to covariantise this action it is useful to write:Sch[ τ ( u ) , u ] = F [ A, u ] ≡ ∂ u A u − A u A u , A = ∂ u log ∂ u τ ( u ) du . (4.10)Notice that A transforms as a connection under changes of coordinates u → f ( u ). Providedan affine connection Γ, it is now easy to covariantise this action in terms of A = A − Γ. In amanifold with a metric ˜ h ( u ), Γ is given byΓ = ˜ h − / ∂ u ˜ h / du . (4.11)The associated covariant Schwarzian action becomes:Sch ˜ h [ τ ( u ) , u ] = ˜ h − F [ A , u ] ≡ ˜ h − (cid:20) D u A u − A u A u (cid:21) (4.12)where D u ≡ ∂ u − Γ u is the usual covariant derivative and we have included an inverse metric˜ h − to make the Schwarzian a scalar density. Now something interesting happens as we expandout this expression: F [ A , u ] = ∂ u A u − A u A u − ∂ u Γ u + 12 Γ u Γ u = F [ A, u ] − F [Γ , u ] (4.13)Putting all this together we can write a fully covariant action as: S [ τ, Γ , ˜ h ] = − κ (cid:90) du (cid:112) ˜ h ˜ φ b ˜ h − ( F [ A, u ] − F [Γ , u ]) . (4.14)It is clear from the above expression that regardless of the original sign of κ we obtain thedifference of two identical actions with different signs. These can of course differ on the actualvariables of integration in the path integral. As expected in first order formalisms of gravity, ifwe take Γ to be an independent variable we obtain (4.11) as the equation of motion. We willelaborate on these issues in future work. We now consider the effect of a massless free scalar, with boundary value χ ( u ), to the physicsof the soft mode. The contribution to on-shell action from the matter theory (3.1) is given by: S χ = − (cid:90) du du (cid:32) τ (cid:48) ( u ) τ (cid:48) ( u )sin τ ( u ) − τ ( u )2 (cid:33) ¯ χ ( τ ( u )) χ ( τ ( u )) . (5.1) Incidentally, A also transforms as a gauge connection for the conformal symmetry action on target space τ → σ ( τ ) and F is a weight 2 tensor invariant under the global SL (2 , R ) sub-algebra of this local group.
17n the above expression, we have assumed that the boundary curve C = { τ ( u ) , ρ ( u ) } alwaysremains near the asymptotic AdS boundary. In addition to (5.1), the total on-shell actionalso contains a contribution from the Schwarzian action (2.18) and an interior contribution S int given in (2.21). Thus, the generating function for boundary correlations of the scalar χ contains an additional contribution when compared to the pure Jackiw-Teitelboim theory witha linear dilaton potential. There, the geometry is always pure AdS and the whole problemcan be mapped to a calculation at the boundary [22, 23, 24]. Here, the soft boundary physicscarries an imprint from the bulk interpolating region. This can be seen rather directly from ourperturbative equations (3.6) and (3.20). The contribution to (3.20) coming from the integralover the stress-tensor is equivalent to the equation of motion stemming from the boundarytheory (4.1) plus (5.1). However, there is also a contribution to (3.20) from the second integralin (3.6). This term is due to the interpolating region.It is instructive to obtain an equation for the off-shell curve parameter τ ( u ) ≈ u + δτ ( u ).This can be done by considering the ADM mass of the theory [21] along the curve C : M ADM = √ h ( − n a ∂ a φ + φ b ) . (5.2)We can evaluate M ADM either using the T ρρ or T ρτ equations of motion. Given a slightlyperturbed metric, ds = e ω ( ρ ) (1 + 2 δω ( ρ, τ )) ( dρ + dτ ) , (5.3)we evaluate (5.2) along the curve C = { u + δτ ( u ) , ρ c + δρ ( u ) } , where δρ ( u ) is expressed in termsof δτ ( u ) and δω ( ρ c , u ) by fixing the induced metric h = e ω ( ρ c ) of (5.3). Working to linear orderin δτ ( u ) and δω ( ρ c , u ), and using that φ b = − φ h tan ρ c , we obtain the following expression:˜ φ b (cid:0) ∂ u − ∂ u (cid:1) δτ ( u ) = − κ T τρ ( π/ , u ) + 3 h ˜ φ b ∂ u δω ( ρ c , u ) . (5.4)We also work in the limit ρ c → π/
2, which is well defined so long as δω ( ρ c , u ) goes as O (( ρ c − π/ ) as we approach ρ c = π/
2. The left hand side of (5.4) is given by linearising the equationof motion of the Schwarzian theory (4.1). The first term on the right hand side of (5.4) comesfrom varying (5.1) with respect to τ ( u ). The last term in (5.4) is due to the slightly perturbedWeyl factor and encodes the information of the interior curve S . Upon inserting the on-shellvalue for δω ( ρ, τ ), the solutions of (5.4) are the same as our linear solutions (3.20). Since δω ( ρ, τ )is fixed by T ρτ ( ρ, τ ) on-shell, we end up an equation for δτ ( u ) that is completely determined bythe boundary data χ ( u ). Recall that in momentum space, the linearised solution for δω nearthe boundary can be obtained by expanding (3.12) near ρ = π/ δ ˜ ω q ≡ lim ρ → π/ δω q ( ρ )( ρ − π/ = − i κ ˜ φ b q e − π | q | (1 − q ) T ρτ (0 , q ) . (5.5)18n position space, this becomes a convolution: δ ˜ ω ( u ) = − κ ˜ φ b ∂ u (cid:90) dw G ( u − w ) T ρτ (0 , w ) , G ( u ) ≡ (cid:88) q (cid:54) = {± , } e − π | q | (1 − q ) e iqu . (5.6)Explicitly: G ( u ) = 12 Re (cid:104) sinh (cid:16) π − iu (cid:17) − cosh (cid:16) π − iu (cid:17) − (cid:16) − e − π + iu (cid:17) sinh (cid:16) π − iu (cid:17) − (cid:105) . (5.7)Notice that G ( u ) is an oscillatory function in u . It is of interest to calculate the boundary four-point function for the matter fields which weexpress as χ = ( χ + iχ ) / √
