Inside the Hologram: Reconstructing the bulk observer's experience
IInside the Hologram: Reconstructing the bulk observer’s experience
Daniel Louis Jafferis, a Lampros Lamprou ba Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138,USA b Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA02139-4307,USA
Abstract:
We develop a holographic framework for describing the experience of bulkobservers in AdS/CFT, that allows us to compute the proper time and energy distribu-tion measured along any bulk worldline. Our method is formulated directly in the CFTlanguage and is universal: It does not require knowledge of the bulk geometry as aninput. When used to propagate operators along the worldline of an observer falling intoan eternal black hole, our proposal resolves a conceptual puzzle raised by Marolf andWall. Notably, the prescription does not rely on an external dynamical Hamiltonian orthe AdS boundary conditions and is, therefore, outlining a general framework for theemergence of time. a r X i v : . [ h e p - t h ] S e p ontents Contents1 Inside a quantum system
Suppose your adventurous colleague jumped in the AdS black hole you created in yourlab’s quantum computer by simulating N = 4 Super Yang-Mills. What did their Geigercounter register along their journey and at what age did they meet their inevitable end?Gauge/gravity duality [1] has offered a wealth of insights on the microscopic de-scription of black holes, as observed from an asymptotic frame. These include theirentropy, fast scrambling dynamics [2, 3] and unitarity of the Hawking evaporation pro-cess [4, 5]. In contrast, the infalling observer’s experience remains mysterious, owing– 1 –o the lack of holographic reconstruction techniques that penetrate bulk horizons. Thedifficulty in posing, even in principle, operationally meaningful questions such as theamount of time or energy measured by observers behind horizons highlights a gap in ourunderstanding of AdS/CFT: The absence of a CFT framework for describing physicsin an internal reference frame .To catalyze progress in this direction, we pursue an observer-centric approach tobulk reconstruction. ∗ Even for observers that do not fall into black holes, no generalmethod for determining how their experience is encoded in the CFT is known, partic-ularly without having to solve the bulk theory directly. This is closely related to thefact that CFT operators are attuned to an external description of the quantum system,while the observer is associated to an internal frame of reference.Any observer made out of bulk matter is simply a suitable subsystem of the dualConformal Field Theory † . In our work, this observer will be a black hole entangledwith an external reference; the subsystem available for their experiments consists ofoperators within the black hole “atmosphere” (Section 2). The entangled referenceprovides an external way to describe the frame associated to the observer. The virtueof such a probe black hole is in providing a particularly simple model of a subsystem,related by a unitary transformation to the thermofield double state [8].The probe black hole will be introduced near the boundary, and then allowed topropagate under time evolution before returning to our possession at a later time ‡ . Theonly assumption about the bulk state we make is that it is described by a semi-classicalspacetime, whose features we wish to probe in the classical limit. In this setup, westart by solving the following problem: Assuming CFT knowledge of the atmospheredegrees of freedom at the initial and final timeslices, and of the CFT Hamiltonian, howmuch proper time did the observer’s clock measure and what energy distribution wasdetected by their calorimeter?In Sections 3 and 4, we explain how to read off this information from the boundaryunitary V H (0 , t ) that relates the initial ( t i = 0) and final ( t f = t ) local atmosphereoperators. The result is universal within its domain of validity, and does not requireas an input the solution for the bulk spacetime. The key ingredient is the modularHamiltonian of the black hole, K = − log ρ , defined by its reduced density matrix ρ after tracing out the reference system. In a nutshell, we propose that the decomposition ∗ Related attempts to describe physics from an internal observer’s point of view include [6, 7] † Our construction does not rely on conformal symmetry. This generality is crucial for it to applyin non-vacuum states with semi-classical bulk duals ‡ This precludes exploring behind horizons in the particular setup of this work. Nevertheless, ourframework contains lessons for black hole interior reconstruction which we discuss in Section 5. – 2 –f log V H in terms of (approximate) “eigen-operators” of K has the schematic form: i log V H (0 , t ) = τ ( t )2 π (cid:90) d Ω d − f (Ω) G π (Ω) + τ ( t )2 π K + other zero-modes + O ( e − τ , N − )(1.1)The coefficient τ ( t ) of the modular operator is the proper time . G π (Ω) are the modularscrambling modes, satisfying [ K, G π ] = − πiG π [9] thus growing exponentially undermodular flow, and the coefficient f (Ω) depends on the expectation value of the horizonnull energy flux (cid:82) x + =0 dx − (cid:104) T −− (cid:105) in the frame of the moving black hole. A non-vanishingscrambling mode coefficient in (1.1) is, therefore, a signature of particle detection alongthe bulk worldline. The other zero mode contributions describe the precession of theobserver’s symmetry frame, e.g. a local rotation about the black hole. Our result isvalid when the proper time is shorter than the scrambling time of our probe black hole τ ( t ) (cid:46) log S BH .The basic intuition, from the point of view of the quantum system, is that propertime is measured by the phase, e imτ , of the state, as in Feynman’s path integral for aparticle worldline. In our setup, the analog of the rest mass, m is provided by the localenergy, conveniently given by the Hamiltonian of the reference system which is exactlyAdS Schwarzschild. This corresponds to the modular Hamiltonian for the spacetimebeing probed. An important additional ingredient is that we keep track of the unitary V H (0 , t ) that relates the presumed known operator algebra of the probe black holeatmosphere at the beginning and end of its journey through the bulk, to correctlytranslate between the initial and final states. This is necessary to have a well-definedcomparison without requiring the input of a collection of CFT operators associated totranslations in a particular bulk time slicing.In the bulk language, we use the fact that the modular Hamiltonian acts approxi-mately as the local Schwarzschild time evolution relative to the extremal surface, in asense that we make more precise below. An important point is that because the totalstate is a single sided unitary transformation of the thermofield double, the correctionsto the geometric action of the modular flow can be regarded as transients due to ex-citations of the bulk fields near the probe black hole, which in our setup do not affectthe initial and final atmosphere operators. The only persistent effect is a shift of theextremal surface relative to the probe black hole causal horizon, with respect to whichwe wish to define the atmosphere operators. This is encoded by the scrambling modesin eq. (1.1) and is explained in detail in Section 4.In Section 3.3, we employ this technology to holographically measure the time di-lation witnessed by two twins who embark on separate journeys and compare clocksat their future reunion. In the dual quantum description, the twins select two time-dependent families of modular Hamiltonians, describing the evolution of their state,– 3 –hich are simply related when the siblings meet at the initial and final moments, thusforming a “closed loop”. The proper time difference experienced by the twins is com-puted by two intrinsic properties of this modular loop: the modular Berry holonomy[10–12] (Section 2.2) and the modular zero mode component of the CFT Hamiltonianintegrated along the path.The experience of an observer falling into a black hole is discussed in Section 5.Our framework naturally resolves a conceptual puzzle raised by Marolf and Wall [13]regarding the ability of an observer that enters from one side of an AdS wormhole toreceive signals coming from the other, despite the dynamical decoupling of the twoexterior regions. Moreover, our proposal outlines an interesting perspective on the“problem of time” in quantum gravity in general, an early form of which was anticipatedin [14]. Our eq. (1.1) links the geometric notion of time in General Relativity to thenatural, quantum mechanical clock of the dual theory, the modular clock, obtained bytracing out the “observer”. It, thus, seems to enjoy a degree of universality that mayextend its validity beyond the AdS/CFT context. In this paper, we model a bulk observer as a black hole. It will be a large black hole,in the sense of having a Schwarzschild radius of order the AdS curvature scale, asrequired for having a simple associated modular flow. However, we will take it to bemuch smaller than the features of the spacetime that it is intended to probe. For thisreason our construction captures AdS scale locality and geometric features.It is, of course, very interesting to generalize the approach to small black holes,whose associated modular flows are less universal. The microcanonical eternal blackholes of [15] are the natural candidate for such a generalization, since they are semi-classical configurations black holes whose horizon radius can be parametrically smallerthan L AdS while being thermodynamically dominant within an energy window. Wediscuss this generalization and related subtleties in some detail in Section 5.This Section summarizes the advantages of the probe black hole description andintroduces the concepts we will utilize in articulating our proposal in Sections 3 and 4:the modular Hamiltonian, modular Berry transport and the observer’s code subspacedegrees of freedom.
