Schwarzschild black hole states and entropies on a nice slice
SSchwarzschild black hole statesand entropies on a nice slice
J. A. Rosabal ∗ Asia Pacific Center for Theoretical Physics, Postech, Pohang 37673, Korea
Abstract
In this work, we define a quantum gravity state on a nice slice. The nice slicesprovide a foliation of spacetime and avoid regions of strong curvature. We explore thetopology and the geometry of the manifold obtained from a nice slice after evolving itin complex time. We compute its associated semiclassical thermodynamics entropyfor a 4d Schwarzschild black hole. Despite the state one can define on a nice slice isnot a global pure state, remarkably, we get a similar result to Hawking’s calculation.In the end, we discuss the entanglement entropy of two segments on a nice slice andcomment on the relation of this work with the replica wormhole calculation. ∗ [email protected] a r X i v : . [ h e p - t h ] N ov ontents − Introduction
Any attempt to describe the black hole (BH) evaporation using a low-energy effectivedescription, such as semiclassical quantum gravity, must be formulated on the niceslices [1–3]. These are Cauchy surfaces that foliate spacetime. On these slices, thehigh energy degrees of freedom decouple from the low energy ones, and thus the lowenergy effective description does not break down.Nowadays, a common question asked in the literature associated to BH evapo-ration is, where is the mistake in the Hawking’s original derivation [4, 5]?. If we cancall it a mistake, which perhaps is too strong an asseveration, his mistake was notto use such a slicing to specify the quantum gravity (QG) state of the BH.The starting point of a quantum calculation is the definition of a quantum state.In QG, a state (here we focus on the state produced by complex time evolution) canbe defined on any three-surface embedded in the four-dimensional spacetime [6]. Thenice slicing of a Schwarzschild BH allows us to define QG state on a particular niceslice and perform some semiclassical calculations.Although the existence of these surfaces have been implicitly assumed in someworks , neither a definition of QG state on them has been presented, nor a calculationof its associated entropies assuming the existence of these slices explicitly, exists inthe literature.At this point, we find it appropriate to clarify that by QG state, we mean thestate of the geometry combined with the state of the matter fields. There have beenseveral remarkable and inspiring works studying only the state of the radiation onthe (fixed) BH geometry using the nice slice foliation, in the context of quantum fieldtheory on curved space [7–11].In this work, we define a new QG state for a Schwarzschild BH on a nice slicefollowing the ideas of Hartle and Hawking [6, 12] for the wave function of the universe.Then, using this state, we compute its associated entropies. Rather Remarkably, weget similar results as of that in [13, 14] for the thermodynamic entropy. Nevertheless,the main and more striking difference with [13, 14] is that on these slices, it isimpossible to define a global pure state for BH’s. A direct consequence of thisimpossibility is that we can not use a wave function to describe this state. Instead,we must use a density matrix to describe the global mixed state on a nice slice, inthe same spirit of [15] and [16].Although we do not include matter in this first proposal, we leave windows opento include it in future works. The advantage of this calculation is that we can trustit until a very late time when studying BH evaporation.It is worth to remark that apparently there is a big problem with the nice sliceswhen considering them in studying BH evaporation; see [3] for a discussion about Quoting [1]:
While it is seldom spelled out, the existence of such a set of surfaces is implicitlyassumed in much of the existing literature on black hole evaporation. < S rad ( t ) ∼ cons × t . Werefer to [17] and references therein for a discussion on the time-dependent entropy ofa BH.This unpleasant fact conflicts with the unitary evolution of quantum mechanics.Quantum mechanically, the entropy should grow until some time t p , called Page time,and then decrease to zero when the evaporation is completed, following the so-calledPage curve [18, 19].One might see this problem as an obstruction to use the nice slices in this setup;however, it is not. Recently there have been remarkable proposals where this problemcan be overcome; for AdS space in two dimensions [20, 21], and for asymptotically flatspace [22–25]. In these works, the nice slicing of a BH has been implicitly assumedtoo. The paper is organized as follows. In section 2, we review the construction ofstate produced by complex time evolution, density matrix, and partition functionin QG. Then we exemplify these constructions presenting the Hawking’s calculationof the Schwarzschild BH thermodynamic entropy. In section 2, we introduce theconcept of nice slice and define the state on a particular one. We explore the complexsections’ topology and geometry defined by evolving a nice slice in complex time. Thiscomplex manifold defines a semiclassical global mixed state. Using it, we computeits associated thermodynamic entropy. In section 5, we introduce the density matrixinterpretation of this state. After this discussion, in section 6 we point out therelation of our work with [20–25], and we make some remarks on the entanglemententropy and replica wormholes on a nice slice. Conclusions are presented in section7. This paper explores the QG groud state defined on a nice slice for a SchwarzschildBH. It is defined by complex time evolution. The nice slice where the state is definedcan be placed anywhere in the Kruskal spacetime, even overlapping the horizons.When a nice