2, with χ and χ real. We will set ˜ β = 2 π unless otherwise specified.At tree-level, we must compute the on-shell Euclidean action for the linearized perturbations δτ and χ . It is given by: S bdy [ δτ ( u ) , χ ( u )] = S fluct [ δτ ( u )] − (cid:90) du (cid:32) T ρτ ( π/ , u ) − φ b κ ∂ u δ ˜ ω ( u ) (cid:33) δτ ( u ) , (5.8)with S fluct [ δτ ( u )] containing the fluctuations of the Schwarzian given in (4.9) and δ ˜ ω ( u ) givenin (5.6). Since T ρτ and δ ˜ ω are quadratic in χ i ( u ), we can read off a cubic interaction between χ i ( u ) and δτ ( u ) directly from (5.8). Moreover, given that: T ρτ ( π/ , u ) = − (cid:32) ∂ u χ ( u ) (cid:90) dw ¯ χ ( w )sin u − w + h.c. (cid:33) , (5.9)it follows that (5.8) is non-local in u , when viewed as a functional of χ ( u ) and δτ ( u ). Similarly, T ρτ (0 , u ) = − (cid:32) ∂ u ˜ χ ( u ) (cid:90) dw ¯˜ χ ( w )sin u − w + h.c. (cid:33) , (5.10)where we have further defined:˜ χ ( u ) ≡ (cid:90) dw sinh π/ χ ( w )cosh π/ − cos( u − w ) . (5.11)The reason for taking a convolution of the boundary matter source χ ( u ) is that there is arelative factor of e −| q | π/ for the Fourier modes of χ ( ρ, τ ) when comparing ρ = 0 to ρ = π/ χ ( u ). As a result we can rewrite the19econd term in (5.8) as: − (cid:90) du du sin u − u [ χ ( u ) ¯ χ ( u ) B ( u , u ) + ˜ χ ( u ) ¯˜ χ ( u ) E ( u , u ) + h.c. ] , (5.12)where we have defined: E ( u , u ) ≡ (cid:32) ∂ u δ T ( u ) + ∂ u δ T ( u ) − δ T ( u ) − δ T ( u )tan u − u (cid:33) , (5.13) B ( u , u ) ≡ (cid:32) ∂ u δτ ( u ) + ∂ u δτ ( u ) − δτ ( u ) − δτ ( u )tan u − u (cid:33) , (5.14)and δ T ( w ) ≡ (cid:82) du G ( w − u ) ∂ u δτ ( u ), with G ( u ) given in (5.6).From the action of small soft mode fluctuations (4.9) with κ >
0, we can extract the propa-gator of the mode δτ ( u ) which in Fourier space reads: (cid:104) δτ m δτ n (cid:105) = κ ˜ φ b δ m + n m + m . (5.15)Notice that m = 0 is a zero mode that must be excised from the configuration space. Goingback to position space, the propagator becomes: (cid:104) δτ ( u ) δτ (0) (cid:105) = κ ˜ φ b (cid:18) ( | u | − π ) − π sinh π cosh( | u | − π ) + 1 − π (cid:19) . (5.16)We display the δτ ( u ) propagator in figure 4. This result is valid for u ∈ [ − π, π ) and periodicallycontinued for other u . To obtain the contribution to the χ i four-point function from (5.8), we - - - - - ϕ ˜ b κ - < δ τ ( u ) δ τ ( ) > Figure 4:
The propagator in equation (5.16) as a function of u is shown in green. In red, we plot the propagatorfor the perturbative Schwarzian action resulting near the boundary of the hyperbolic disk. must integrate out δτ ( u ). To leading order, this is given by calculating the tree-level exchangediagram of δτ ( u ). 20 .2 Four-point functions We have now collected all the ingredients necessary to calculate the tree-level, connected, four-point function. It is given by: (cid:104) χ ( u ) χ ( u ) χ ( u ) χ ( u ) (cid:105) c = 116 (cid:42)(cid:32) B ( u , u )sin u − u + ˜ E ( u , u ) (cid:33) (cid:32) B ( u , u )sin u − u + ˜ E ( u , u ) (cid:33)(cid:43) , (5.17)where we have defined:˜ E ( u , u ) ≡ (cid:90) dw dw sinh π/ π/ − cos( u − w ) E ( w , w )sin w − w sinh π/ π/ − cos( w − u ) . (5.18)The calculations are somewhat lengthy, but straightforward. In appendix C we give somedetails of the calculations. The time ordering u < u < u < u is the one relevant for out-of-time-ordered correlations [28] upon suitable analytic continuation. This can be done in manydifferent ways. A particularly simple configuration is given by placing the four operators atequally-spaced points along the thermal circle, and then evolving only the two diametricallyopposed ones along the real-time axis: F ( t ) ≡ (cid:104) χ ( π/ χ ( it ) χ ( − π/ χ ( − π + it ) (cid:105) c = Tr ˆ ρ [ˆ y ˆ χ ˆ y ˆ χ ( t ) ˆ y ˆ χ ˆ y ˆ χ ( t )] , (5.19)where ˆ ρ ≡ e − ˜ β ˆ H and ˆ y ≡ ˆ ρ / . It has been argued that quantum systems in thermal equilibriumwith a large number of degrees of freedom, N , and a parametrically large separation betweendissipation and scrambling time obey [29]: F ( t ) = f − f N exp( λ L t ) , with λ L ≤ π ˜ β . (5.20)Here, λ L is the Lyapunov exponent, f and f are positive order-one constants (independent of N , ˜ β , t ), and ˜ β (cid:28) t (cid:28) ˜ β log N . It turns out that black holes in Einstein gravity saturate thisbound. The AdS black hole with running dilaton also saturates the chaos bound.We are interested in the four-point function F ( t ) for the interpolating geometry. The im-portant point is that the δτ ( u ) propagator (5.16) is built from hyperbolic functions, which uponanalytic continuation become oscillatory. This already implies that the piece of F ( t ) stemmingpurely from the B ( u, w ) contribution will oscillate in t . Upon reinstating ˜ β , the frequency ofoscillation is 2 π/ ˜ β , which is the same as the Lyapunov coefficient. Analytically: (cid:68) B ( π/ , − π/ B (2 πit/ ˜ β, − π + 2 πit/ ˜ β ) (cid:69) c = 2 κ ˜ β ˜ φ b (cid:18) π csch π πt ˜ β − (cid:19) . (5.21)The piece of F ( t ) stemming from the E ( u, w ) contribution contains trigonometric functions bothin (5.7), as well as in the convolution (5.10). Recalling that we must calculate correlators of21 E ( u, w ) in (5.18), the only piece that is subject to analytic continuation is the cos( u − w ) in thedenominator of (5.18). All the relevant integrals appearing in the convolutions are finite andwell defined. Consequently, upon setting u = π/ u = − π/ u = it and u = − π + it , thecontribution from ˜ E ( u, v ) will either oscillate or decay exponentially at large t . In appendix Cwe provide numerical evidence of this behaviour.In conclusion, the boundary out-of-time-ordered four-point function of χ ( u ) for the interpo-lating geometry does not display the exponential Lyapunov behaviour (5.20) observed for blackholes in AdS. Instead, we observe oscillatory