The textbook observer in General Relativity is a local probe: They have zero size,no gravitational field and they travel along a worldline whose local neighborhood is– 4 – igure 1 : An illustration of our setup. The red line represents the worldline of an ideal observer.We replace them by a small black hole of radius r BH much smaller than the spacetime’s curvaturefeatures which we thermally entangle with a reference system, assumed to be another AdS black holefor simplicity. The black hole propagates along the original geodesic due to the equivalence principle.The green tube of radius (cid:96) around the black hole represents its “atmosphere”: The operators ourobserver can manipulate at any given time. approximately flat. This is a convenient idealization in the classical theory which, how-ever, is unavailable quantum mechanically. Quantum observers are physical systemsthat should be included in the wavefunction of the Universe. They have finite massand occupy volume of some linear size (cid:96) necessarily larger than their own Schwarzschildradius but smaller than the curvature scale of the spacetime they live in. Their gravita-tional field changes the local geometry around them to approximately Schwarzschild intheir local inertial frame and quantum mechanical evolution entangles them with theirenvironment —a potential decoherence they need to protect themselves against to stayalive. Last but not least, observers have agency: They can manipulate and measuredegrees of freedom in their vicinity to perform experiments.A general treatment of internal observers in AdS/CFT is a hopeless task but,fortunately, an unnecessary one. All we need is a more appropriate idealization ofthe physical observer above. In this paper, we squeeze our observer down to their– 5 –chwarzschild radius, collapsing them to a black hole of size r BH which we thermallyentangle with an external reference system (fig. 1). This black hole “observer” willapproximately propagate along the same worldline, leaving the physics at scales muchlarger than r BH unaffected. The only difference is that excitations that originallyintercepted the worldline are now absorbed by our black hole: They get detected byour observer’s device!The entangled external system provides a precise meaning to the reference frameof the observer. In particular, one could consider gravitational operators probing thesystem spacetime whose diffeomorphism dressing is attached to the reference boundarythrough the Einstein-Rosen bridge. These must be complicated operators acting onboth system and reference quantum systems, whose action in a (non-factorized) codesubspace approximates that bulk description.In this work, we will be able to phrase our final results without needing such op-erators, which agrees with the fact that the proper time along the probe’s trajectorybetween known near-boundary initial and final configurations is diffeomorphism invari-ant without requiring the observer’s frame. As such, only system framed atmosphereoperators and the probe black hole’s density matrix appear in formula for the propertime. However, it is interesting to compare the result to the clock of the referencesystem, which gives a simple way to understand our results.Assuming the reference is a copy of the original CFT for simplicity, and that r BH ∼ L AdS , the initial global state describing a black hole inserted somewhere in aclassical, asymptotically AdS background has the general form: | Ψ (cid:105) = Z − / (cid:88) n e − βE n / U sys | E n (cid:105) sys | E n (cid:105) ref (2.1)where U sys is a unitary transformation that excites the bulk fields and metric to createthe background our black hole will propagate in. § Note that any state which looks likethe eternal black hole in the reference system must be of this form.Crucially, collapsing our observer to a black hole offers protection from decoherence:Its degrees of freedom will remain spatially localized in the bulk for a time-scale of orderthe evaporation time which we will assume to be much larger than the time-scales ofthe experiments we will perform.We choose a black hole “atmosphere” of size (cid:96) to be the observer’s lab: Localoperators in the atmosphere can be directly manipulated or measured. We will assume r BH (cid:46) L AdS (cid:28) (cid:96) (cid:28) L where L the scale of the spacetime curvature perturbations § Black holes with r BH (cid:28) L AdS could also in principle be discussed in our formalism, with thedifference that the system-reference state would be more complicated since small black holes do notdominate the canonical ensemble. See Section 5 for related discussion. – 6 –bout AdS. The CFT duals of the atmosphere operators at different times select atime-dependent subsystem of the boundary theory. This family of (abstract) CFTsubsystems serves as a quantum notion of our observer’s local frame which will beimportant in our formalism and is explained further in Section 2.3.
An observer’s description of the Universe they inhabit does not include a descriptionof themselves. The degrees of freedom that make up the observer, which are generallyentangled with the rest of the bulk, ought to be traced out. For the idealized eternalblack hole observer of Section 2.1 this corresponds to tracing out the reference system.
Modular Hamiltonian
Entanglement leads to uncertainty about a subsystem’squantum state. In our setup, ignorance of the state of the reference results in a mixedstate ρ = Tr ref [ | Ψ (cid:105)(cid:104) Ψ | ] for our black hole system. The Hermitian operator K = − log ρ is the modular Hamiltonian and it defines an automorphism of the operator algebra.The action of this automorphism is generally non-local, except in special situations,and it can be thought of as generating the “time” evolution with respect to which thesubsystem is in equilibrium. Somewhat more formally [16], the modular Hamiltonianof a quantum field theory subsystem in the state Ψ is defined as K = − log ∆ Ψ where∆ Ψ is the “KMS operator” of Ψ satisfying: (cid:104) Ψ | φ † ∆ Ψ φ | Ψ (cid:105) = (cid:104) Ψ | φ φ † | Ψ (cid:105) (2.2)for all correlation functions of the subsystems’s operator algebra.For our class of states (2.1), the modular Hamiltonian is unitarily equivalent to thedynamical Hamiltonian of the boundary theory: K = 2 π U sys HU † sys (2.3)where we have renormalized H to β π H , for convenience. Recall that this is actuallythe most general type of state in which the full spacetime is identical to empty AdS-Schwarzschild on the reference side. In Schrodinger picture, time evolution acts on thestate, resulting in a time-dependent modular Hamiltonian K ( t ).A simple example we will frequently use to illustrate the ideas of this paper is aboosted black hole, which bounces back and forth in AdS. This is prepared by actingon a static black hole with the asymptotic boost symmetry of the spacetime, generatedby the conformal boost B in the CFT. Its dual state corresponds to (2.1) with thechoice U sys = e − iBη , where η the black hole’s rapidity, and the time dependent modularHamiltonian reads explicitly:(2 π ) − K ( t ) = cosh η H + sinh η ( P cos t + B sin t ) (2.4)– 7 –here H, P, B the conformal generators satisfying the usual SL (2 , R ) algebra:[ B, H ] = iP , [ B, P ] = iH , [ H, P ] = iB (2.5) Modular Berry Wilson lines
A continuous family of modular Hamiltonians K ( t )—like eq. (2.4)— selects a continuous family of bases in the Hilbert space, consisting ofthe eigenvectors of K ( t ). The local generator D ( t ) of the basis rotation can generallybe obtained as the solution to the problem: ∂ t K ( t ) = − i [ D ( t ) , K ( t )] (2.6)Equation (2.6) by itself determines D ( t ) only up to modular zero modes Q ( t ), generatingsymmetries of the reduced state [ Q ( t ) , K ( t )] = 0. This reflects the freedom to chooseat will the local modular frame, e.g. phases of eigenstates of different K ( t ), which isan example of a Berry phase, discussed in detail in [12]. A canonical map between thebases that is intrinsic to the family K ( t ) can be constructed following Berry’s footsteps[17], by defining the modular parallel transport operator as the solution to the problem(2.6) supplemented by the condition: P t [ D ( t )] = 0 (2.7)where P t is the projection of the Hermitian operator D ( t ) onto the subspace of zeromodes of K ( t ). Eigenframes of different K ( t ) are related by modular Wilson lines, thepath ordered exponential of the parallel transport D ( t ) W ( t , t ) = T exp (cid:20) − i (cid:90) t t dt (cid:48) D ( t (cid:48) ) (cid:21) (2.8) W can also be thought of as a canonical unitary automorphism of the operator algebraof our system: O W ( t ) = W (0 , t ) O W † (0 , t ) (2.9)with the properties: K ( t ) = W (0 , t ) K (0) W † (0 , t ) (2.10) (cid:104) Ψ( t ) | O (1) W ( t ) . . . O ( n ) W ( t ) | Ψ( t ) (cid:105) = (cid:104) Ψ | O (1) . . . O ( n ) | Ψ (cid:105) (2.11)When K ( t ) is obtained via the time evolution of the system —as in the exampleof the previous Subsection— the modular parallel transport is related to the, possiblytime-dependent, Hamiltonian H ( t ) via: D ( t ) = H ( t ) − P t [ H ( t )] (2.12)– 8 –n our boosted black hole example with modular Hamiltonian (2.4), the parallel trans-port problem can be solved explicitly by a straightforward group theory exercise. Aconvenient way to express the solution is: D ( t ) = ˙ x ( t ) P + ˙ η ( t ) e − iP x ( t ) B e iP x ( t ) − b ( t ) K ( t ) (2.13) x ( t ) = tanh − (tanh η (0) sin t ) (2.14) η ( t ) = sinh − (sinh η (0) cos t ) (2.15) b ( t ) = 12 π ˙ x ( t ) sinh η ( t ) (2.16)where x ( t ) , η ( t ) measure the black hole’s geodesic distance from the AdS origin and itsrapidity in the global frame respectively, while the last term enforces the vanishing ofthe zero-mode component of D ( t ). The modular parallel transport operator (2.8) thenreads: W boosted BH (0 , t ) = T exp (cid:20) − i (cid:90) t dt (cid:48) D ( t (cid:48) ) (cid:21) = e − iP x ( t ) e − iB ( η ( t ) − η (0)) e i π K (0) (cid:82) t dt (cid:48) ˙ x sinh η ( t (cid:48) ) (2.17) Modular holonomies.