slide overlaps the horizons, we can reach null infinity and perform somesemiclassical calculations in this region. It is the region where we can compute, forinstance, the time-dependent entanglement entropy for an evaporating BH.The new geometry we present here does not correspond to a semiclassical globalpure state. It is a global mixed state whose description is supplied by a densitymatrix ρ (cid:2) h + ij , φ +0 ; h − ij , φ − (cid:3) , where ( h + ij , φ +0 ) and ( h − ij , φ − ) are the values of the three-metric and the matter fields on the boundaries of the complex-extended manifold.3hese boundaries correspond to the slice of the Lorentzian space where the state isdefined.When evolving a nice slice in complex time, we find that the metric on it iscomplex, and the topology of the complex-extended manifold resembles a cylinder.In other words, the boundaries of the density matrix are connected by a surface.This fact supports that the state one can define on any nice slice is a global mixedstate. Higher genus topologies can be considered too; however, we do not explorethose geometries here.Remarkably, the semiclassical state described by this geometry leads to a thermo-dynamic entropy which corresponds to the expected one for a two-sided BH, despitethe state is not a global pure state, in contrast to the Hartle Hawking state. We havefollowed similar steps to those in the original Hawking’s derivation for performing allthe calculations.The complex time evolution of a nice slice is not straightforward; the main reasonfor this is that a portion of the nice slice remains fixed inside the horizon. This portiongrows in Lorentzian time but does not evolve forward. For this portion, the complexextension is driven only by the metric’s boundary values on the boundaries of theportions that explicitly depend on time.One of the exciting features this geometry presents is that it intersects theLorentzian space in two surfaces. This feature allows us to split the complex-extendedmanifold in two manifolds. Each of these manifolds have a density matrix associated,and they can be regarded as the building blocks of the original density matrix. Inother words, the density matrix factorizes as ρ (cid:2) h + ij , φ +0 ; h − ij , φ − (cid:3) = (cid:90) D h ij D φ ρ + (cid:2) h + ij , φ +0 ; h ij , φ (cid:3) ρ − (cid:2) h ij , φ ; h − ij , φ − (cid:3) . (1.1)Each of these manifolds represent semiclassical amplitudes from a surface in the pastto a future surface. These two surfaces can overlap the past and future horizon, inwhich case the building blocks of ρ (cid:2) h + ij , φ +0 ; h − ij , φ − (cid:3) can be regarded as S matrices.This paper also discusses the entanglement entropy associated to the semiclas-sical QG state described above. We explain how the replica manifold must be built.Due to some ambiguities in extending the nice slice’s portion that does not evolveforward in Lorentzian time, we find that the density matrix associated with the repli-cated manifold contains contributions from the disconnected as well as connectedgeometries. The concept of replica wormhole naturally arises in this setup.Following a similar logic to that in the construction of the density matrix of theuniverse [15, 16], we construct the most general density matrix we can associate tothe replicated manifolds. It is given by˜ ρ ( n ) = ˜ ρ n disconnected + ˜ ρ n connected , (1.2)4here n is the number of replicas. Within this logic, we argue how, from the definition(1.2), the so-called factorization problem [21] can be avoided and, hence, no ensembleaverage is needed to make the setup consistent. In the end, we argue on how thisconstruction must be extended to account for the proper definition of informationflux at null infinity. In QFT, the state of a system can be specified by giving its wave functional if thestate is pure or its associated density matrix if the state is mixed. A state of interestin QFT is the ground state. It can be defined by a path integral [6], with boundaryon a given spacelike three-surface Σ of spacetime labeled by some time, t = t . Thefour-manifold that defines the state can be obtained by evolving in complex time thethree-surface Σ. Equivalently it can be simply stated as t → t − i τ . After choosingsome boundary conditions on the boundary of the four-manifold, the state can bewritten as Ψ[ φ ( x ) , t ] = (cid:90) D φ ( x, τ )exp (cid:0) iI[ φ ( x, τ )] (cid:1) , (2.1)where I[ φ ( x, t )], is the action of the system. The wave functional Ψ[ φ ( x ) , t ], givesthe amplitude that a particular field configuration φ ( x ), happens to be on the space-like surface t = t . The path integral is over all fields for τ <
0, which match φ ( x ),on the surface τ = 0, (the τ = 0, surface corresponds to the t = t , slice of space-time). Having the state on the t -slice one can evolve the state to a different t -slice.Formally it can be stated asΨ[ φ ( x ) , t ] = exp (cid:2) − i( t − t ) ˆH (cid:3) Ψ[ φ ( x ) , t ] , (2.2)where ˆH is the Hamiltonian operator of the system. Expression (2.2) can be regardedas a formal solution of a Schr¨odinger-like equation for the wave functional Ψ[ φ ( x ) , t ],i ∂ t Ψ[ φ ( x ) , t ] = ˆHΨ[ φ ( x ) , t ] , (2.3)with initial conditions Ψ[ φ ( x ) , t ].In QG, as there is no well-defined measure of the location of a particular spacelikesurface in spacetime, the state’s definition differs from that in QFT. Despite this fact,a state can be defined by complex time evolution. Following [6] one can define a wavefunctional for a state of a gravitational system asΨ[ h ij , φ ] = (cid:90) D g D φ exp (cid:0) iI[ g, φ ] (cid:1) . (2.4)Where, now I[ g, φ ], is the gravitational and the matter action defined over a complexsection of the original space . The integration, in this case, is over