behaviour in Lorentzian time. This is an interestingdistinction between de Sitter-like and black-hole-like horizons whose microscopic interpretationwe will discuss in future work. Having discussed several aspects of the Euclidean problem, we move on to some features ofperturbations in Lorentzian signature. Specifically, we will consider the effect of a null energeticpulse travelling from the AdS boundary through the dS horizon. It is convenient to work in light-cone coordinates, such that our gauge-fixed background is givenby: ds = e ω ( x + , x − ) dx + dx − . (6.1)The relation to the Schwarzschild like coordinates (2.11) in the x + > x − > ± log x ± = t ± (cid:90) ρ drN ( r ) . (6.2)The Ricci scalar is given by R = − e − ω ∂ + ∂ − ω . The non-vanishing Christoffel symbols are:Γ +++ = 2 ∂ + ω , Γ −−− = 2 ∂ − ω . (6.3)The background solution is given as follows. The dS region is covered by the range − 1. The metric and dilaton take the form: e ω ( x + ,x − ) = 4(1 + x + x − ) , ¯ φ ( x + , x − ) = | φ h | (cid:18) x + x − − x + x − + 1 (cid:19) . (6.4)The dS horizon resides at x + x − = 0. The future boundary in the interior of the dS horizonresides at x + x − = − 1. The transition region occurs near x + x − = 1. The AdS region is covered22y 1 < x + x − < e π and described by: e ω ( x + ,x − ) = 1 x + x − sec log x + x − , ¯ φ ( x + , x − ) = | φ h | tan log x + x − . (6.5)The AdS boundary resides at x + x − = e π . The matter content we consider will be a free massless scalar described by the action: S χ = − (cid:90) d x √− gg ab ∂ a χ∂ b χ . (6.6)The solutions to the massless wave-equation are given by a sum of left and right moving waves: χ ( x + , x − ) = χ + ( x + ) + χ − ( x − ) . (6.7)In Lorentzian space we can turn on a purely chiral excitation. This would correspond to acomplex, holomorphic solution in Euclidean space. Let us consider the case χ − ( x − ) = 0. Theequations of motion are (see for example [30]): e ω ( x + ,x − ) ∂ + e − ω ( x + ,x − ) ∂ + φ = − κ T χ ++ ( x + ) , (6.8) e ω ( x + ,x − ) ∂ − e − ω ( x + ,x − ) ∂ − φ = 0 , (6.9)4 ∂ + ∂ − φ − e ω ( x + ,x − ) V ( φ ) = 0 . (6.10)Let us further consider the case where the geometry is transitioning sharply from AdS todS . The pulse emanates from the AdS boundary and we can solve for φ by integrating the++-equation of motion from the past boundary to the interpolating region. In this region wefind: ∂ + φ ( x + , x − ) = ∂ + ¯ φ ( x + , x − ) − κ e ω ( x + ,x − ) (cid:90) x + −∞ d ˜ x + e − ω (˜ x + ,x − ) T χ ++ (˜ x + ) . (6.11)We can consider the shockwave limit where T χ ++ ( x + ) = αδ ( x + ). The deviation from the back-ground value of the dilaton is: δφ ( x + , x − ) = − ακ e − ω (0 ,x − ) Θ( x + ) (cid:90) x + du e ω ( u,x − ) = − ακ (cid:18) x + x + x − + 1 (cid:19) Θ( x + ) . (6.12)It is also useful to consider the following coordinate transformation:˜ x + = x + Θ( − x + ) + 2 x + ακ x + Θ( x + ) , ˜ x − = x − − ακ Θ( x + ) / , (6.13)23pplied to the region where φ ( x + , x − ) < α → α/ | φ h | . This mapsthe metric and dilaton in the region φ ( x + , x − ) < ds = 4 d ˜ x + d ˜ x − (1 + ˜ x + ˜ x − ) + 2 ακδ (˜ x + )( d ˜ x + ) , φ (˜ x + , ˜ x − ) = | φ h | (cid:18) ˜ x + ˜ x − − x + ˜ x − + 1 (cid:19) . (6.14)In this coordinate system, the dilaton exhibits no jump in the dS region, whereas the metrichas a characteristic δ -function singularity along the shockwave. An analogous procedure can bedone for the remaining AdS region.If the integral (6.12) is such that ¯ φ + δφ becomes zero for some x + , then we must make sureto switch from the dS to the AdS background geometry. This defines a curve in the region x + > x + (cid:0) x − − ακ (cid:1) = 1 . (6.15)As expected, when α = 0 this becomes x + x − = 1. We schematically depict the geometries infigure 5. ̂ x + x − = e π x + ( x − − κ α ) =1 x + x − (a) κ α < κ = 0 x + x − =0 x + x − = e π x + x − =1 x + x − = e π x + x − =0 x + x − =1 x + x − (b) κ α = 0 ̂ x + x − = e π x + ( x − − κ α ) =1 x + x − (c) κ α > Figure 5: Penrose diagrams for interpolating geometries after a shockwave perturbation. The induced metric h ++ along the curve is given by: ds = − ακ x + ) (cid:18) dx + x + (cid:19) . (6.16)It is also useful to compute the extrinsic curvature along the curve (6.15). If we parameterizethe curve as ( x + ( u ) , x − ( u )), the tangent vector is given by: T µ ( u ) = ( ∂ u x + ( u ) , ∂ u x − ( u )) . (6.17)The normal vector n µ obeys T µ n µ = 0 and n µ n µ = 1. Explicitly: n µ ( u ) = 1 | x + ( u ) x − ( u ) | (cid:32)(cid:115) − ∂ u x − ( u ) ∂ u x + ( u ) , (cid:115) − ∂ u x + ( u ) ∂ u x − ( u ) (cid:33) . (6.18)24he extrinsic curvature is given by: K uu = T µ T ν ∇ µ n ν . (6.19)Evaluating it on the curve (6.15), we find that K uu = 0. The normal derivative of the scalaralong (6.15): n µ ∂ µ φ = 1 . (6.20)To continue the solution to the AdS region, we must smoothly glue a constant negative curvaturemetric across the curve (6.15).It is convenient to consider a chiral coordinate transformation:log ˆ x + = (cid:90) x + ακu ) duu = log (2 + ακ ) x + ακx + , (6.21)such that the induced metric on the curve takes the simpler form ds = − ( d ˆ x + / ˆ x + ) . Therange of ˆ x + is the positive real axis. Thus, we can smoothly glue (up to first derivatives) halfof global AdS with coordinates: e ω (ˆ x + ,x − ) = 1ˆ x + x − sec log ˆ x + x − , (6.22)along the curve ˆ x + x − = 1. The normal direction then becomes the standard radial direction inglobal AdS . The dilaton is given by (6.5) with x + replaced by ˆ x + . Notice that for matter obeying the null energy condition, i.e. α > 0, the