A key property of the modular Berry transport is that itgenerally leads to non-trivial holonomies —a fact that will play an important role inour subsequent discussion. We can consider two families of modular Hamiltonians, K ( t ) , K ( t ) for t ∈ [0 , T ], which coincide at the initial and final times: K (0) = K (0)and K ( T ) = K ( T ). These could correspond to two distinct worldlines for our blackhole that begin and end at the same spacetime location and with the same momentum. K ( t ) , K ( t ) then form a closed “loop” K ( t ) = (cid:40) K ( t ) 0 ≤ t ≤ TK (2 T − t ) , T ≤ t ≤ T (2.18)and the property (2.10) becomes: K (0) = W loop (0 , T ) K (0) W † loop (0 , T ) (2.19)which implies that the modular Wilson loop, W loop (0 , T ) will be a, generally non-trivial,element of the modular symmetry group, generated by the zero modes Q i (0): W loop (0 , T ) = exp (cid:34) − i (cid:88) i d i Q i (0) (cid:35) (2.20)This is a modular Berry holonomy, an example of which we will see below in ourdiscussion of time dilation between observers.– 9 – .3 The observer’s code subspace Up to this point, we have treated the bulk observer as a physical system, entangledwith their environment in the global wavefunction, which results in a time-dependentmodular Hamiltonian K ( t ) upon tracing them out. Another defining characteristicof an observer, however, is their ability to control some degrees of freedom in theirUniverse to learn about Its state. In our model, these will be the local bulk fields ina small atmosphere of size (cid:96) around the black hole, denoted by φ i with i an abstractindex, and O (1) − degree polynomials built out of them and their derivatives.The atmosphere degrees of freedom on a particular bulk timeslice Σ t which asymp-totes to boundary time t form a set of observables S t ≡ { φ i ( x i ) , φ j ( x j ) φ l ( x l ) , ∂φ k ( x k ) , . . . (cid:12)(cid:12) x i ∈ Σ t and | x i − x H | ≤ (cid:96) } (2.21)By acting with elements of S t on the background state (2.1) we obtain the observer’sinstantaneous code subspace [18]: The subspace of the CFT Hilbert space the observercan explore with their apparatus at a given bulk time. Crucially, this code subspace isnot generally preserved by time evolution. The observer moves in the bulk hence theoperators in their vicinity (and their boundary duals) differ at different times, resultingin an evolution of the CFT subspace that the observer can probe.As remarked in our introductory Section, we will assume knowledge of the CFTduals of the atmosphere operators at an initial and a final timeslice, S t i and S t f re-spectively —but not in-between. This is physically reasonable when studying processeswhere the black hole is introduced far out in the asymptotically AdS region and returnsto it at some later boundary time, in which case the familiar HKLL prescription for anAdS black hole can be employed for the initial and final reconstruction. Dressing S t i , S t f refer to local operators in a theory of gravity so it is important toclarify their gravitational framing. The choice of an initial and final timeslices Σ t i , Σ t f in the definition of the operator sets is a selection of a bulk gauge, at least in the vicinityof the black hole. Since the black hole is assumed to be near the AdS boundary at thosemoments, its local neighborhood is diffeomorphic to an AdS-Schwarzschild geometry.These local AdS-Schwarzschild coordinates serve as the analog of the local inertial frameabout an idealized observer’s worldline. We are interested in describing the operatorsin the black hole’s reference frame, thus we choose Σ t i , Σ t f to both be constant timewith respect to the corresponding local time-like killing vector within the atmosphere(fig. 2). Operators φ ∈ S t i or S t f can then be labelled by their location in this localAdS-Schwarzschild coordinate system.These are operators that are dressed with respect to the AdS boundary with theproperty that their action within the code subspace results in their insertion at given– 10 –ositions relative to the “local horizon”, namely the place where the horizon would formif no matter were absorbed in the future. This “local horizon” may generally differ for S t i and S t f , as we will see in Section 4. Using standard HKLL, we can construct suchoperators that work in an entire family of perturbative excitations about a black holeof a given temperature [19, 20].It is important to note that when acting within the code subspace of small pertur-bations ¶ around a given semi-classical spacetime state, bulk operators with differentdressings that result in insertion at the same point in the original spacetime are equal atleading order. The associated states φ | Ψ (cid:105) would appear to have different gravitationalfield configurations associated to the energy of the particle produced by φ , howeverthese are subleading to the quantum fluctuations in the ambient gravitational field.This can be seen by explicitly computing the overlap of two such states with differ-ent dressings for φ that classically result in the same insertion point. The states areidentical as quantum states up to G N corrections. Example
For illustration, we return to our boosted black hole example (2.4). Theatmosphere operators at the t = 0 global AdS timeslice, when the black hole is locatedat the AdS origin and has rapidity η , can be obtained from the standard HKLL op-erators in a static AdS-Schwarzschild metric via the action of a boundary conformalboost: φ ( r, Ω) = e − iBη φ static ( r, Ω) e iBη where: (cid:96) P l (cid:28) r − r BH < (cid:96) (2.22)After global time t the black hole has moved to a new location x ( t ) and has a localrapidity η ( t ) given in (2.14) and (2.15). By the previous reasoning, the atmosphereobservables, in the Schrodinger picture , are given by: φ t ( r, Ω) = e − iP x ( t ) e − iBη ( t ) φ static ( r, Ω) e iBη ( t ) e iP x ( t ) where: (cid:96) P l (cid:28) r − r BH < (cid:96) (2.23) Proper time evolution
The unitary that relate the atmosphere operators S t atdifferent times t is, by definition, the proper time evolution along the black hole’sworldline. We will denote this unitary by V S ( t , t ) or V H ( t , t ) depending on whetherwe represent it in the Schrodinger or Heisenberg picture. The goal of the remainderof this paper is to understand the construction of V directly in the CFT language,without reference to bulk reconstruction —except at the initial and final moments ofour probe black hole’s history. ¶ The perturbations must be small at the specified time, to avoid exciting scrambling modes thatlead to large deviations —as we explain in Section 4. – 11 – igure 2 : Our black hole is introduced geometrically by cutting a hole of size (cid:96) around the idealobserver’s worldline in the initial Cauchy slice and a small time band Σ ( (cid:15) ) about it and replacingthe interior with a black hole metric. The local killing vector generating the worldline’s proper timeis glued to the local generator of Schwarzschild time which, in turn, is modular time. As long asnothing falls in the black hole, this identification is valid everywhere along the worldline, suggestingthat modular time is correlated to proper time. With all the necessary concepts in place, we are ready to present the advertised con-nection between modular time and proper time, when no matter gets absorbed by ourprobe black hole; the case of particle absorption is postponed for Section 4.
We now provide three complementary perspectives on our main claim. We start witha bulk geometric argument that offers some useful intuition and then make the casequantum mechanically, using both Heisenberg and Schrodinger picture reasoning, eachof which illuminates different aspects of the physics.
An intuitive geometric argument
In the bulk, our black hole observer can beunderstood via the geometric construction of fig 2. We start with an idealized probeobserver of some small mass m and we choose an initial Cauchy slice Σ which is“constant time” in their local inertial frame, as well as the Cauchy slices within an– 12 – → ( (cid:15) ) around it. The geometry of this time band near theobserver’s location reads, in local inertial coordinates: ds obs r (cid:29) µ ≈ − dτ + dr + r d Ω + µr d − ( dτ + dr ) + O (cid:0) ( Lx ) (cid:1) (3.1)where τ is the proper length of the worldline, r the radial distance from it and L is thescale of the curvature features of the surrounding spacetime.We cut a hole of size (cid:96) in Σ ( (cid:15) ), with µ d − (cid:28) (cid:96) (cid:28) L , around the worldline andreplace its interior with the black hole geometry (cid:107) : ds BH r s <(cid:96) = − (cid:18) − µr d − s (cid:19) dt s + (cid:18) − µr d − s (cid:19) − dr s + r s d Ω (3.2)To ensure a smooth gluing at (cid:96) , the radial coordinates must be identified r = r s sincethey control the size of the transverse sphere at r = r s = (cid:96) . Similarly, the localSchwarzschild killing vector ξ s = ∂ t s at r = (cid:96) − δ gets identified with the correspondinglocal inertial frame killing vector ξ τ = ∂ τ at r = (cid:96) + δ for δ →
0. We then feed theseinitial conditions to the Einstein equations and evolve the system.By the equivalence principle, this black hole will propagate along the original world-line, as long as the scale of bulk curvature features is much larger than r BH . Due to ourassumption of no infalling energy, its atmosphere will also remain locally diffeomorphicto Schwarzschild, so at any point along its path, and at distances r BH (cid:28) r ∼ (cid:96) (cid:28) L the Schwrarzschild clock t S will coincide with the local inertial clock τ of the idealizedobserver.To complete the argument, we need to relate the Schwarzschild clock to modulartime. The CFT modular Hamiltonian for a global state | Ψ (cid:105) is identified with the bulkmodular Hamiltonian [21, 22] which, in turn, is defined via the KMS operator (2.2) (cid:104) Ψ | φ † e − K φ | Ψ (cid:105) = (cid:104) Ψ | φ φ † | Ψ (cid:105) (3.3)As long as particles do not cross paths with our black hole’s trajectory, the state ofthe black hole atmosphere will remain in an approximate local thermal equilibrium:Expectation values of atmosphere observables will be approximately given by theirthermal ones, with the Wick rotation of the local timelike killing direction t s playingthe role of the thermal circle. By virtue of the usual KMS condition then, we have (cid:104) Ψ | φ † e − K φ | Ψ (cid:105) = (cid:104) Ψ | φ φ † | Ψ (cid:105) ≈ (cid:104) Ψ | φ † e − πP ξs φ | Ψ (cid:105) (3.4) (cid:107) The geometry in the atmosphere of the L AdS sized black holes described the thermofield doublestate (2.1) is, instead, given the AdS-Schwarzschild metric which needs to be glued to the local AdSframe of the observer’s neighborhood. This subtlety has no effect on the argument of this section andis only omitted for clarity. – 13 –here: P ξ s the geometric generator of the geometric flow of ξ s . Hence, within the localthermal atmosphere r (cid:46) (cid:96) , K acts like the geometric generator 2 πP ξ s which coincideswith the worldline proper time generator 2 πP ξ τ at r ∼ (cid:96) . Heisenberg picture