all matter fields The usual prescription in QG, as in QFT, is to set a foliation of the space labeled by some time t , and then pick a particular slice Σ which corresponds to t = t , and evolve it in imaginary time,i.e., t → t − i τ , see for instance [26]. φ ( x ), and the induced three-metric h ij , on aboundary Σ that belongs to the real space (Lorentzian manifold). Up to this point,we consider that Σ divides the Lorentzian manifold in two parts.Another important quantity in QFT and QG is the probability P [ h ij , φ ] that aparticular field configuration occurs on Σ. It is defined as P [ h ij , φ ] = Ψ[ h ij , φ ]Ψ ∗ [ h ij , φ ] . (2.5)Combining (2.5) and (2.4) one can regard the probability as a path integral over thefour metrics defined on the manifold resulting from gluing the original four-manifold(with a boundary Σ) that defines Ψ[ h ij , φ ], with another copy of itself, albeit withan opposite orientation. They share the same boundary, and the path integral isobtained by integrating over the field configurations defined on the resulting manifoldwhich match ( h ij , φ ), on ΣThe total probability Z (or the partition function) is given byZ = (cid:90) D h ij D φ P [ h ij , φ ] , (2.6)where the integration is over the values of the fields on Σ. From (2.6) and (2.4) onecan see that the total probability is a path integral over the four metrics definedon the manifold resulting from gluing the original four-manifold with another copy(with opposite orientation) of itself.Z = (cid:90) D g D φ exp (cid:0) iI[ g, φ ] (cid:1) . (2.7)In addition it is possible to define a density matrix ρ (cid:2) h + ij , φ +0 ; h − ij , φ − (cid:3) = Ψ (cid:2) h + ij , φ +0 (cid:3) Ψ ∗ (cid:2) h − ij , φ − (cid:3) . (2.8)As it clearly factorizes, it is associated to a pure state. The diagonal elements givesus the probability as in (2.5), P (cid:2) h ij , φ (cid:3) = ρ (cid:2) h ij , φ ; h ij , φ (cid:3) ; and its trace gives usthe partition function, similarly to (2.6),Z = Tr[ ρ ] = (cid:90) D h ij D φ ρ (cid:2) h ij , φ ; h ij , φ (cid:3) . (2.9)In the previous definition of the density matrix, we first have to perform an in-tegration over two disjoint manifolds. Then, the trace operation glues them togetherover the surface Σ.Soon after this proposal for the gravitational state came out Hawking and Page[15, 16] realized that in principle one can include contributions from geometriesthat connect the boundaries of the density matrix. For recent applications of thisprocedure see [27, 28]. We shall call them connected geometries . In this case, the6tate can not be considered pure, hence the only object available to describe it wouldbe a density matrix of the form ρ (cid:2) h + ij , φ +0 ; h − ij , φ − (cid:3) = (cid:88) m,n C mn Ψ m (cid:2) h + ij , φ +0 (cid:3) Ψ ∗ n (cid:2) h − ij , φ − (cid:3) , (2.10)where C mn does not factorizes i.e., we can not find a basis where the matrix C mn factorizes as C mn = c m c n . In fact, by allowing connected geometries, the boundaryΣ does not need to divide the real space into two parts.As more involved geometries are allowed the density matrix can have more thantwo boundaries on the real space. The trace operation over the non-observableboundaries gives rise to what could be regarded as a reduced density matrix withboundary values on the real space, on the remaining boundary. Like for the purestate, a trace over the remaining boundary (observable boundary) gives us the par-tition function Z of the system. This can be regarded as a path integral over thedisconnected and connected geometries resulting from gluing the boundaries of themanifolds that previusly defined a density matrix with several boundaries [15, 16]Z = Tr (cid:2) ρ (cid:3) = (cid:90) Dh ij (cid:90) Dφ (cid:88) m,n C mn Ψ m (cid:2) h ij , φ (cid:3) Ψ ∗ n (cid:2) h ij , φ (cid:3) = (cid:88) disconnected+ connected (cid:90) D g D φ exp (cid:0) iI[ g, φ ] (cid:1) . (2.11)We end the discussion of the section with the time evolution in quantum gravity.Any state in QG must be a solution to the Wheeler-Dewitt equation [29]. Thisequation states that ˆHΨ[ h ij , φ ] = 0 , (2.12)or ˆH ρ [ h + ij , φ +0 ; h − ij , φ − ] = δ (cid:2) h + ij , h − ij (cid:3) δ (cid:2) φ +0 , φ − (cid:3) . (2.13)It is a Schr¨odinger-like equation for the wave functional, or the density matrix of agravitational system, and ˆH is the gravitational Hamiltonian operator including thematter contribution. This Schr¨odinger-like equation differs enormously from that inQFT (2.3). Note that time does not appear explicitly in the equations above.From (2.12) or (2.13), it is not difficult to see that the unitary evolution ofQFT (2.2) does not apply to quantum gravity. Nevertheless, in the semiclassicalapproximation, when a metric is fixed and a foliation specified, one can recover aSchr¨odinger-like equation for the state of the matter in that particular metric [30]. These are the boundaries which do not correspond to the boundary where we are interested incomputing the observables [15]. .1 Hawking’s calculation As an example of the previous constructions, we review the Hawking’s calculationof the Schwarzschild black hole entropy. For simplicity we focus only on the grav-itational contribution [13, 14], which leads to the famous formula S BH = A4 , whereA = 16 π M , is the area of the horizon.Let us compute the partition function of a gravitational system in vacuum. Ourstarting point is the gravitational stateΨ[ h − ij ] = (cid:90) D g exp (cid:0) iI[ g ] (cid:1) , (2.14)where I[ g ] = (16 π ) − (cid:90) V dx √− g R + (8 π ) − (cid:90) ∂V dx √− h (cid:2) K (cid:3) , (2.15)with (cid:90) ∂V dx √− h (cid:2) K (cid:3) = (cid:90) ∂V dx √− h (cid:0) K − K (cid:1) . (2.16)The boundary term in (2.16), also known as the Gibbons–Hawking–York term, playsa crucial role in finding the entropy of a black hole. To define state we must specify onwhich three-surface Σ we want to define it and the asymptotic behaviour at spatialinfinity (after the complex extension) of the metrics we are integrating over. Forthis case we consider those metrics which are asymptotically flat. To specify Σ it isconvenient but not necesary to specify a folliation of the space.In Kruskal coordinates usually we take a folliation that corresponds to an ob-server at spatial Lorentzian infinity Fig. 1,T = (cid:114) ( r2M − r4M sinh( t