shift in the dilatonvalue due to the stress-tensor in (6.11) depends on the sign of the κ . For κ > κ < | φ | increases (decreases) in the region x + > | φ | increasesacross the shockwave, particles going across the shockwave will experience a Shapiro time-delay.This situation is similar to the case of black holes [31] – a flux of positive null energy into thehorizon causes a Shapiro time delay.What is novel is the case κ < | φ | decreases across the shockwave, causing aShapiro time advance. For ordinary black holes, this can only happen if we send in matterthat violates the null energy condition [32]. But for a de Sitter type horizon this behaviour isallowed without any violation of the null energy condition. In appendix B we connect the κ < , for which the dilaton is related to the size ofthe celestial circle. A more general statement is due to a theorem of Gao and Wald [33] whichstates that perturbations obeying the null energy condition make the de Sitter Penrose diagrammore vertical, hence allowing access to otherwise causally disconnected regions of space. Whatis often called the horizon re-entry of super-horizon modes in inflationary cosmology is closely25elated to these phenomena. Releasing energy from the AdS boundary We would like to make a final remark about the clock appearing in the Lorentzian interpolatinggeometry: ds = cos − ρ (cid:0) − dt + dρ (cid:1) , ρ ∈ ( (cid:15), π/ , cosh − ρ (cid:0) − dt + dρ (cid:1) , ρ ∈ ( −∞ , − (cid:15) ) . (6.23)Imagine that we release a small (in Planck units) energy (cid:15) at some early time t i (cid:28) − boundary of the Lorentzian interpolating geometry. Near the boundary, the timelikeisometry is approximately that of the global AdS clock. It is also the one that generates time-translations of the inertial clock at ρ = 0. Thus, the energy in a local frame at ρ = 0 is alsoof order (cid:15) . The energy (cid:15) will consequently enter the dS static patch region at the origin, andeventually, near the dS horizon at ρ = −∞ , the particle will experience a Rindler geometrywhere the local frame energy will be enhanced. We are interested in how the energy (cid:15) dependson t i , when measured in the local frame.It is useful to recall the transformation between the static dS coordinates ( t, ρ ) and theglobal dS coordinates ( T, ϕ ):cosh T = 1 + sech ρ sinh t , sin ϕ = tanh ρ ρ sinh t , (6.24)where we have set (cid:96) = 1. Translations in t comprise a dS isometry. Near the dS horizon,sech ρ ≈ δ with δ (cid:28) t (cid:29) 1, the global and static patch clocks are related in the usualRindler sense T ≈ δ sinh t . Unlike the relation between the Rindler and Minkowski clock,however, the exponential relation between the global and static dS clocks is true only in theregime δe t (cid:28) 1. In the limit δe t (cid:29) 1, the clocks are linearly related and hence there is noexponential effect. This is in stark contrast to what happens when releasing some energy in astandard AdS black hole with temperature β . There, the Rindler effect persists for β (cid:28) t (cid:28) β log S BH and the local frame energy acquires a factor of ∼ e πt/β near the horizon such that theconnection to the shockwave and the saturation of the Lyapunov exponent follows immediately. We end with some general remarks on our results and how they may tie into a holographicpicture. Since our solutions have an AdS like boundary, the natural candidates for holographicduals will be (0 + 1)-dimensional, i.e. quantum mechanical theories. As a reminder the global dS metric is ds = − dT + cosh T dϕ with ϕ ∼ ϕ + 2 π . S horizon and chaos The chaotic nature of AdS black holes has been heavily explored in recent literature. On theother hand, dS horizons have been less explored from a more modern, holographic perspective(see however [16, 34, 35, 36, 37, 38, 39, 40, 41, 42]). In constructing interpolating solutionsbetween AdS and dS we have a potential framework to discuss these questions using thestandard tools of AdS/CFT, such as correlators at the AdS boundary. Using the Hartle-Hawking construction we can build a global state from the Euclidean saddle. To do so, we cutthe disk in half such that a state is prepared along the t = 0 spatial slice. In the absence of anyperturbations, the Lorentzian geometry becomes a two-sided interpolating geometry which issmooth across the dS horizon. Assuming time-reversal invariance, this solution can be continuedall the way to the past. For standard black holes, this state is the thermofield-double state whichis built purely out of entangled, non-interacting CFTs. Perturbing this state for black holes,while obeying the null-energy condition, leads to the two sides becoming less connected, andcan be viewed as a geometrization of decorrleation due to chaos [43].The negative κ shockwave solution (6.14), which behaves qualitatively similar to shockwavesin pure dS, indicates that perturbing the two-sided interpolating geometry leads to more corre-lation rather than less [3, 44]. Signals may now arrive from the previously causally inaccessibleregion, something that can only be achieved in the standard black hole case by turning on aninteraction between the two CFTs [32]. From the perspective of a single sided interpolating ge-ometry, perhaps the mechanism is analogous to that of [46], which might suggest an interestingrole of state-dependent operators in the context of dS. Perhaps an analogy can be drawn toglassy states which are statically indistinguishable from liquid states but dynamically very dif-ferent. We view these phenomena as features rather than bugs. They are part of the definitionof