To justify our proposal quantum mechanically, it is simplest towork in the Heisenberg picture. Hamiltonian evolution of the system is described by theunitary rotation of the operator basis, while the state and by extension the modularHamiltonian remain fixed. The atmosphere operator set S t of Section 2.3, however,does not simply consist of the Heisenberg evolved elements of S , because their correctevolution, which we denote by the unitary V H (0 , t ), needs to also reflect the motion ofthe black hole in the gravity dual: φ tH ( x ) = V H (0 , t ) φ H ( x ) V † H (0 , t ) (3.5)where the subscript H is introduced to make the Heisenberg picture explicit. Givenour assumption that1. the black hole on the initial and final timeslices Σ , Σ t is located in the asymptoticAdS region and is thus locally diffeomorphic to AdS-Schwarzschild, with the statebeing approximately invariant under the local killing time-like vector2. no energy is absorbed by our probe black hole —an assumption we lift in Section4we conclude that correlation functions of Heisenberg operators in S and S t are identicalin the background state | Ψ (cid:105) : (cid:104) Ψ | φ tH, . . . φ tH,n | Ψ (cid:105) = (cid:104) Ψ | φ H, . . . φ H,n | Ψ (cid:105) (3.6)The meaning of these operators are bulk fields, dressed to the AdS boundary in sucha way that in the subspace under consideration they are inserted in the atmosphere aslabeled by coordinates relative to the extremal surface. Due to the Schwarzschild time-like isometry of the near horizon region, one needs to additionally specify a timeslice,anchored to the AdS boundary. We can do this because we assume that the black holebegins and ends its journey in understood regions near the boundary.Due to (3.6), the isomorphism V H (0 , t ) must be a “modular symmetry”, whenacting on the observer’s code subspace. Such a unitary can be generated by two classesof operators: • zero modes Q a of the modular Hamiltonian projected onto the observer’s codesubspace : [ Q a , P code K (0) P code ] = 0 , where: H code = { O | Ψ (cid:105) , ∀ O ∈ S } (3.7)– 14 – operators G λ = G † λ that are eigenoperators of the code subspace K with imaginaryeigenvalues [ P code K (0) P code , G λ ] = − iλG λ (3.8)The latter necessarily annihilate state | Ψ (cid:105) since otherwise G λ | Ψ (cid:105) would constitute aneigenstate of the modular Hamiltonian with imaginary eigenvalue which contradictsthe Hermiticity of K . A special class of these imaginary eigenvalue operators is thosewith λ = ± π . These were dubbed modular scrambling modes in [9] because theysaturate the bound on modular chaos and they were argued to generate null translationsnear the entangling surface. The simplest example of such a scrambling mode is theAveraged Null Energy operator (cid:82) dx + T ++ (Ω) at the horizon of a static AdS black holein equilibrium, where the eigenvalue 2 πi follows from the near horizon Poincare algebra.We claim that G λ do not contribute to the unitary V H when no particles getabsorbed by our black hole. This is not true for cases with non-vanishing infalling energyflux which, as we show in Section 4.1, results in a scrambling mode G π contribution.Modes with | λ | > π are forbidden by the modular chaos bound [9, 23], as we reviewin Section 4.2. We are unaware of any situations where G λ with − π < λ < π appear,thus we tentatively suggest they are, also, absent in general —leaving a more thoroughinvestigation of this issue for future work. With some foresight, we can return to thecase with no absorption and express the evolution operator in (3.10) as: V H (0 , t ) = exp (cid:34) − i τ ( t )2 π K (0) − i (cid:88) a d a ( t ) Q (cid:48) a (cid:35) (3.9)where we separated the modular Hamiltonian from the rest of the zero modes Q (cid:48) a . Wepropose that the coefficient of the modular Hamiltonian τ ( t ) measures the proper timealong the bulk observer’s worldline, in units of the black hole temperature β/ π . Theother zero modes Q (cid:48) a describe the precession of the symmetry frame of the observer,e.g. a certain amount of rotation of the local reference frame.The intuition for identifying τ ( t ) with proper time is as follows. Within the codesubspace, the action of the atmosphere φ is, at leading order, identical to bulk operatorsthat are framed to the reference boundary, at an appropriate time. Evolution underthe reference Hamiltonian moves the anchor point of those operators, and this gives thelocal Schwarzschild evolution in the atmosphere region. Thus the proper time alongthe trajectory is exactly the amount of modular evolution required to relate the initialand final atmosphere operators, where we equate operators with equal projection ontothe code subspace. Schrodinger picture
It is illuminating to present the same argument in the Schrodingerpicture, where the black hole state in what evolves under the Hamiltonian evolution,– 15 – igure 3 : Illustration of the three different flows appearing in our discussion. H is the CFTHamiltonian generating global AdS evolution. V H is modular flow which maps the t i = 0 atmosphereoperators (green disk on t i = 0 slice) to the Heisenberg picture atmosphere operators at t f = t . V S describes the evolution of the atmosphere operators in the Schrodinger picture and captures themotion of the black hole relative to the boundary. as encapsulated in a time-dependent K ( t ). While, now, the operator basis does notevolve, the atmosphere operator set S t does, due to the motion of the bulk black holerelative to the AdS boundary. The Schrodinger picture atmosphere operators in S and S t are related by a unitary V S (0 , t ): φ t ( x ) = V S (0 , t ) φ ( x ) V † S (0 , t ) (3.10)where the subscript S is a reminder that we are working in the Schrodinger picture.The Schrodinger version of eq. (3.6) is that correlation functions of operators in S in the initial state | Ψ (cid:105) are equal to correlation functions of the final atmosphereoperators S t in | Ψ( t ) (cid:105) : (cid:104) Ψ( t ) | φ t . . . φ tn | Ψ( t ) (cid:105) = (cid:104) Ψ | φ . . . φ n | Ψ (cid:105) (3.11)By virtue of eq. (2.11), property (3.11) the isomorphism (3.10) can be identifiedwith the modular Berry transport W , up to a symmetry Z Q of the observer’s codesubspace correlators: V S (0 , t ) = W (0 , t ) Z Q [ c a ( t )] (3.12)where: Z Q [ c a ( t )] = exp (cid:34) − i (cid:88) a c a ( t ) Q a (0) (cid:35) (3.13)– 16 –nd, as before, Q a are the code subspace modular zero-modes (3.7).As explained in Section 2.2, for a family K ( t ) obtained by Hamiltonian time evo-lution W is generated by (2.12) so (3.12) becomes: V S (0 , t ) = T exp (cid:20) − i (cid:90) t dt (cid:48) ( H − P t (cid:48) [ H ]) (cid:21) Z Q [ c a ( t )]= e − iHt exp (cid:104) i (cid:90) t dt (cid:48) e iHt (cid:48) P t (cid:48) [ H ] e − iHt (cid:48) (cid:105) Z Q [ c a ( t )] (3.14)Substituting (3.14) in eq. (3.10) and switching to the Heisenberg picture we get thefollowing relation between the atmosphere operators: φ tH ( x ) = V H (0 , t ) φ H ( x ) V † H (0 , t )where: V H (0 , t ) = exp (cid:104) i (cid:90) t dt (cid:48) e iHt (cid:48) P t (cid:48) [ H ] e − iHt (cid:48) (cid:105) Z Q [ c a ( t )] (3.15)The unitary V H is now obtained by the product of two contributions, one coming fromthe zero-mode projection of the CFT Hamiltonian and the other from the code sub-space symmetry transformation Z Q in (3.12). These two terms have distinct physicalinterpretations which we discuss in the context of our AdS example below. This decom-position will be important in our discussion of the relative time between two observersin Section 3.3, where the Z Q contributions will give rise to a modular Berry holonomy,providing a conceptually clean way of organizing the CFT dual of time dilation. As an illustration of the idea, we focus on black holes moving in empty AdS along arbitrary worldlines and compute their proper time using our proposed method.
AdS Black holes in inertial motion
Consider the case of the boosted black hole,propagating along an AdS geodesic. In the CFT, it is characterized in the Schrodingerpicture by the time-dependent modular Hamiltonian (2.4), with atmosphere operatorson the initial and final timeslices given by (2.22) and (2.23) respectively. The unitary V S (0 , t ) in eq. (3.10) is equal to: V S (0 , t ) = e − iP x ( t ) e − iB ( η ( t ) − η (0)) (3.16)Recalling the expression (2.17) for the modular parallel transport in this example, V S can be written as: V S (0 , t ) = W boosted BH (0 , t ) exp (cid:20) − i (2 π ) − K (0) (cid:90) t dt (cid:48) ˙ x ( t (cid:48) ) sinh η ( t (cid:48) ) (cid:21) (3.17)– 17 –qually straightforwardly, we can compute the projection of the dynamical Hamiltonianon the modular zero modes of K ( t ), which reads: P t [ H ] = 12 π cosh η (0) K ( t ) (3.18)Combining the results (3.17) and (3.18) in expression (3.15) for the proper timeevolution operator V H (0 , t ) we find: V H (0 , t ) = exp (cid:20) − i (2 π ) − K (0) (cid:90) t dt (cid:48) ( ˙ x ( t (cid:48) ) sinh η ( t (cid:48) ) − cosh η (0)) (cid:21) (3.19)The coefficient of the modular Hamiltonian, using the expressions (2.14) and (2.15) for x ( t ) and η ( t ), reads: τ ( t ) = tan − tan t cosh η (0) (3.20)which is indeed the proper length of the black hole’s worldline between the 0 and t global AdS timeslices. A worldline interpretation of the result
At a sufficiently coarse-grained level,our black hole behaves like a particle, whose propagation in the bulk spacetime followsfrom extremization of its worldline action, i.e. its proper length S worldline [ x µ ( t )] = (cid:90) dτ = (cid:90) t dt (cid:48) L ( x µ ( t ) , ˙ x µ ( t ) | g ) (3.21)which can alternatively be written as a Legendre transform of the worldline energy E [ x µ ( t )]: S worldline [ x µ ( t )] = (cid:90) t dt (cid:18) ˙ x µ δ L δ ˙ x µ − E [ x µ ( t )] (cid:19) (3.22)It is instructive to observe that the two zero mode contributions to V H in eq. (3.15)have different physical interpretations. The zero mode of the CFT Hamiltonian (3.18)measures the the energy of the black hole E [ x µ ( t )], namely the worldline Hamiltonianevaluated on-shell, while the zero mode contribution to (3.17) in the chosen gauge isequal to the quantity ˙ x µ ( t ) δ L δ ˙ x µ along the trajectory. The two are combined in eq. (3.19)to give an amount of modular evolution equal to the on-shell worldline action for ourprobe black hole. Accelerating AdS black holes