4M ) , X = ± (cid:114) ( r2M − r4M cosh( t
4M ) . (2.17)Coordinates (2.17) cover only the left and right wedges in Fig. 1. Note that so far,we have not specified the metric of the spacetime, only the foliation. Now we pick aparticular spacelike slice. The most popular is T = 0, ( t = 0), where we can definethe so-called Hartle-Hawking (HH) state for a black hole [31]. This state is not purefor the portion of space X ≥
0. However, it is obtained from the density matrixassociated with the global pure state (2.14) (where we can also include the mattercontribution) after tracing over the degrees of freedom on X < = t X T t = = r Figure 1 . Kruskal spacetime and the ( t, r ) foliation of it. After the complex extension , t → t − i τ , T → T, and X → X,T = (cid:114) ( r2M − r4M sinh( t − i τ
4M ) , X = (cid:114) ( r2M − r4M cosh( t − i τ
4M ) . (2.18)Where, the periodicity of the τ direccion, τ ∼ τ + 8 π M, follows from (2.18). Atleast formally, now we can define the state (2.14), where h − ij , is the boundary valueof the path integral on the slice t = 0. This state can be geometrically representedas in Fig. 2. Note that we have used (T , X ), for the real variables in Kruskal X T ψ [ h - ij ] Figure 2 . Geometric representation of the BH state defined on the T = 0 slice. coordinates Fig. 1 and (T , X), for the complex ones of the complexified space (2.18).The (T , X ), variables will be reserved only for the imaginary part, for example T ,in Fig. 2. The space for the particular choice t = 0, is called the Eucliedan section.Note also that the axis X , (T = 0, or T = 0) is common for both, the Lorentzian In this case t = 0, which corrsponds with T = 0. Having defined the state we are in a condition of computing the partition func-tion. For that we define the density matrix ρ [ h + ij , h − ij ] = Ψ[ h + ij ]Ψ[ h − ij ] (we do not takethe complex conjugate because in this case the wave functional is real). It is repre-sented geometrically in Fig. 3. This density matrix factorizes in two wave functionals X T ψ [ h + ij ] ψ [ h - ij ] ρ [ h + ij , h - ij ] = Figure 3 . The two disjoint geometries that geometrically represent the density matrixassociated with the state in Fig. 2. and it is defined through a path integral over two disjoint geometries, which meansthe state is pure. The partition function of the system is given byZ = Tr[ ρ ] = (cid:90) Dh ij ρ [ h ij , h ij ] . (2.19)Geometrically the trace operation on the density matrix amounts to gluing the twosemi-disk in Fig. 3. The partition function can be geometrically represented as inFig. 4, where δ ∞ represents the boundary of the disk geometry. In the end of thecalculation we send it to infinity.As discussed above, after combining (2.19) and (2.14), it is not difficult to seethat the path integration in the partition function (2.19) is over the metrics definedon the resulting manifold in Fig. 4 with flat boundary conditions at spatial infinity.In the semiclassical aproximation we just evaluate the path integral on a classicalsolution g c , extracted from the Einstein’s equations,Z = Tr[ ρ ] = (cid:90) Dh ij ρ [ h ij , h ij ] = (cid:90) D g exp (cid:0) iI[ g ] (cid:1) ∼ exp (cid:0) iI[ g c ] (cid:1) . (2.20) At the end of the conclusions, we comment on a possible issue in this procedure. T δ ∞ Z = Figure 4 . The disk geometry that geometrically represents the partition function associ-ated with the state in Fig. 2.
At this point is where we fix the metric by solving the Einstein’s equation on thedisk Fig. 4. In the (T , X ), coordinates, the vacuum solution takes the formds = 32M r e − r2M (cid:0) dT + dX (cid:1) + r dΩ , (2.21)r = 2M (cid:0) ( X + T e ) (cid:1) . (2.22)Which is the Wick rotated version of the Kruskal metric with r ≥ τ, r), coordinates (2.18)ds = (1 − τ + dr (1 − ) + r dΩ , (2.23)where τ ∼ τ + 8 π M. The peridodicity β = 8 π M, of the τ direcction indicates thesemiclassical state that is described by this geometry is thermal with a temperature T = β − .In the Euclidean section these two set of coordinates (T , X ), and ( τ, r), coverthe same space which correspods to the whole disk geometry. Since it is a vacuumsolution R µν = 0. The only contribution to the action comes from the boundaryterm. It is given byI[ g c ]( β ) = (8 π ) − (cid:90) r=r ∞ →∞ dx √− h (cid:2) K (cid:3) = 4 π iM = i β π , (2.24)then Z( β ) = exp (cid:2) − β π (cid:3) . (2.25)11ow, using S BH = (cid:0) − β∂ β (cid:1) lnZ( β ) (cid:12)(cid:12)(cid:12) β =8 π M , see [13, 14], (2.25) leads to the famousrelation S BH = A4 . (2.26)Instead of choosing the slice t = 0, one could have chosen t (cid:54) = 0. For this case,(2.18) would be complex. One might see this fact as an obstruction for choosing otherslices to define state; however, as we will see in the next section, it is not. In thecase under discussion ( t (cid:54) = 0), it is not needed to perform any further calculation ifwe want to find the partition function on a different t -slice. Using only the rotationsymmetry of the metric (2.21) (boost symmetry for the Kruskal metric on the realspace), one concludes that the partition function and the entropy are invariant undertime translations. Although, the state might differ from the one defined on the slice t = 