static patch geometry of dS.In section 5, we computed the out-of-time ordered correlator for these interpolating geome-tries, in complete analogy to computations in the AdS case [22, 23, 24]. The result is surprising,showing an oscillatory correlator (rather than exponential with maximal Lyapunov exponent).It is worth noting, however, that the propagator G ( u ) in equation (5.6) contains a piece thatresembles the Schwarzian propagator that appears at the boundary of the AdS black hole. Thismight suggest that the maximally chaotic behaviour of the Rindler horizon may be encoded inmore subtle correlators. It may also be of interest to study this question for the more generalbackground presented in Appendix D. In that case, it seems that γ ∈ [ − , 1] acts as a parametertuning the behaviour of the out-of-time ordered correlator: for γ > 0, the result is oscillatory;for γ < 0, the correlator is exponential with maximal chaos at γ = − 1; and, at γ = 0, there isan intermediate behaviour where it behaves as a power law, i.e., F ( t ) ∼ t . These features mayserve as a guiding principle for the construction of dual microscopic SYK-type models.27 ole of null-energy violation More generally, given the current reassessment of the role of null-energy in the bulk, it may beinteresting to reconsider previous attempts to construct dS in a higher dimensional AdS [14, 15].There, the main stumbling block was that the Raychaudhuri equation plus null-energy obeyingmatter forbade any dS region to reside in a causally accessible part of the geometry. In view ofrecent developments [32, 45], perhaps one can construct an accessible dS region by turning onweak interactions between the two sides. dS fragmentation In our analysis of anisotropic perturbations we observed the tendency of the region separatingpositive and negative curvature to separate. In other words, there was only a restricted regimein the space of sources for which one observed a piece of dS in the interior.More generally, we might imagine that the interior Euclidean dS region could fragment intoseveral disconnected regions. Perhaps we should interpret these observations as indications thatthe static patch cannot be arbitrarily sharply defined in and of itself. Holography of a feature Though our discussion was specific to the interpolating geometries of [3], there is no reasonwhy it could not be applied to more general cases. Understanding the interplay of features inthe interior of an asymptotically AdS spacetime and the boundary soft modes seems like aninteresting general question, and may pertain to broader issues like bulk locality in AdS/CFT. Relation to dS/CFT? Finally, it is natural to ask how our approach might be related to the standard dS/CFT picture[47, 48, 49, 50]. There, the dual theory resides at the future boundary. From the perspectiveconsidered in this paper, the future boundary resides within the horizon of the interpolatinggeometry and thus, one would have to reconstruct it from the boundary AdS degrees of freedom.It would seem that there are far fewer degrees of freedom at the AdS boundary to account forthose at the future boundary. Perhaps this is an indication that there are unexpected relationsamong degrees of freedom at the future boundary, as was observed in higher spin models [51]. Acknowledgements We gratefully acknowledge discussions with Frederik Denef, Sean Hartnoll, Ben Freivogel, JuanMaldacena, Rob Myers, Daniel Roberts, Douglas Stanford, Kyriakos Papadodimas, Evita Ver-heijden, Herman Verlinde, and Erik Verlinde. D.A.’s research is partially funded by the RoyalSociety under the grant “The Atoms of a deSitter Universe” and the ∆ ITP. The work of D.A.G.28nd D.M.H. is part of the ∆ ITP consortium, a program of the NWO that is funded by theDutch Ministry of Education, Culture and Science (OCW). D.A. would like to thank ICTS,Bengaluru for their kind hospitality during completion of this work. D.A.G. would like to thankthe Galileo Galilei Institute for Theoretical Physics for hospitality and ACRI and INFN forpartial support during the completion of this work. This project has received funding fromthe European Research Council (ERC) under the European Unions Horizon 2020 research andinnovation programme (grant agreement No 715656) A Jackiw-Teitelboim gravity with κ < In this appendix we discuss some solutions to Jackiw-Teitelboim gravity with action: S = S top − κ (cid:90) d x √ gφ ( R + 2) − κ (cid:90) ∂ M √ h φ K , (A.1)where S top is given by (2.2). The metric is fixed to be AdS due to the dilaton equations ofmotion. The smooth solution with single boundary is: ds = dρ + dτ sinh ρ , τ ∼ τ + 2 π , ρ ≥ . (A.2)with τ -independent dilaton: φ ( ρ ) = φ h coth ρ , φ h ∈ R . (A.3)There is also a static solution on the Euclidean cylinder: ds = dρ + dτ cos ρ , φ ( ρ ) = φ h tan ρ , (A.4)where now ρ ∈ ( − π, π ). One can periodically identify τ with any periodicity in (A.4) withoutspoiling the smoothness of the solution.Since the sign of φ h is not fixed, the theory admits solutions with an increasingly negativedilaton near the boundary. As discussed in the main text, at the semi-classical level what werequire is that ( φ + φ/κ ) > 0. The negative φ solution is equivalent to a positive φ solutionwith κ → − κ . If we view the theory as a dimensional reduction of Einstein-Maxwell theory, thesolutions correspond to near-extremal Reissner-Nordstrom black holes. The solutions with neg-ative dilaton correspond to a static, non-asymptotically flat solution with a timelike singularityat the origin which is surrounded by a horizon. Though singular, the ‘wrong