The example can be extended to arbitrary acceler-ating black holes. A simple example is a black hole that starts at the AdS origin at t = 0 with rapidity η (0) and at some boundary time t receives a kick that changes itsrapidity, e.g. flips it from η ( t ) to − η ( t ). The black hole returns to the origin at global– 18 –ime t = 2 t when its internal clock is showing τ (2 t ) = 2 tan − t cosh η (0) , according tothe bulk calculation.The modular Wilson line associated to the corresponding family of modular Hamil-tonians can be computed straightforwardly from its defining equations (2.6), (2.7): W (0 , t ) = T e − i (cid:82) t dt (cid:48) D ( t (cid:48) ) = W boosted BH ( π − t , π ) exp (cid:2) iB x ( t ) η ( t ) (cid:3) W boosted BH (0 , t ) (3.23)where W boosted BH is given by (2.17), and the instantaneous boost B x ( t ) = e − iP x ( t ) B e iP x ( t ) accounts for the t = t discontinuity in the operator family K ( t ) due to the kick of theblack hole. This discontinuity is, of course, an artifact of our approximation that wouldbe absent from any realistic accelerating black hole.On the boundary, the local atmosphere fields at t = 0 and t = 2 t are related by φ t = e iBη (0) φ e − iBη (0) (3.24)In view of (3.23), the map V S (0 , t ) = e iBη (0) in (3.24) can be shown to be equal to V S (0 , t ) = W (0 , t ) exp (cid:20) − i (2 π ) − K (0) (cid:90) t dt (cid:48) ˙ x ( t (cid:48) ) sinh η ( t (cid:48) ) (cid:21) (3.25)Extracting the proper time requires computing the Heisenberg picture evolutionoperator (3.15). The zero mode component of the CFT Hamiltonian is once againgiven by (3.18) so the final result reads V H (0 , t ) = exp (cid:20) − i tan − tan t cosh η (0) K (0)2 π (cid:21) (3.26)which agrees with the bulk geometric computation.By an appropriate dense sequence of small kicks like the one studied here, an ar-bitrary worldline can be constructed, allowing our method to correctly compute theproper length of any timelike path in AdS. This construction guarantees that our pre-scription works in all weak curvature perturbations of Anti-de Sitter spacetime. The proper time measured by a bulk observer is a gauge dependent quantity, being afunction of the initial and final points between which the proper length of the worldlineis computed. This fact was reflected in our previous discussion in the choice of the bulkslices Σ t i and Σ t f on which the atmosphere operators are defined. Waiving the needfor the latter requires asking a gauge invariant question.– 19 – igure 4 : LEFT: A black hole in AdS that receives a kick at t . Arbitrary trajectories in AdScan be generated by a dense sequence of such instantaneous kicks, allowing us to describe proper timeevolution in any weakly curved spacetime. RIGHT: Twin black holes. The left twin is static whilethe right twin is the accelerated black hole of the LEFT panel. The time dilation experienced by thetwins is computed by the modular Berry holonomy of the “loop” of modular Hamiltonians describingthe two trajectories and the integral of the zero mode projection of the CFT Hamiltonian along theloop via eq. (3.28), (3.29). In this Section, we are interested in computing the relative time, or time dilation,between two twin observers who follow different paths through spacetime until theymeet at a later boundary time t. Each observer is described in the CFT by a familyof modular Hamiltonians K ( t ) and K ( t ). At their meeting events t i = 0 and t f = t ,the two black holes are near each other so their local atmosphere operator sets S , and S , t are related by simple unitaries U (0) and U ( t ) respectively (fig. 4), whichwe assume known.Working in the Schrodinger picture, the operators S t at the final meeting time canbe obtained from S via the map (3.10), in two different ways, depending on whetherwe propagate them along the worldline of the first or the second twin. The two pathsare distinguished quantum mechanically by whether V S (0 , t ) in (3.12) is constructedfrom the modular Wilson line for the family K ( t ) or from the Wilson line of K ( t )with the appropriate inclusion of U (0), U ( t ). Equivalence of these two proceduresimplies that the two modular Wilson lines satisfy: V (1) S (0 , t ) = U † ( t ) V (2) S (0 , t ) U (0) ⇒ W (0 , t ) e − i (cid:80) a c ( t ) Q a (0) = U † ( t ) W (0 , t ) U (0) e − i (cid:80) a c ( t ) Q a (0) ⇒ U † (0) W † U ( t ) W = e − i (cid:80) a c ( t ) Q a (0) e i (cid:80) a c ( t ) Q a (0) (3.27)where Q a (0) are the zero modes of K (0) and, in the second line, we used the factthat Q a (0) = U (0) Q a (0) U † (0). The two families of modular Hamiltonians in thisproblem, together with the unitaries that relate them at the initial and final moments,– 20 –orm a closed operator “loop”, therefore, the L.H.S of eq. (3.27) is an example of amodular Berry holonomy W loop discussed in Section 2.2.According to our proposal, each observer’s proper time is the coefficient of themodular Hamiltonian in the evolution operators V (1) H (0 , t ), V (2) H (0 , t ) given by eq. (3.15).To measure the time dilation between the two observers we have to look at the coefficientof K in the operator U † (0) V (2) H U (0) V (1) † H which by virtue of (3.15) and (3.27) becomes: U † (0) V (2) H U (0) V (1) † H = exp (cid:20) i (cid:90) t dt (cid:48) U † (0) e iHt (cid:48) P (2) t (cid:48) [ H ] e − iHt (cid:48) U (0) (cid:21) W loop exp (cid:20) − i (cid:90) t dt (cid:48) e iHt (cid:48) P (1) t (cid:48) [ H ] e − iHt (cid:48) (cid:21) (3.28)The result (3.28) is a unitary operator generated by modular zero modes of K (0)that depends only on the CFT Hamiltonian and an intrinsic property of the two blackholes: the families of modular Hamiltonians K ( t ), K ( t ) describing the time evolutionof their state and the relation of their instantaneous frames at their meeting points U (0) , U ( t ). As per our proposal in Section 3.1, the proper time is identified with thecoefficient of the modular Hamiltonian in the modular eigenoperator decomposition of − i log (cid:104) U † (0) V (2) H U (0) V (1) † H (cid:105) = (2 π ) − ∆ τ K (0) + (cid:88) a c (cid:48) a Q a (0) (3.29) Exercise
The reader is encouraged to use the technology explained in Section 3.2 tocompute the left hand side of (3.29) for the twin black holes of fig. 4 and confirm that∆ τ yields the correct time dilation. Up to this point, our black hole was guaranteed an undisturbed journey: no particleswere allowed to cross its path. Under this condition, we argued, modular flow of itsatmosphere operators amounts to proper time evolution along the worldline of the blackhole, in the classical background it lives in. This ceases to be true in the presence ofinfalling excitations, since the atmosphere is defined relative to the apparent horizon,which becomes shifted (fig. 6) with respect to the extremal surface when particles getabsorbed.In this Section we explain that in order to describe proper time evolution ofthe atmosphere fields, modular flow needs to be corrected by a modular scramblingmode G π contribution: an operator that exponentially grows under modular flow– 21 – iKτ G π e − iKτ = e πτ G π with an exponent that saturates the modular chaos bound of[9, 23]. This physically describes the null shift of the causal horizon of the final blackhole relative to the extremal surface. Its coefficient measures the infalling null energyflux at the horizon. This establishes our advertised formula (1.1): proper time andinfalling energy distribution can be extracted from the unitary relating the initial andfinal atmosphere operators, by expanding it in the modular eigenoperator basis. Suppose we make a boundary perturbation to a static AdS black hole, so that someparticles later fall in. The state of the Universe is then | Ψ J (cid:105) = U J | T F D (cid:105) = Z − / (cid:88) E e − βE/ U J | E (cid:105) sys | E (cid:105) ref (4.1)where U J = e − i (cid:80) i (cid:82) J i (Ω ,r ) φ i ( r, Ω ,t =0) inserts the small perturbation of the supergravityfields φ i , with i an abstract flavor index, on an initial bulk Cauchy slice Σ . We alsoassume that the perturbation is introduced far from our probe black hole so that U J isinitially spacelike separated from the “lab”, the operators within a radius (cid:96) from theblack hole [ U J , φ ( ρ, Ω)] = 0 , for: 0 < ρ < (cid:96) (4.2)The absorption of the perturbative particle, of course, does not affect the proper lengthof the black hole’s worldline at leading order in 1 /N , which in this case coincides withthe global time separation of the worldline’s endpoints τ = ∆ t .In order to understand this example in our formalism, we start by choosing twotimeslices Σ and Σ t , where we assume that on Σ t the U J excitation has alreadybeen absorbed by the black hole, namely that it has reached the stretched horizonin Schwarzschild frame. The absorption causes the black hole to grow, resulting in asmall perturbation in the near horizon metric at Σ t .The local atmosphere fields are gravitationally dressed to the local horizon , asexplained in Section 2.3, with time set from the boundary by the slice Σ. This meansthat the operator φ ( ρ, Ω) inserts a particle on Σ at a particular distance ρ from thehorizon, when acting on a CFT state dual to the original black hole geometry | Ψ J (cid:105) or small fluctuations about it. Since the metric on Σ t is only perturbatively differentfrom that of Σ (since now the black hole is assumed to remain stationary at the centerof AdS), the Schrodinger picture atmosphere operators at the final slice φ t will be the same as φ : Acting with φ t ( ρ, Ω) = φ ( ρ, Ω) on e − iHt | Ψ J (cid:105) introduces an excitation atthe same distance ρ from the new local horizon. Switching to the Heisenberg picture– 22 – igure 5 : Free field vs shock contributions to the modular flow of a local “atmosphere” operator φ in the state (4.1) we then have: φ tH ( ρ, Ω) = e iHt φ ( ρ, Ω) e − iHt (4.3)Proper time evolution V H (0 , t ) is generated by the CFT Hamiltonian in this case.According to our proposal, to read off the proper time we need to express V H (0 , t )in terms of modular flow. The modular Hamiltonian for our system, after tracing outthe reference, reads: K J = 2 πU J HU † J (4.4)and the corresponding evolution of the atmosphere fields gives φ K J ( t ) = e i π K J t φ e − i π K J t = (cid:40) φ tH ∀ t : [ φ tH , U J ] = 0 U J φ tH U † J ∀ t : [ φ tH , U J ] (cid:54) = 0 (4.5)At sufficiently small t modular and time evolutions coincide, so our prescription worksas in Section 3.2. It fails, however, once time evolution inevitably moves φ tH insidethe lightcone of U J , after which modular flow and proper time flow of φ differ by U J [ φ tH , U † J ]. Understanding this commutator is the goal of this Section. At leadingorder in N there are two contributions of interest: The free field contribution and theShapiro delays due to the highly blueshifted infalling particles near the horizon. Wediscuss them in order. The free field contribution