0. In this section, we shall introduce the concept of nice slices [1–3]. On these slices,the semiclassical QG calculations for an evaporating black hole do not break downuntil a very late time. We shall also define the QG state on a nice slice. Then, usingit, we will compute its associated partition function and thermodynamic entropy.The nice slices are a set of Cauchy surfaces which foliate spacetime. The surfacesavoid regions of strong spacetime curvature (close to singularities) but cut throughthe infalling matter and the outgoing Hawking radiation. Importantly, infallingmatter and the outgoing Hawking radiation should have low energy in the local co-ordinates on each slice. We also require that the slices be smooth everywhere, withsmall extrinsic curvature compared to any microscopic scale. With these require-ments, we ensure that the effective QG description does not break down, and usingthis foliation, we can follow the evaporation of a black hole until a very late time.Conveniently, one can chose slices that agree with slices of constant Schwarzschildtime in the asymptotic region. A particular set of nice slices is depicted in Fig. 5.In Kruskal coordinates we use Schwarzschild time t to parameterize them,Σ − : cosh( t
4M ) T + sinh( t
4M ) X = R ; X < − R sinh( t
4M ) , Σ : X − T = − R ; − R sinh( t
4M ) < X < − R sinh( t
4M ) , Σ + : cosh( t
4M ) T − sinh( t
4M ) X = R ; X > R sinh( t
4M ) . The constant R is assumed to be large by comparison with any microscopic scale,but small enough to keep the slices far from the singularity. Note that Σ (the red12 = = ∞ t = t X T T = R Σ + Σ - Σ T = = r r = Figure 5 . Nice slice foliation of a Schwarzschild BH in Kruskal coordinates. line in Fig. 5) only grows as we evolve forward in Schwarzschild time but it is fixedat a constant r < = 32M r e − r2M (cid:0) − dT + dX (cid:1) + r dΩ , (3.1)where r = 2M (cid:0) ( X − T e ) (cid:1) , (3.2)with W the Lambert function.As in the usual folliation (2.17) of the Schwarzschild space, in the nice slicefoliation we can change from the coordinates (T , X ), to the coordinates ( t, r). Forchanging coordinates, for example, on X < − T = ( r2M − r2M , (3.3)Σ − : cosh( t
4M ) T + sinh( t
4M ) X = R . In the first line of (3.3) we have inverted the relation (3.2). The solution of thissystem of equations is given byT = + ρ sinh( t
4M ) + R cosh( t
4M ) , X = − ρ cosh( t
4M ) − R sinh( t
4M ) , with ρ = (cid:114) ( r2M − r2M + R . (3.4)In the ( t, r), coordinates the metric takes the formds = − (1 − t + 2R (cid:113) ( r2M − r2M + R d t dr+ dr − (1 − R e − r2M ) +r dΩ . (3.5)13ere, 0 ≤ t < ∞ , and r < r < ∞ , where r , in a solution to the equation X − T =( r − r02M = − R , at the boundaries of Σ − and Σ + , i.e.,1 − (1 − R e − r02M ) = 0 = ⇒ r = 2M (cid:0) ( − R e ) (cid:1) < . (3.6)The metric (3.5) can be rewritten in a more suggestive form, making manisfestthe foliation and the canonical structure of this geometryds = − N d t + h ab (dx a + V a d t )(dx b + V b d t ) . (3.7)In the ADM form (3.7), [32–34], we have: N = 1 − (1 − R e − r2M ), V = R (cid:0) (cid:1) e − r4M N, h ab = diag(N − , r , r sin ( θ )), and dx a = (dr , d θ, d φ ). Note that the lapse functionN , is non-negative Fig. 6, and N (r ) = 0, see equation (3.6). r r Figure 6 . Lapse function.
Like in the Hawking’s calculation above, now we can pick an slice and performthe complex extension. Picking the slice t = t and extending it, t → t − i τ ,T → T, X → X, yields toT = + ρ sinh( t − i τ
4M ) + R cosh( t − i τ
4M ) , X = − ρ cosh( t − i τ
4M ) − R sinh( t − i τ
4M ) . (3.8)The periodicity of the τ direcction follows from (3.8), τ ∼ τ +8 π M. If we are going toconsider that the geometry we are building describes a semiclassical state, this statewould have a temperature T = β − = π M . Note that X − T = ( r2M − r2M ∈ R ,which implies that r ∈ R . Also, that τ = 0, corresponds to the t = t , slice in Fig.5. For the sake of generality we want to consider t (cid:54) = 0, in (3.8). Note that when t → ∞ , Σ − , and Σ + sit on the horizons, on null infinity, i.e., on I + in thePenrose diagram.
14e can see a clear difference when we compare (3.8) with the Euclidean sectionof the Schwarzschild space (2.18) (recall that in (2.18) t = 0). The section definedin (3.8) is complex. Moreover, the state defined by (3.8) does not lead to the HHstate. As we will see below, now it is more convenient to define a density matrixassociated with a global mixed state to describe it.Finally the metric on the complex sections corresponding to the extension of theslices denoted by Σ − and Σ + isds = 32M r e − r2M (cid:0) − dT + dX (cid:1) + r dΩ , (3.9)with (T , X), defined over the complex surfacesT = + (cid:0) ρ sinh( t − i τ
4M ) + R cosh( t − i τ
4M ) (cid:1) , X = ± (cid:0) ρ cosh( t − i τ
4M ) + R sinh( t − i τ
4M ) (cid:1) , (3.10)where the minus sign in the second line of (3.10) corresponds to the extension of Σ − ,and the plus sign to the extension of Σ + , and r = 2M (cid:0) ( X − T e ) (cid:1) . In the ( τ, r)coordinates the metric takes the formds = N d τ + h ab (dx a − iV a d τ )(dx b − iV b d τ ) , (3.11)which follows directly from (3.7). Note that no subscripts appear in the differentialforms of the metric (3.9). Metrics (3.9) or (3.11) are complex, however this is notan issue in this kind of calculation. Complex metrics have been explored (used) inseveral guises [20, 21, 30, 35]. This section shall study the topology and geometry of the manifolds obtained by thecomplex extension. We shall call them δ − ρ for the extension of Σ − , and δ + ρ for theextension of Σ + .Expression (3.8) defines a 2d surface, δ − ρ : { T = T + iT , X = X + iX } (cid:46) T = +R ( ρ ) cos( τ
4M ) , T = − R ( ρ ) sin( τ
4M ) , X = − R ( ρ ) cos( τ
4M ) , X = +R ( ρ ) sin( τ