sign’ solutions aresupported by standard null-energy preserving matter. The boundary dynamics will also be theSchwarzian, at least in the regime φ (cid:29) ( φ b /κ ) (cid:29) 1. However, the Schwarzian will have theopposite sign corresponding to a ‘negative’ specific heat. From the higher dimensional perspec-tive, this sign indicates that the size of the celestial sphere shrinks toward the UV part of the29eometry where time flows the fastest. At a qualitative level, this is also what happens for thestatic patch of de Sitter space. B Shockwaves in dS In this appendix we discuss a shockwave solution in dS and connect it to the analysis in section6. The following Lorentzian shockwave geometry: ds (cid:96) = 4 dx + dx − (1 + x + ( x − + α Θ( x + ) / + µ (cid:18) − x + ( x − + α Θ( x + ) / x + ( x − + α Θ( x + ) / (cid:19) dϕ , (B.1)with ϕ ∈ [0 , π ], ( x + , x − ) ∈ R , solves the three-dimensional Einstein’s equations with a positivecosmological constant Λ = +1 /(cid:96) in the presence of a stress energy tensor T ++ = αδ ( x + ) (wherewe have set 8 πG = 1). The null-energy condition enforces α > 0. The worldlines of theNorthern and Southern static patch observers are at x + x − = 1. The future and past boundariesare at x + x − = − 1. The parameter µ ∈ (0 , µ = 0 andthe pure dS universe at µ = 1. In the absence of a shockwave the Penrose diagram is thatof two conical defects sitting at the North and South poles of dS . These cause the horizonto be smaller than that of pure dS , with the limiting case being µ = 1. A useful coordinatetransformation is: ˜ x − = x − + α Θ( x + ) / , (B.2)for which we have the geometry: ds (cid:96) = 4 dx + d ˜ x − (1 + x + ˜ x − ) + µ (cid:18) − x + ˜ x − x + ˜ x − (cid:19) dϕ − αδ ( x + )( dx + ) , (B.3)The deformed geometry, with α (cid:54) = 0, can be viewed as a de Sitter universe with a boostedcircular shell from the North pole static patch worldline. We can compare the metric (B.3) to(6.14). The absolute value of the dilaton is equivalent to the size of the ϕ -circle. Notice thatit corresponds to the case κ < δ -function piece is negative. This impliesthat a light particle traveling near the shock will experience a Shapiro time-advance, allowing itto enter the region of dS that was out of causal contact in the absence of the shockwave.Analogous solutions can be considered in higher dimensions [52, 53, 54]. The simplest isa geometry connecting two dS d +1 regions with a shock that lives exactly on the cosmologicalhorizon. One simply replaces the circle with a d -sphere for the µ = 1 geometry in (B.1).30 Some details for out-of-time-ordered correlators Here we present some explicit results for the ordered and out-of-time ordered 4-points functionsin the centaur background. Recall that the full connected 4-points function is given by (cid:42)(cid:32) B ( u , u )sin u − u + ˜ E ( u , u ) (cid:33) (cid:32) B ( u , u )sin u − u + ˜ E ( u , u ) (cid:33)(cid:43) . (C.1)There are then, four different contributions to the correlator, that schematically we name (cid:104)BB(cid:105) , (cid:104)B ˜ E(cid:105) , (cid:104) ˜ EB(cid:105) and (cid:104) ˜ E ˜ E(cid:105) .The first contribution can be computed exactly as an analytic function of the four points u , u , u , u . We can consider first the ordered correlator with u < u < u < u . Taking κ/ ˜ φ b = 1, this gives (cid:104)B ( u , u ) B ( u , u ) (cid:105) ord = π csch π (cosh ( u + π ) + cosh ( u + π ) + cosh ( u + π ) + cosh ( u + π ))+ cot ( u / 2) (2 u + π csch π (sinh ( u + π ) − sinh ( u + π ) + sinh ( u + π ) − sinh ( u + π )))+ cot ( u / 2) (2 u + π csch π ( − sinh ( u + π ) − sinh ( u + π ) + sinh ( u + π ) + sinh ( u + π )))+ cot ( u / 2) cot ( u / π csch π ( − cosh ( u + π ) + cosh ( u + π ) + cosh ( u + π ) − cosh ( u + π )) − u u ) − , where we introduced the usual notation u ij ≡ u i − u j . In the main text, we are interested inthe out-of-time-ordered correlator (otoc). In this case, u < u < u < u , and taking again κ/ ˜ φ b = 1, we obtain: (cid:104)B ( u , u ) B ( u , u ) (cid:105) otoc = (cid:104)B ( u , u ) B ( u , u ) (cid:105) ord + 2 πu cot ( u / 2) cot ( u / 2) +2 iπe − u − u ( e u − e u ) (cid:0) e u (cid:0) ie iu + e iu (cid:1) (cid:0) ie iu + e iu (cid:1) + e u (cid:0) e iu + ie iu (cid:1) (cid:0) e iu + ie iu (cid:1)(cid:1) ( e iu − e iu ) ( e iu − e iu ) . (C.2)Given this formula, it is straightforward to compute the contribution of this term to the real-time 4-points function that appears in equation (5.19) in the main text, that results in (5.21).Moreover, a completely analogue calculation using the propagator that corresponds to the AdS black-hole case gives the saturation of the Lyapunov exponent for quantum chaotic systems[9, 22].In the case of the interpolating geometries considered in the main text, we have three ad-ditional contributions to the out-of-time-ordered four-point function. As argued in the maintext, these contributions either oscillate, or are exponentially suppressed as a function of realtime t . Here, we provide numerical evidence to support this claim. In figure 6 we display thefull contribution to (cid:104)B ˜ E(cid:105) and (cid:104) ˜ E ˜ E(cid:105) – they behave as ∼ e − πt/ ˜ β . The (cid:104) ˜ EB(cid:105) -terms oscillate with31requency 2 π/ ˜ β . None of these contributions grow exponentially with time. 10 12 14 16 18 20 - - < ℰ ˜ ℬ > (a) (cid:104) ˜ EB(cid:105) - - - | < ℬ ℰ ˜ > | (b) (cid:104)B ˜ E(cid:105) - - < ℰ ˜ ℰ ˜ > (c) (cid:104) ˜ E ˜ E(cid:105) Figure 6: Three contributions to the otoc in the case of the interpolating geometry. The blue dots correspondto the numerical