At leading order in N , the bulk theory is a free QFTon a semi-classical geometry. In this approximation, φ tH inserted at time t can be– 23 –xpressed in terms of t = 0 fields by usual causal propagation φ tH ( x ) = (cid:90) dy (cid:0) ∂ t G ret ( x, t | y, φ ( y ) + G ret ( x, t | y, π ( y ) (cid:1) (4.6)where ( φ, π ) a symplectic pair of QFT degrees of freedom. The commutator of interest,in the free field approximation, becomes: U J (cid:104) φ tH ( x ) , U † J (cid:105) (cid:12)(cid:12)(cid:12) free = − i (cid:90) dy G ret ( x, t | y, U J δδφ ( y ) U † J = (cid:90) dy G ret ( x, t | y, J ( y )(4.7)= −(cid:104) Ψ J | φ tH ( x ) | Ψ J (cid:105) (4.8)In contrast to the geometric proper time evolution, modular flow removes the expec-tation value that φ acquires in the reference state. This is a version of the “frozenvacuum” problem, inherent in many entanglement based approaches to bulk recon-struction. The operators of interest to us are located near a black hole horizon sothe relevant G ret is controlled by the quasi-normal modes and decays exponentially inproper time (cid:104) Ψ J | φ tH | Ψ J (cid:105) ∼ e − t (4.9)after crossing the future lightcone of U J . With the assumption that our chosen finalmoment is at least a few thermal times later than the last infalling quantum, we cansafely neglect this contribution to modular flow.It is, of course, possible to consider more general bulk QFT excitations, for example: U (cid:48) J = exp (cid:20) i (cid:90) dx dx J ( x , x ) φ ( x ) φ ( x ) + . . . (cid:21) (4.10)The free field contribution (4.8) follows from the same reasoning and yields the non-local operator U (cid:48) J (cid:104) φ tH ( x ) , U (cid:48) † J (cid:105) (cid:12)(cid:12)(cid:12) free = (cid:90) dy dy G ret ( x, t | y , J ( y , y ) φ ( y ) + . . . (4.11)The important observation is that, once again, these operator contributions are ex-ponentially decaying in time, after crossing the lightcone of U (cid:48) J , due to the retardedpropagator contribution to the smearing function U (cid:48) J (cid:104) φ tH ( x ) , U (cid:48) † J (cid:105) (cid:12)(cid:12)(cid:12) free ∼ e − t (4.12)These non-local contributions, therefore, also become negligible, by assuming enoughproper time separation between Σ t and the last infalling particle.– 24 – he shock contribution In the absence of a black hole, the free field result (4.8)would have been the dominant contribution to the commutator, since all higher ordercorrections coming from interactions would be suppressed by powers of 1 /N . Thelarge redshift of the near horizon metric, however, accelerates the infalling quantumexponentially as it approaches r BH . This exponential increase of its energy in the localSchwarzschild frame competes with the G suppression of gravitational interactions andresults in a non-trivial change in the propagation of the atmosphere operators [3].This gravitational intuition is reflected quantum mechanically in the observationthat the overlap of the state φ K J ( t, r, Ω) | Ψ J (cid:105) with φ tH ( r, Ω) | Ψ J (cid:105) is, in this case, equalto a familiar, out-of-time-order bulk correlation function [24] in the thermofield doublestate: (cid:104) Ψ J | φ tH ( r, Ω) φ K J ( t, r, Ω) | Ψ J (cid:105) = (cid:104) U † J φ tH ( r, Ω) U J φ tH ( r, Ω) (cid:105) T F D (4.13)In theories of gravity, for t sufficiently large, the scattering of φ tH and the infallingparticle U J takes place very close to the horizon. Due to the near horizon geometry, theinfalling particle’s null energy is exponentially blueshifted in the frame of the particle φ tH , (cid:104) P + (cid:105) = e t δE where δE ∼ O (1) is the null energy of U J in the t = 0 frame,when the excitation was introduced. The effect of such a blueshifted infalling particleon the propagation of φ tH can be approximated by a null shockwave with some spatialdistribution along the transverse directions Ω, which results in a null translation of φ H [25, 26]: (cid:104) Ψ J | φ tH φ K J ( t ) | Ψ J (cid:105) ≈ (cid:104) φ tH exp (cid:20) − i (cid:90) d Ω ∆ x − ( t, Ω) P − (Ω) (cid:21) φ tH (cid:105) T F D (4.14)where: ∆ x − ( t, Ω) = (cid:90) d Ω (cid:48) f (Ω , Ω (cid:48) ) (cid:104) U † J P + (Ω (cid:48) ) U J (cid:105) T F D (4.15) P ± (Ω) = (cid:90) dx ± T bulk ±± (Ω , x ∓ = 0) (4.16)∆ x − ( t, Ω) is the Shapiro time delay caused by the infalling U J which grows as e t for1 (cid:28) t (cid:28) log N , and the smearing function G (Ω , Ω (cid:48) ) is a transverse propagator alongthe horizon, satisfying ( ∇ − f (Ω , Ω (cid:48) ) = − πδ (Ω , Ω (cid:48) ). We have assumed here thatthe perturbation U J results in a semi-classical spacetime, so that ∆ x − can be replacedby its expectation value at leading order.The exponentially growing Shapiro delay results in the exponential decay of theoverlap (4.14) and the states φ K J ( t ) | Ψ J (cid:105) , φ tH | Ψ J (cid:105) become nearly orthogonal after thescrambling time. This implies that modular evolution is not a good approximationto the geometric proper time evolution when there is infalling energy. Nevertheless,eq. (4.14) shows how to fix this. Consider the operators G π (Ω) = U J P − (Ω) U † J which– 25 –bey: [ K J , G π (Ω)] = − πi G π (Ω) (4.17) G π were called modular scrambling modes in [9] and are discussed further in Section4.2. It straightforwardly follows from (4.14) that (cid:104) Ψ J | φ tH e i (cid:82) d Ω ∆ x − ( t, Ω) G π (Ω) φ K J ( t ) e − i (cid:82) d Ω ∆ x − ( t, Ω) G π (Ω) | Ψ J (cid:105) ≈ φ so that the state φ | Ψ J (cid:105) is normalized to 1. The result
Our observation (4.18) illustrates that proper time evolution of the “lab”degrees of freedom φ continues to be related to modular flow, at leading order in N ,but the two no longer coincide; modular evolution needs to supplemented by scramblingmode contributions to account for the infalling particle’s backreaction on the relativelocation of the atmosphere and the extremal surface: φ tH ≈ (cid:40) φ K J ( t ) ∀ t : [ φ tH , U J ] = 0 e i (cid:82) d Ω∆ x − ( t, Ω) G π (Ω) φ K J ( t ) e − i (cid:82) d Ω∆ x − ( t, Ω) G π (Ω) + O (cid:0) e − t , N (cid:1) ∀ t : [ φ tH , U J ] (cid:54) = 0(4.19)We can now combine the scrambling mode and modular flows above, using theBaker-Campbell-Hausdorff relation, the commutator (4.17), the Shapiro delay (4.15)and the fact that (cid:104) Ψ J | P + (Ω) | Ψ J (cid:105) = δE (Ω) e t where δE (Ω) is the local averaged nullenergy at the horizon in the frame of the t = 0 timeslice Σ , to obtain φ tH = V H (0 , t ) φ V † H (0 , t ) V H (0 , t ) ≈ exp (cid:20) i π (cid:90) δE (Ω (cid:48) ) f (Ω , Ω (cid:48) ) G π (Ω) t + i π K J t + O (cid:0) e − t , N − (cid:1)(cid:21) (4.20)Eq. (4.20) is an example of our general claim (1.1) advertised in the introduction:Proper time evolution along the worldline of our black hole V H can be organized interms of operators of definite modular weight, with the coefficient of the modularHamiltonian measuring proper time and the coefficient of the scrambling mode G π measuring the infalling null energy distribution at the horizon. Moving Black Holes.
The generalization of the result (4.20) to the Section 3.2scenario of black holes in a general semi-classical asymptotically AdS spacetime isstraightforward. For example, starting with the state for the boosted black hole inempty AdS and exciting infalling bulk QFT modes as before we get | Ψ J,η (cid:105) = U J e − iBη | T F D (cid:105) (4.21)– 26 –here B is the generator of the boost symmetry of AdS, giving our black hole rapidity η , while U J = e − i (cid:80) i (cid:82) J i (Ω ,r ) φ i ( r, Ω ,t =0) .The CFT representation of the Heisenberg picture atmosphere operators on theinitial ( t i = 0) and final ( t f = t ) Cauchy slices are given by (2.22) (2.23), with φ static being the HKLL formula for a local bulk field in a static black hole background, witha perturbative gravitational dressing to the local horizon: φ H = e − iBη φ static ( r, Ω) e iBη (4.22) φ tH = e iHt e − iP x ( t ) e − iBη ( t ) φ static ( r, Ω) e iBη ( t ) e iP x ( t ) e − iHt (4.23)The functions x ( t ) , η ( t ) , τ ( t ) describe the location, momentum and proper time of theblack hole in the AdS background.Following the previous reasoning, it can be shown that the proper time evolutioncan be expressed in terms of a flow generated by modular eigenoperators as: V H (0 , t ) = exp (cid:20) i π (cid:90) δE (Ω) f (Ω , Ω (cid:48) ) G π (Ω (cid:48) ) τ ( t ) + i π K J,η τ ( t ) + O (cid:0) e − t , N − (cid:1)(cid:21) (4.24)where: K J,η = 2 π U J e − iBη H e iBη U † J δE (Ω) = (cid:104) Ψ J,η | e − iBη P + (Ω) e iBη | Ψ J,η (cid:105) τ ( t ) = tan − tan t cosh η [ K J,η , G π ] = − πiG π (4.25)This is another illustration of our main claim, where τ ( t ) is the proper length of theblack hole’s trajectory and δE (Ω) is the average null energy crossing the causal horizonat angle Ω in the frame of the initial Σ bulk timeslice. Let us now place the results of Section 4.1 within a more general framework. It wasargued in [9, 23] that analyticity properties of general QFT modular Hamiltonians, K , imply an upper bound on the modular weight of δK , where the latter denotes thefirst order perturbation of K due to a state excitation or an infinitesimal change ofthe subalgebra of interest. This bound on modular chaos can be articulated as thecondition: lim (cid:28)| τ |(cid:28) log N (cid:12)(cid:12)(cid:12) ddτ log |(cid:104) O | e iK Ψ τ δK e − iK Ψ τ | O (cid:105)| (cid:12)(cid:12)(cid:12) ≤ π , ∀ | O (cid:105) = O | Ψ (cid:105) (4.26)– 27 – igure 6 : LEFT: Two null separated Rindler wedges in Minkowski space and the correspondingvacuum state modular flows, generating boosts about the boundary of the corresponding wedge. Sat-uration of modular chaos is manifested in the exponential deviation of the two trajectories. RIGHT:Backreaction of a black hole spacetime due to an infalling particle and the comparison between theaction of the modular flowed operator φ K J and the proper time evolved operator φ tH on the state | Ψ J (cid:105) .The former preserves the distance of the excitation from the RT surface whereas the later preserversthe distance from the local horizon, up to exponentially decaying corrections. The exponential devia-tion of the two trajectories at late times reflects the Shapiro shift of the location of the horizon whichis manifested quantum mechanically in the saturation of the modular chaos bound—in direct analogyto the physical interpretation of maximal modular chaos in the LEFT panel. where the O operators act within the bulk code subspace. The modular scramblingmodes are operators that saturate this bound as τ → ±∞ and can be extracted from δK formally as: G ± = ± π lim τ →±∞ e − π | τ | e iK Ψ τ δK e − iK Ψ τ (4.27)Note that the limit here should be taken after the large N limit, so that τ remains lessthan or of order the scrambling time.The prototypical example of maximal modular chaos involves two Rindler wedgesin Minkowski space, with their entanglement surfaces being separated by a null defor-mation (fig. 6). The modular Hamiltonians for two wedges in the vacuum state, whichequal Rindler boost generators about the two entanglement surfaces, form the Poincarealgebra [27, 28]: [ K , K ] = 2 πi ( K − K ) (4.28)Eq. (4.28) continues to hold for arbitrary null deformations of the Rindler wedge andthe corresponding subalgebras are said to form a modular inclusion . The operator– 28 – π = K − K saturates the bound (4.26) and can be shown to generate a locationdependent null shift at the entangling surface.Maximal modular chaos, therefore, reflects the geometric structure of the QFTbackground: Saturation of (4.26) for τ → ±∞ is a diagnostic of the inclusion propertiesof spatial subalgebras, and the corresponding scrambling modes (4.27) encode the localPoincare algebra near the region’s edge. This motivated [9] to propose the use ofmodular chaos in holography, where the structure bulk spacetime is not a priori known,as a principle for extracting the local Poincare algebra, directly from the CFT. Theresults of the previous Section can be understood in this framework, as we now explain. Maximal modular chaos from infalling particles