4M ) , (4.1)15here R ( ρ ) = ρ sinh( t
4M ) + R cosh( t
4M ) , R ( ρ ) = ρ cosh( t
4M ) + R sinh( t
4M ) . (4.2)We can think about this 2d surface as embedded in C or R . Either way we can seethat the surface has the topology of an annulus. For each constant ρ = ρ , the curve δ − ρ is a circumference on a Clifford torus or on S . The surface has two boundaries,one at r = r , ( ρ = 0), and the other at r = r ∞ → ∞ , ( ρ → ∞ ), note that ρ (r ) = 0. Σ − In R , for a constant ρ = ρ , we can define the torus T ( ρ ) = S × S : (T , X , T , X )= (cid:16) R ( ρ ) cos( θ
4M ) , R ( ρ ) sin( θ
4M ) , − R ( ρ ) sin( θ
4M ) , − R ( ρ ) cos( θ
4M ) (cid:17) . (4.3)Now we pick the curve on the torus, parameterized by τ , θ = θ = τ , we shall call it δ − ρ . To see that each curve δ − ρ , is an S we use the representation of the torus in Fig.7, where the lines of the same color are idenfied. From Fig. 7 it is straightforward θ θ θ = θ = τ Figure 7 . Topological representation of the torus. The red and blue lines are identified.The diagonal dashed line represents a circle on the torus. to see that the dashed diagonal line is indeed a circle. Finally, joining all the circles δ − ρ from each torus T ( ρ ) ( ρ ranges from zero to infinity), we can easily see thatthe resulting surface is exactly δ − ρ , (4.1) or (3.8).16nder similar considerations one can get a 2d surface δ + ρ , from the complexextension of Σ + δ + ρ : (cid:16) R ( ρ ) cos( τ
4M ) , − R ( ρ ) sin( τ
4M ) , − R ( ρ ) sin( τ
4M ) , R ( ρ ) cos( τ
4M ) (cid:17) . (4.4)So far, we can view this space as two disjoint annulus; or as portions of twodisjoint cigar geometries, each one with a boundary at r = r , and the other atinfinity, when t (cid:54) = 0. When t = 0, these two spaces touch each other at r = r ,( ρ = 0), on the boundaries δ − : (cid:16) R cos( τ
4M ) , +R sin( τ
4M ) , , (cid:17) ,δ +0 : (cid:16) R cos( τ
4M ) , − R sin( τ
4M ) , , (cid:17) . Note that δ − ≡ δ +0 , but they have diffrent orientation.The picture so far is: for t = 0, see Fig. 8, while for t (cid:54) = 0, see Fig. 9. It is worthto emphasize that Fig. 8 and Fig. 9 are just 2d representations of two-dimensionalsurfaces embedded in R . Figure 8 . Two-dimensional representation for t = 0 , of the 2d surfaces δ − ρ and δ + ρ emmbeded in four dimensions. No red slice appear in this case. Now we have the task of extending the portion of the t -slice, denoted by Σ (thered line in Fig. 5). For this portion, the extension is less obvious since the variabletime does not appear explicitly on Σ as it does on Σ − and Σ + . We shall call thissurface δ ζ , and its boundaries δ − t , and δ t .To extend Σ , we should note that the only time dependence of these slicesappears at the boundaries. As mentioned, these slices only grow in time, but theyare fixed at r = r . 17 igure 9 . Two-dimensional representation for t (cid:54) = 0 , of the 2d surfaces δ − ρ and δ + ρ , andthe Σ slice emmbeded in four dimensions. First, let us parameterize the Σ silces using a new parameter ζ . On Σ we coulduse the parametrization T = R cosh( ζ ), and X = R sinh( ζ ), with − t ≤ ζ ≤ t .However, we find it more convenient to make the distinction Σ − for X ≤
0, andΣ for X ≥
0, Σ = Σ − ∪ Σ ; and use the following parametrization: on Σ − T = +R cosh( ζ
4M ) , X = − R sinh( ζ
4M ) , ≤ ζ ≤ t ; (4.5)while on Σ T = +R cosh( ζ
4M ) , X = +R sinh( ζ
4M ) , ≤ ζ ≤ t ; (4.6)see Fig. 9. Note the dependence of t , on the boundaries δ − t , and δ t , of Σ .Plugging (4.5) and (4.6) in (3.1) we can find the induced metric on Σ = Σ − ∪ Σ . In the coordinates ( ζ, θ, φ ), it is given byds = 2Mr e − r02M R d ζ + r dΩ . (4.7)18n the other hand, the induced metric at r = r , i.e., on the boundaries δ − and δ +0 ,which would correspond to the boundary of δ t (see Fig. 9) isds = − e − r02M R d τ + r dΩ . (4.8)Although r = r , is a coordinate singularity for the metric in the form (3.11), (4.8)can be obtained directly from (3.11), or more easily from the Wick rotated versionof (3.5) and the relation (3.6).With this in mind we conclude that the extension of Σ will be driven only bythe boundaries values of the metric on δ − t , and δ t , which match the boundariesvalues of the metric on δ − and δ +0 (4.8), respectively. Also, by the condition that at τ = 0, the induced metric of the complex extension matches the induced metric onthe real slice Σ (4.7), see Fig. 9. Therefore, the solutions we are seeking are thosefour-geometries that satisfy the boundary conditions (4.7) and (4.8).The ansatz for the surface δ ζ takes the form: for the extension of Σ − ,T = +R( τ, ζ ) cosh( ζ − i τ
4M ) , X = − R( τ, ζ ) sinh( ζ − i τ
4M ) , ≤ ζ ≤ t , (4.9)while for the extension of Σ ,T = +R( τ, ζ ) cosh( ζ − i τ
4M )X = +R( τ, ζ ) sinh( ζ − i τ
4M ) , ≤ ζ ≤ t , (4.10)where R( τ, ζ ), is a real function and τ ∼ τ + 8 π M.The vacuum solution of the Einsten’s equations on δ ζ has the same form as in(3.9), but (T , X) are defined on the complex surface δ ζ given by (4.9) and (4.10).Plugging (4.9) and (4.10) in (3.9) we get a family of complex metrics, wherer = 2M (cid:16) (cid:0) − R( τ, ζ ) e (cid:1)(cid:17) ∈ R . (4.11)Naively one may think that R( τ, ζ ) = R, is the simplest solution. The obstacle tosuch a choice is that a constant R( τ, ζ ), leads to a non invertible metric.In order to avoid possible metric singularities on δ ζ and, as we necesarilyneed a non constant function R( τ, ζ ), now we have to move the conditions on R tothe function R( τ, ζ ). In other words, we consider only solutions with small (smallenough but not infinitesimal) deviations from the constant value R, i.e., R( τ, ζ ) =R + s ( τ, ζ ) <<
1, with s ( τ, ζ ) ∼ Notice that if R( τ, ζ ) = 1, (4.11) would vanish. Recal that W ( − e − ) = −
1, and r = 0, is asingular point for the metric (3.9).