solution while the dashed red lines correspond to the best fit by functions of the form Ae − Bt forthe decaying terms and A cos( Bt + C ) for the oscillating one. We set ˜ β = 2 π and κ/ ˜ φ b = 1. A, B, C are constantsto be fitted. In all cases, the best fit gives B ≈ D The γ -theory and γ -Schwarzian In this appendix we generalise the dilaton potential to V ( φ ) = 2( | φ − φ | − φ ). The solutionfor φ < φ is now given by: ds = dρ + dτ cosh ρ , φ ( ρ ) = − φ h tanh ρ , (D.1)with φ h < 0. Smoothness requires τ ∼ τ + 2 π . At tanh ρ c = − φ /φ h , the geometry interpolatesto one with negative curvature.The general negative curvature solution is given by: ds = γ sin √ γR ( dR + dτ ) , φ ( R ) = α cot ( √ γR ) + 2 φ . (D.2)For γ > R ∈ (0 , π/ √ γ ) whereas for γ < R ∈ (0 , ∞ ). Locally we canrescale R and τ , and thus (D.2) is diffeomorphic to the standard hyperbolic metric on the disk.However, since we are fixing the periodicity of τ the above geometry is globally distinct fromthe standard hyperbolic metric. Indeed, for γ < φ = 2 π (1 − √− γ ). Thus, for γ = − γ < − γ → − , the geometry tends to ds = dτ + dR R , τ ∼ τ + 2 π . (D.3)Here the geometry develops an infinite throat as R → ∞ .We need to glue the two geometries (D.1) and (D.2) so that both the metric and the dilatonare smooth up to first derivatives. This is achieved by gluing the negative curvature solution at32 a) φ < φ = 0 (c) φ > Figure 7: The different interpolating geometries as a function of φ . In general, negative φ solutions contain asmaller part of the sphere than the φ = 0, while φ > a specific R g given by tan √ γR g = (cid:114) γ − γ , γ = 1 − φ /φ h . (D.4)This is valid only for positive ρ c and positive γ . This completes the smooth gluing at theinterpolating region. Finally, boundary conditions must be imposed near the AdS boundaryof (D.2) to complete the Dirichlet problem. We give a schematic depiction of the differentgeometries in figure 7. Case (i): φ > φ h > solution, that for κ > κ < 0) will be the dominant (sub-dominant) saddle. For φ h < ρ c > ρ c (cid:46) . 88. In these cases, the interpolating geometry will have a greater portion of thesphere compared to the φ = 0 case. As we move along the φ h negative values, we change theinterpolating radius. The limiting case φ h = −√ φ corresponds to a geometry with γ → 0, i.e.one that is about to be disconnected. This is shown in figure 8. For greater values of φ h , onlythe AdS geometry exists. So, for φ > 0, only interpolating solutions with γ ∈ [0 , 1] exist. Case (ii): φ < φ < 0, the interpolating geometry contains a smaller piece of two-sphere. The interpolatingradius ρ c is always negative and it is possible to have γ ∈ [ − , 1] in this case. A transition occursat γ = 0, where φ h = √ φ , but the interpolating geometry remains smooth and connected. Inthe limit of φ h → φ , the sphere is lost completely. Note that these solutions do not have a zerotemperature limit. D.1 The γ -Schwarzian theory It is interesting to consider the boundary mode actions that emerge for boundary fluctuationsin the geometries (D.2). In equation (2.18), we saw that for global AdS ( γ = 1) the boundaryaction is a Schwarzian plus a kinetic term that has the opposite sign to the usual AdS -black33 igure 8: A sketch of the interpolating geometry close to γ → + with φ > 0. As γ approaches to zero, theinterpolating geometry develops an infinitely large throat until it gets disconnected for γ > hole. This, in turn, generates an oscillating OTOC as opposed to the black hole where theOTOC grows exponentially and saturates the chaos bound.It is interesting to see how the Schwarzian action is modified by considering boundary fluc-tuations of a general negative curvature geometry (D.2). We find: S bdy = φ b κ (cid:90) du (cid:16) γ ∂ u τ ( u )) − Sch [ τ ( u ) , u ] (cid:17) . (D.5)Here τ ( u ) maps u to a circle of period 2 π . Note that γ = 1 gives the global AdS action alreadydiscussed in section 4, while γ = − τ ( u ) = 2 πu/ ˜ β , such that u ∼ u + ˜ β . Considering anexpansion as τ ( u ) = 2 πu/ ˜ β + δτ ( u ), the action for this γ -Schwarzian becomes (to quadraticorder in δτ ), S bdy = φ b κ (cid:90) ˜ β du (cid:32) γ δτ (cid:48) ( u ) + ˜ β π δτ (cid:48)(cid:48) ( u ) (cid:33) . (D.6) References [1] G. W. Gibbons and S. W. Hawking, “Cosmological Event Horizons, Thermodynamics, andParticle Creation,” Phys. Rev. D , 2738 (1977).[2] S. A. Hartnoll, “Lectures on holographic methods for condensed matter physics,” Class.Quant. Grav. , 224002 (2009) [arXiv:0903.3246 [hep-th]].[3] D. Anninos and D. M. Hofman, “Infrared Realization of dS in AdS ,” Class. Quant. Grav. , no. 8, 085003 (2018) [arXiv:1703.04622 [hep-th]].[4] A. Lopez-Ortega, “Quasinormal modes of D-dimensional de Sitter spacetime,” Gen. Rel.Grav. , 1565 (2006) [gr-qc/0605027].[5] H. Nariai, “On a new cosmological solution of Einstein’s field equations of gravitation,”General Relativity and Gravitation , 963 (1999).346] S. Sachdev and J. Ye, “Gapless spin fluid ground state in a random, quantum Heisenbergmagnet,” Phys. Rev. Lett. , 3339 (1993) [cond-mat/9212030].[7] O. Parcollet and A. Georges. Non-Fermi-liquid regime of a doped Mott insulator - 1999.Phys.Rev.,B59,5341 [cond-mat/9806119].[8] A. Kitaev. A simple model of quantum holography - 2015. KITP stringsseminar and Entanglement program (Feb. 12, April 7, and May 27).http://online.kitp.ucsb.edu/online/entangled15[9] J. Maldacena and D. Stanford, “Remarks on the Sachdev-Ye-Kitaev model,” Phys. Rev. D , no. 10, 106002 (2016) [arXiv:1604.07818 [hep-th]].