The two protagonists of thispaper have been the CFT modular flow, e iKτ , and the proper time evolution of theatmosphere fields along the black hole worldline, V H . In absence of infalling energy inthe state | Ψ (cid:105) the two were argued to coincide. Infalling particles whose backreactionaway from the probe black hole can be neglected are included by acting with a unitary W , so that our state | Ψ (cid:105) = W | ˜Ψ (cid:105) where | ˜Ψ (cid:105) describes the same spacetime without anyparticles that fall into the probe, the proper time evolution is given by the modularHamiltonian ˜ K associated to | ˜Ψ (cid:105) . The bound (4.26), therefore, applies to the differencebetween modular and proper time Hamiltonians.Our results (4.20), (4.24) show that a state excitation that introduces an amountof infalling energy flux through H + , leads to a modular Hamiltonian perturbation thatsaturates (4.26). This guarantees that no operators with higher modular weight canappear in the modular eigenoperator expansion of log V H . The bound (4.26) is saturatedin theories in which the bulk dual is Einstein gravity, in the sense that there is a largehigher spin gap. Then the Averaged Null Energy distribution at the horizon can, then,be extracted from log V H by taking the limit12 π (cid:90) d Ω d Ω (cid:48) δE (Ω (cid:48) ) f (Ω (cid:48) , Ω) G π (Ω) τ ( t ) = lim s → + ∞ e − π | s | e iKs i log V H (0 , t ) e − iKs (4.29)Conversely, the vanishing of the R.H.S. of eq. (4.29) signifies that no particles crossedthe horizon.The physical interpretation of (4.29) is very analogous to the Rindler example ofmaximal modular chaos above. The original modular flow, e iKt , continues to boostthe atmosphere fields about the Ryu-Takayanagi surface even past the shockwaveand (at least) until the scrambling time τ (cid:46) log S BH —up to exponentially decay-ing corrections— whereas proper time evolution V H = e i ˜ Kt preserves the location ofthe fields in the local AdS-Schwarzschild frame which gets non-trivially shifted in thenull direction after crossing the shock (fig. 6). The geometric action of the two modular– 29 –ows, K and ˜ K , is reminiscent of the case of included algebras in flat space and thesaturation of the bound (4.26) is a manifestation of this inclusion property [29], with G π implementing the relevant null shift. The additional feature of the present case isthat the null separation of the two “included wedges” is given by the, appropriatelysmeared, null energy of the absorbed particles. We may now return to the opening question of Section 1: The experience of an observerfalling into a black hole which we will take to be an eternal, two sided AdS blackhole. This bulk configuration is described holographically by two decoupled conformalCFT L × CFT R in the highly entangled, thermofield double state.As first emphasized by Marolf and Wall [13] in the early days of the firewall debates,this setup presents us with a conceptual puzzle: Entanglement wedge reconstructionallows us to introduce an observer somewhere in the right black hole exterior by actingonly with CFT R operators. Since the two CFTs are decoupled, we are guaranteed thatthe observer is composed by CFT R degrees of freedom for the entire boundary time evolution and, thus commutes with all CFT L operators. On the other hand, ER=EPRsuggests that the bulk dual to | T F D (cid:105) is an Einstein-Rosen bridge with a smooth interiorgeometry. The bulk observer’s trajectory crosses the right black hole horizon at finite proper time and after horizon crossing, the observer can receive signals sent from theleft exterior which implies that its degrees of freedom do not commute with the CFT L operator algebra. Proper time evolution must, therefore, couple the two CFTs, despitethe absence of a microscopic dynamical coupling! This seemingly bizarre conclusionappears to suggest that either the two decoupled CFTs in | T F D (cid:105) cannot predict theexperience of the infalling observer beyond the horizon without further specifying somecoupling between the two sides [31, 32], or that the | T F D (cid:105) does not actually describea connected geometry and our observer’s experience can be reconstructed entirely fromCFT R , while their detection of particles coming from the left is merely a mirage.The puzzle is resolved quite elegantly in our framework. The right bulk observeris introduced in our setup by thermally entangling a subset of the CFT R degrees offreedom with an external reference and collapsing them into a black hole, somewherenear the right asymptotic boundary. Our proposal says that, as long as nothing fallsin our probe black hole, proper time evolution of the atmosphere operators, even pastthe horizon, is generated by the Left-Right system’s modular Hamiltonian, obtainedby tracing out the reference. Insofar as ρ RL = Tr ref (cid:2) | ψ (cid:105) L,R,ref L,R,ref (cid:104) ψ | (cid:3) is not Left-– 30 – igure 7 : The modular Hamiltonian K of the probe black hole propagates the local atmosphereoperators in proper time along its worldline, even past the Rindler horizon. K can be expressed as alinear combination (5.1) of the two (decoupled) Rindler Hamiltonians B L and B R which preserve theleft and right wedge, respectively, and the ANEC operators P ± that shift the RT surface along nulldirections and, therefore, mix the two operator algebras, resolving the Marolf-Wall puzzle. Right separable, it is clear that modular flow ρ iτLR will generically mix the Left andRight algebras, thus naturally evading the puzzle. In other words, for states that aresufficiently entangled to describe a short bulk wormhole connecting the two exteriors,entangling the reference with degrees of freedom of CFT R , necessarily entangles it withCFT L as well, and the observer’s modular flow couples the two sides, allowing propertime evolution to access the common interior.For a simple illustration of the resolution, consider the Rindler decomposition ofAnti-de Sitter spacetime, where we introduce our probe black hole inside the Rightwedge and near the asymptotic boundary ρ → ∞ , far away from the Rindler horizon(fig. 7). Upon tracing out the reference system, the modular Hamiltonian of our systemat t = 0 reads: K (0) = e − iP ρ He iP ρ ∼ e ρ B L − B R + P + + P − ) (5.1) B L and B R are the Rindler Hamiltonians of the left and right Rindler wedges, respec-tively. These generate automorphisms of the corresponding algebras and, consequently,do not mix the two sides of the hyperbolic black hole. In contrast, P + and P − are theANE operators along the two Rindler horizons which are related to the global Hamli-– 31 –onian H and AdS translation isometry P by H = P + + P − and P = P + − P − . Theygenerate null shifts of the bifurcation surface, resulting in a flow that mixes the leftand right algebras.In particular, evolution of the atmosphere operators φ in proper time would, ac-cording to our formalism, correspond to modular flow: φ K ( τ ) = e iKτ φe − iKτ = e iP − e τ e iBτ φe − iBτ e − iP − e τ + O ( e − τ )= e iP − e τ φ B ( τ ) e − iP − e τ + O ( e − τ ) (5.2)The key thing to observe is the appearance of the exponentially growing null shift P − which will translate the Rindler evolved field φ B ( τ ) past the Rindler horizon after a finite proper time τ .By analogy, we hypothesize that, in the background of an eternal black hole, evolv-ing operators in the right asymptotic region by the modular Hamiltonian of an infallingobserver introduced in CFT R , as in fig. 7, schematically reads, at sufficiently late propertime τ : φ K ( τ ) = e iKτ φe − iKτ ∼ e iP − e τ φ H ( τ ) e − iP − e τ + O ( e − τ ) (5.3)where the exponentially growing, ANE operator contributions P ± to the observer’smodular Hamiltonian K = − log ρ LR will appear in the form of left/right operatorproducts O L O R , as in [30]. Such products are expected to appear due to the entan-glement of the two CFTs. Preliminary calculations of the modular Hamiltonian of aninfalling observer in an SYK setup similar to [33] confirm this hypothesis for AdS black holes, where the ANE operator contributions to K appear in the form of the sizeoperator of [35]. Our prescription, as outlined above, is an explicit method for reconstructing the op-erators in the black hole interior. Importantly, it is also extremely simple, utilizingonly the CFT dual of the atmosphere operators on the initial slice and the modularHamiltonian K = − log ρ LR . Its relation to the Papadodimas-Raju proposal for theblack hole interior [34] will be discussed in related upcoming work [36].However, this prescription, as it currently stands, suffers from the “frozen vacuum”problem [37]: We can use the modular flowed operators (5.3) to create particles in theblack hole interior, or detect excitations of the initial state we used to construct themodular Hamiltonian, but we cannot measure excitations already present in the initialstate. This is a consequence of the fact that modular evolution of an operator preservesits expectation value in the given state and is therefore blind to the causal effect thatoriginally spacelike separated excitations can have on the operator at later times.– 32 –n our particular setup, we were able to evade the frozen vacuum problem byassuming knowledge of the local atmosphere operators on the final timeslice. Thecomparison of the initial and final operators in the CFT allowed us to reconstructour black hole’s “history”, including whether it encountered any energy on its path.Such knowledge of the final operators, however, can not reasonably be assumed for anobserver that falls inside a black hole. Resolution of the frozen vacuum in this caseseems to require some new conceptual element. In the main body of this work we considered fairly large probe black holes, with horizonradii of the order of the AdS scale L AdS . These black holes are simplest to describebecause they dominate the canonical ensemble and, therefore, thermally entanglingthem with a reference is described by the “canonical” thermofield double (2.1) betweenthe system and the reference CFTs. The resulting system modular Hamiltonian, then,reads K = U sys H CF T U † sys which has a relatively simple action. The price we pay withthis simplification is that we can only probe features of the AdS universe at cosmologicalscales.In order to probe the bulk geometry at sub-AdS scales, we need smaller black holes.Black holes with R H < L AdS cannot be described by the canonical ensemble, due totheir negative specific heat. There exists, however, a parametrically large window ofsmaller than L AdS black holes that dominate the microcanonical ensemble [38]. Thiscan be seen by a back-of-the-envelope calculation. The thermodynamic behavior ofsmall black holes in