19. R( τ, ζ ) = R( τ + 8 π M , ζ ) ∈ R ,2. R(0 , ζ ) = R,3. R( τ, t ) = R,4. ∂ ζ R(0 , ζ ) = 0,5. ∂ τ R( τ, t ) = 0.Now we are in a condition to represent the full picture of the geometry of thecomplex extension of a nice slice Fig. 10. Figure 10 . Full picture of the manifold obtained after the complex extension of the t -slice. We have found a family of manifolds that matches continuously with δ − ρ and δ + ρ ,but this is not the end of the story. In order to fully determine the solution we alsohave to impose smoothness at the matching surfaces. For that we must compute theextrinsic curvature defined asK ab = − ( ∂ a γ µb ˆn µ + Γ µνρ ˆn µ γ νa γ ρb ) . (4.12)As we are interested in computing the extrinsic curvature at a constant value of acoordinate, either r = r , or ζ = t , or on the asymptotic boundaries at r = r ∞ , K ab ,reduces to K ab = − Γ µνρ ˆn µ γ νa γ ρb , (4.13)20here γ µ = (1 , , , ,γ µ = (0 , , , ,γ µ = (0 , , , , ˆn µ = (0 , , , (cid:112) (cid:15) g , (4.14)with (cid:15) = ±
1, according to the signature of the metric.In what follows we use the superscripts 0 − , 0+, − and +, in the tensor K ab toindicate on which boundary we are computing the extrinsic curvature according tothe superscripts of δ − t , and δ t , δ − , and δ +0 respectively.First, for consistency, we have checked that at ζ = 0 both spaces (4.9), and(4.10) match smoothly, see Fig. 10K ab (cid:12)(cid:12) ζ =0 = K ab (cid:12)(cid:12) ζ =0 . (4.15)The extrinsic curvature on the boundaries δ − t , and δ t , is given byK − ab (cid:12)(cid:12) ζ = t = K ab (cid:12)(cid:12) ζ = t = sign( ∂ ζ R( τ, t ))(2M − r ) × diag (cid:16) Mr (cid:0) − R ∂ τ ∂ τ R( τ, t ) (cid:1) , r , r sin ( θ ) (cid:17) . (4.16)While on the boundaries δ − and δ +0 , isK − ab (cid:12)(cid:12) r=r0 = K + ab (cid:12)(cid:12) r=r0 = − (2M − r ) diag (cid:16) Mr , r , r sin ( θ ) (cid:17) . (4.17)Requering that on the boundaries δ − t , δ − , and δ t , δ +0 , see Fig. 10, both spacesmatch smoothly K − ab (cid:12)(cid:12) ζ = t = K − ab (cid:12)(cid:12) r=r0 , K ab (cid:12)(cid:12) ζ = t = K + ab (cid:12)(cid:12) r=r0 , (4.18)leads to the extra conditions6. ∂ τ ∂ τ R( τ, t ) = 0,7. ∂ ζ R( τ, t ) ≤
0. 21he functions that satisfy the conditions listed above areR( τ, ζ ) = R + ∞ (cid:88) n =1 a n ( ζ )sin( n τ ) , a n ( t ) = 0; (4.19)with a n ( ζ ), such that ∂ ζ R( τ, t ) ≤
0, holds, and R( τ, ζ ) << (cid:90) r=r ∞ K √ h d x = − π iM(2r ∞ − . (4.20)Therefore, the only contribution to the action (2.15) isI[ g c ]( β ) = 2 × (8 π ) − (cid:90) r=r ∞ →∞ [K] √ h d x = 2 × π iM = i β π , (4.21)where the factor 2 appears because there are two asymptotics boundaries, δ −∞ and δ + ∞ .Like in the Hawking’s calculation in section 2.1, using thermodynamics arguments(4.21) leads to S BH = A . (4.22)We have computed the thermodynamics entropy of a black hole on a nice slice;however, we do not know yet which state leads to such an entropy. Before moving tothe next section, where we discuss the density matrix interpretation of the calculationpresented above, we shall point out another feature of the geometry we have obtained.From (3.8) and (4.19) we can see this geometry intersects the Lorentzian spacein two differents surfaces. The surface τ = 0, which corresponds to the slice t = t ,on the real space, and the surface τ = 4 π M, which corresponds to the T -reflectedslice of t = t , see Fig. 11.Let us stress one more point. In this section, we have considered that the Σ slice extension leads to a manifold that is topologically equivalent to two cylindersjoined at their boundaries, with opposite orientation (similar to Fig. 8 but elongatedin the ζ direction). In principle, we could consider contributions from manifoldswith higher genus topologies. As long as the Σ slice belongs to these manifolds,the definition of state on the Lorentzian space will remain untouched. Withoutconsidering the matter fields, there will not be a semiclassical contribution to theaction coming from these manifolds because they would be solutions of the vacuumEinstein’s equations. However, the situation would be different if matter fields aretaken into account. 22 T τ = τ = τ = π M τ = π M Figure 11 . Intersections of the complex-extended manifold with the real space. The bluelines represent the intersections of the δ − ρ surface with the real space. At the same time,the magenta lines represent the intersections of δ + ρ . For instance, starting from the blueline on the upper left at τ = 0 , and evolving it in complex time up to τ = 4 π M, it reachesthe blue line on the lower right. Similarly, for the red lines.