[10] S. Sachdev, “Bekenstein-Hawking Entropy and Strange Metals,” Phys. Rev. X , no. 4,041025 (2015) [arXiv:1506.05111 [hep-th]].[11] J. Polchinski and V. Rosenhaus, “The Spectrum in the Sachdev-Ye-Kitaev Model,” JHEP , 001 (2016) [arXiv:1601.06768 [hep-th]].[12] D. Anninos, T. Anous, P. de Lange and G. Konstantinidis, “Conformal quivers and meltingmolecules,” JHEP , 066 (2015) [arXiv:1310.7929 [hep-th]].[13] D. Anninos, T. Anous and F. Denef, “Disordered Quivers and Cold Horizons,” JHEP ,071 (2016) [arXiv:1603.00453 [hep-th]].[14] B. Freivogel, V. E. Hubeny, A. Maloney, R. C. Myers, M. Rangamani and S. Shenker,“Inflation in AdS/CFT,” JHEP , 007 (2006) [hep-th/0510046].[15] D. A. Lowe and S. Roy, “Punctuated eternal inflation via AdS/CFT,” Phys. Rev. D ,063508 (2010) [arXiv:1004.1402 [hep-th]].[16] D. Anninos, S. A. Hartnoll and D. M. Hofman, “Static Patch Solipsism: Conformal Sym-metry of the de Sitter Worldline,” Class. Quant. Grav. , 075002 (2012) [arXiv:1109.4942[hep-th]].[17] S. Leuven, E. Verlinde and M. Visser, “Towards non-AdS Holography via the Long StringPhenomenon,” JHEP , 097 (2018) [arXiv:1801.02589 [hep-th]].[18] T. Banks and M. O’Loughlin, “Two-dimensional quantum gravity in Minkowski space,”Nucl. Phys. B , 649 (1991).[19] M. Cavaglia, Phys. Rev. D , 084011 (1999) doi:10.1103/PhysRevD.59.084011 [hep-th/9811059].[20] R. Jackiw and C. Teitelboim, in: Quantum Theory of Gravity, S. Christensen ed. (AdamHilger, Bristol, 1984). 3521] D. Grumiller and R. McNees, “Thermodynamics of black holes in two (and higher) dimen-sions,” JHEP , 074 (2007) [hep-th/0703230 [hep-th]].[22] J. Maldacena, D. Stanford and Z. Yang, “Conformal symmetry and its breaking intwo dimensional Nearly Anti-de-Sitter space,” PTEP , no. 12, 12C104 (2016)[arXiv:1606.01857 [hep-th]].[23] K. Jensen, “Chaos in AdS Holography,” Phys. Rev. Lett. (2016) no.11, 111601[arXiv:1605.06098 [hep-th]].[24] J. Engelsy, T. G. Mertens and H. Verlinde, “An investigation of AdS backreaction andholography,” JHEP (2016) 139 [arXiv:1606.03438 [hep-th]].[25] D. Bagrets, A. Altland and A. Kamenev, “Sachdev-Ye-Kitaev model as Liouville quantummechanics,” Nucl. Phys. B , 191 (2016) [arXiv:1607.00694 [cond-mat.str-el]].[26] J. B. Hartle and S. W. Hawking, “Wave Function of the Universe,” Phys. Rev. D , 2960(1983).[27] E. Witten, “A Note On Boundary Conditions In Euclidean Gravity,” arXiv:1805.11559[hep-th].[28] A. I. Larkin and Y. N. Ovchinnikov, “Quasiclassical method in the theory of superconduc-tivity,” Sov.Phys.JETP,28,1200-1205 (1969).[29] J. Maldacena, S. H. Shenker and D. Stanford, “A bound on chaos,” JHEP , 106 (2016)[arXiv:1503.01409 [hep-th]].[30] A. Almheiri and J. Polchinski, “Models of AdS backreaction and holography,” JHEP ,014 (2015) [arXiv:1402.6334 [hep-th]].[31] T. Dray and G. ’t Hooft, “The Gravitational Shock Wave of a Massless Particle,” Nucl.Phys. B , 173 (1985).[32] P. Gao, D. L. Jafferis and A. Wall, “Traversable Wormholes via a Double Trace Deforma-tion,” JHEP , 151 (2017) [arXiv:1608.05687 [hep-th]].[33] S. Gao and R. M. Wald, “Theorems on gravitational time delay and related issues,” Class.Quant. Grav. , 4999 (2000) [gr-qc/0007021].[34] T. Banks, “Holographic spacetime,” Int. J. Mod. Phys. D , 1241004 (2012).[35] M. K. Parikh and E. P. Verlinde, “De Sitter holography with a finite number of states,”JHEP (2005) 054 [hep-th/0410227]. 3636] N. Goheer, M. Kleban and L. Susskind, “The Trouble with de Sitter space,” JHEP ,056 (2003) [hep-th/0212209].[37] E. P. Verlinde, “Emergent Gravity and the Dark Universe,” SciPost Phys. , no. 3, 016(2017) [arXiv:1611.02269 [hep-th]].[38] X. Dong, E. Silverstein and G. Torroba, “De Sitter Holography and Entanglement Entropy,”JHEP , 050 (2018) [arXiv:1804.08623 [hep-th]].[39] X. Dong, B. Horn, E. Silverstein and G. Torroba, “Micromanaging de Sitter holography,”Class. Quant. Grav. , 245020 (2010) [arXiv:1005.5403 [hep-th]].[40] D. Anninos and T. Anous, “A de Sitter Hoedown,” JHEP , 131 (2010).[arXiv:1002.1717 [hep-th]].[41] D. Anninos, “De Sitter Musings,” Int. J. Mod. Phys. A , 1230013 (2012) [ arXiv:1205.3855[hep-th]].[42] D. Anninos, T. Anous, I. Bredberg and G. S. Ng, “Incompressible Fluids of the de SitterHorizon and Beyond,” JHEP , 107 (2012) [arXiv:1110.3792 [hep-th]].[43] S. H. Shenker and D. Stanford, “Black holes and the butterfly effect,” JHEP , 067(2014) [arXiv:1306.0622 [hep-th]].[44] D. C. Dai, D. Minic and D. Stojkovic, “A new wormhole solution in de Sitter space,”[arXiv:1810.03432 [hep-th]].[45] J. Maldacena and X. L. Qi, “Eternal traversable wormhole,” [arXiv:1804.00491 [hep-th]].[46] J. De Boer, S. F. Lokhande, E. Verlinde, R. Van Breukelen and K. Papadodimas, “On theinterior geometry of a typical black hole microstate,” arXiv:1804.10580 [hep-th].[47] A. Strominger, “The dS / CFT correspondence,” JHEP , 034 (2001) [hep-th/0106113].[48] J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field infla-tionary models,” JHEP , 013 (2003) [astro-ph/0210603].[49] E. Witten, “Quantum gravity in de Sitter space,” hep-th/0106109.[50] D. Anninos, T. Hartman and A. Strominger, “Higher Spin Realization of the dS/CFTCorrespondence,” Class. Quant. Grav. (2017) no.1, 015009 [arXiv:1108.5735 [hep-th]].[51] D. Anninos, F. Denef, R. Monten and Z. Sun, “Higher Spin de Sitter Hilbert Space,”[arXiv:1711.10037 [hep-th]].[52] K. Sfetsos, “On gravitational shock waves in curved space-times,” Nucl. Phys. B , 721(1995) [hep-th/9408169]. 3753] M. Hotta and M. Tanaka, “Shock wave geometry with nonvanishing cosmological constant,”Class. Quant. Grav. , 307 (1993).[54] M. Hotta and M. Tanaka, “Gravitational shock waves and quantum fields in the de Sitterspace,” Phys. Rev. D47