AdS can be approximated by that of their flat space cousins. A d + 1 − dimensional black hole with energy E such that R H ∼ (cid:96) d − d − pl E d − < L AdS has anentropy: S BH ∼ (cid:18) R H (cid:96) pl (cid:19) d − ∼ (cid:96) pl ( (cid:96) pl E ) d − d − (5.4)On the other hand, the competing configuration, a thermal gas of supergravity excita-tions of the same total energy, has an entropy that scales like a gas of massless particlesin a box of size L AdS ∗∗ : S gas ∼ ( L AdS E ) dd +1 (5.5)The two configurations exchange dominance when S BH ∼ S gas which happens at energy ∗∗ The same formula also applies when an internal manifold is present, assuming its size is O ( L AdS ).The only difference is that the box along the internal manifold directions has periodic —instead ofreflective— boundary conditions. – 33 – ∼ (cid:96) − pl (cid:16) L AdS (cid:96) pl (cid:17) d ( d − d − , when the black hole radius reaches: R H ∼ (cid:96) pl (cid:18) L AdS (cid:96) pl (cid:19) d d − (5.6)The important observation is that small black holes entropically dominate over a ther-mal gas of the same energy, for horizon radii that are parametrically smaller than L AdS for any dimension d > L AdS /(cid:96) pl → ∞ limit of the ratio R H L AdS ∼ (cid:18) L AdS (cid:96) pl (cid:19) − d d − → withcurvature radius comparable to the Hubble length L AdS ∼ m and Planck length (cid:96) pl ∼ − m. The smallest microcanonically stable black hole has a Schwarzschildradius R H ∼
100 m, comparable to the size of a physics department!As explained in more detail in [38], the microcanonical equilibrium states in theenergy window (cid:96) − pl (cid:18) L AdS (cid:96) pl (cid:19) d ( d − d − (cid:46) E (cid:46) (cid:96) − pl (cid:18) L AdS (cid:96) pl (cid:19) d − (5.8)should be understood as a coexistence phase between small black holes and thermalgas, with most of the total energy stored in the black hole. Due to its negative specificheat, a small black hole in AdS will initially radiate some of its energy but the entropicargument above suggests that it will quickly equilibrate with its thermal atmosphere,as long as we keep the total energy fixed. “Microcanonical” thermofield double The previous thermodynamic argumentsuggests that small probe black holes thermally entangled with a reference can be de-scribed quantum mechanically by the microcanonical version of the thermofield doublestate [15]: | T F D (cid:105) micro = Z − / (cid:88) n e − bE n f ( E n ) | E n (cid:105) sys | E n (cid:105) ref (5.9)where f ( E ) a smooth function of energy that effectively projects the coherent sum ontoa microcanonical window of width σ about a fixed energy E . A simple example of sucha function is a Gaussian f ( E ) ∝ exp [ − ( E − E ) /σ ]. Note that the coefficient b > defined via β = ∂S∂E .The gravitational duals of the microcanonical wormholes (5.9) were studied indetail in [15], where it was shown that the bulk Euclidean path integral preparation of– 34 –his state is dominated by a semi-classical saddle configuration describing a small blackhole, as long as the width of the energy window satisfies:1 (cid:28) σ (cid:28) G − / N (5.10)For energy windows that are too narrow, σ (cid:46) O (1) the uncertainty principle implieslarge quantum fluctuations in the relative time of the two exteriors ∆ t > O (1) so theclocks at the two ends of the wormhole are decohered, and the state does not describea semi-classical wormhole. On the other hand, a wide window effectively takes us backto the canonical ensemble and our small black hole becomes unstable.In order to introduce a small “black hole observer” in a general spacetime wetherefore simply have to replace (2.1) with the analogous unitary excitation of (5.9): | Ψ (cid:105) = Z − / (cid:88) n e − bE n f ( E n ) U sys | E n (cid:105) sys | E n (cid:105) ref (5.11) Code subspace modular Hamiltonian
The central ingredient in our proposalfor describing the proper time propagation of the atmosphere fields was the modularHamiltonian of the system obtained after tracing the reference. More specifically, weonly cared about its projection onto the bulk code subspace S , roughly consistingof excitations with O (1) energy about the background state (5.11). Given that thefunction f ( E ) is approximately constant within an energy window that can scale with N (5.10), K = − log ρ sys on the code subspace acts simply as a unitary rotation ofdynamical CFT Hamiltonian up to G N corrections K ∝ U sys H U † sys + O ( G N ) . (5.12)If the state (5.11) is described by a dual semiclassical black hole geometry with atimelike killing vector near the black hole horizon, the modular Hamiltonian (5.12) willact geometrically within the black hole atmosphere, up to the corrections from infallingparticles discussed in Section 4. In this case, the reasoning of Section 3 goes through,extending the validity of our prescription to small observers and offering a useful toolfor exploring sub-AdS locality.However, there is a subtlety with the assumption that the reduced state obtainedfrom (5.11) describes a semiclassical black hole. The problem can be seen by recall-ing that localized wavepackets in flat space spread out in time in a diffusive fashion∆ r ∼ (cid:113) tm , where m is the particle’s mass. Similarly, the wavefunction of an ini-tially localized small black hole will tend to spread over an L AdS sized region in time.The microcanonical equilibrium state obtained from (5.11) by tracing out the refer-ence, therefore, does not describe a single classical geometry but rather an ensemble– 35 – igure 8 : A Wheeler-de Witt patch in AdS corresponding to the CFT state at a fixed boundarytime t CF T = 0. Our observer (red worldline) travels between the two Cauchy slices of the WdW patchlabelled by ˜ t i,f . The proper length of the worldline can be computed via eq. (1.1), even though nodynamical evolution of the full quantum system takes place. of macroscopically distinct spacetimes with the black hole located at different pointswithin an L AdS region. Notice that this is not an issue for the canonical black holeswith R H ≥ L AdS because the gravitational potential preserves the localization of thewavepacket.In order to construct a classical bulk observer, therefore, the simple state (5.11)does not suffice: We need to further “measure” the black hole location, i.e. project thestate onto a localized wavepacket. This could conceivably be done by performing thecorresponding measurement on the reference side, where the black hole lives in an emptyAdS Universe and the localization can be achieved by exploiting the AdS isometries.The resulting state will be only in approximate equilibrium with the corrections set bythe rate of the wavepacket spreading. We leave a detailed exploration of this interestingconstruction of sub-AdS scale observers for future study.
Our proposal serves as a step towards demystifying the internal notion of time in holo-graphic, gravitational systems. The central idea is simple: The observer is a physicalsystem, entangled with the world. Tracing out the observer, endows the rest of the sys-tem with a modular Hamiltonian which defines the time flow in their reference frame,insofar as the observer remains undisturbed —and with corrections of the type dis-cussed in Section 4 for perturbative disturbances. By its very construction, this is an– 36 –nherently relational clock that becomes available due to the quantum entanglementbetween the observer and the environment, adding one more entry to the growing listof gravitational concepts whose roots can be traced to ubiquitous features of quantumsystems [11, 31, 39–44]. The importance of entanglement between the clock and itsenvironment, and of the modular automorphism in particular, in the emergence of timehas been discussed in the past, [14, 45]. Our work descends from the same conceptuallineage.Crucially, our “proper time Hamiltonian” (1.1) does not rely on the existence ofany global notion of time evolution. In our AdS example, the initial and final times-lices Σ ˜ t i , Σ ˜ t f could be chosen to be Cauchy slices in the same Wheeler-de Witt patch,asymptoting to the same CFT time (fig. 8). The construction of the proper time evo-lution V H (˜ t i , ˜ t f ) would follow identical steps to those of Sections 3 and 4, with the onlydifference that the contribution from the zero mode projection of the CFT Hamiltonianin eq. (3.15) would be absent. In addition, we made very limited use of the asymptoticAdS boundary. We may, therefore, be optimistic that our approach can serve as theseed for a more general framework of emergent time in cosmological quantum gravitymodels. Acknowledgments
LL is grateful to Jan de Boer for collaboration on related topics and numerous illumi-nating conversations and to Bartek Czech for discussions and feedback on the draft. Wewould also like to thank Raphael Bousso, Tom Faulkner, Juan Maldacena, Jamie Sully,Lenny Susskind, Erik Verlinde, Herman Verlinde. for discussions. DLJ and LL bothacknowledge the hospitality of the Aspen Center for Theoretical Physics, the “Ams-terdam String Workshop 2019” and the KITP program “Gravitational Holography”,where parts of this work were completed. DLJ is supported in part by DOE AwardDE-SC0019219. LL is supported by the Pappalardo Fellowship.
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