At this point one might be tempted to define a density matrix ρ (cid:2) h + ij , φ +0 ; h − ij , φ − (cid:3) =Ψ (cid:2) h + ij , φ +0 (cid:3) Ψ ∗ (cid:2) h − ij , φ − (cid:3) , where Ψ ∗ (cid:2) h − ij , φ − (cid:3) and Ψ (cid:2) h + ij , φ +0 (cid:3) are defined on two disjointgeometries with boundary values on the surfaces τ = 0 − , ( h − ij , φ − ); and τ = 8 π M ∼ + , ( h + ij , φ +0 ); and associate it to the geometry above to describe a semiclassical state.The issue is that in this geometry the density matrix does not factorize. To see this,we can just evolve the slice t = t , in imaginary time τ , and note that the slices τ = 0 − , and τ = 8 π M ∼ + , Fig. 12 are connected by a surface .Moreover, this geometry intersects the real space in two differents surfaces Fig.11. In other words, this geometry divides the real space in more than two parts.As discused in [15, 16] the semiclassical state described by the geometry above is amixed state with an associated density matrix of the form ρ (cid:2) h + ij , φ +0 ; h − ij , φ − (cid:3) = (cid:88) m,n C mn Ψ m (cid:2) h + ij , φ +0 (cid:3) Ψ ∗ n (cid:2) h − ij , φ − (cid:3) , (5.1)fulfilling the equation (2.13), where C mn does not factorize, i.e., C mn (cid:54) = c m c n . Wewould like to stress that (5.1) is not the density matrix associated to the ther-mofield double (TFD) of the HH state [36]. The wave functionals Ψ m (cid:2) h + ij , φ +0 (cid:3) , andΨ ∗ n (cid:2) h − ij , φ − (cid:3) , are defined on the whole nice slice t = t , Fig. 5, and not only onhalf of the space as in the TFD. The state associated to (5.1) is not pure, but yetafter tracing over the boundary values on the t -slice we get the expected entropy, Contrast with Fig. 3. igure 12 . Representation of the complex-extended geometry that connects the boundary τ = 0 − , with the boundary τ = 0 + . as shown in the previous section in equation (4.21), and disscussed in section 2 inequation (2.11),Z( β ) = Tr (cid:2) ρ (cid:3) = (cid:90) Dh ij (cid:88) m,n C mn Ψ m (cid:2) h ij (cid:3) Ψ n (cid:2) h ij (cid:3) = (cid:88) disconnected+ connected (cid:90) D g D φ exp (cid:0) iI[ g, φ ] (cid:1) ∼ exp (cid:0) iI[ g c ] (cid:1)(cid:12)(cid:12)(cid:12) connectedonly = exp (cid:2) − β π (cid:3) , (5.2)where to match the calculation in the previous section we have removed the matterfields appearing in (5.1).Interestingly enough, (5.1) factorizes in two density matrices [15] ρ (cid:2) h + ij , φ +0 ; h − ij , φ − (cid:3) = (cid:90) D h ij D φ ρ + (cid:2) h + ij , φ +0 ; h ij , φ (cid:3) ρ − (cid:2) h ij , φ ; h − ij , φ − (cid:3) . (5.3)The boundary values ( h ij , φ ), match the value of the fields on the T -reflected sliceof t = t , at τ = 4 π M, as discussed in Fig. 11. In this case we can see that ρ − ,and ρ + , do not correspond to pure states since each one comes from a connectedgeometry Fig. 13; and the trace over the non-observable boundary τ = 4 π M, leadsto (5.3). 24 igure 13 . Representation of the manifolds associated to ρ − and ρ + . Now we can regard ρ − , and ρ + , as transition amplitudes. For instance, ρ − couldbe seen as the transition amplitude from the state on the slice t = − t , ( τ = 4 π M)with values ( h − ij , φ − ), to the state on the slice t = t , ( τ = 0) with values ( h − ij , φ − ).In fact, ρ − could be regarded as an S matrix when t → ∞ . Note that the state doesnot depend on the choice of the slice t . In the limit t → ∞ , the futute and pastsegments Σ − and Σ + , lie completely on null infinity, I + and I − , see for instanceFig. 14. It is worth to stress that ρ − , and ρ + , satisfy the equationsˆH ρ − [ h ij , φ ; h − ij , φ − ] = δ (cid:2) h ij , h − ij (cid:3) δ (cid:2) φ , φ − (cid:3) , ˆH ρ + [ h + ij , φ +0 ; h ij , φ ] = δ (cid:2) h + ij , h ij (cid:3) δ (cid:2) φ +0 , φ (cid:3) . In this section, we shall point out the relation of our work, when extended to computethe entanglement entropy, with some recent proposals [20–25]. Here we would seehow following a slightly different logic, we arrive at the concept of replica wormhole.Although we do not consider the matter contribution in the following discussion, wegive a prescription for how the entanglement entropy in QG should be computed fora four-dimensional Schwarzschild black hole on a nice slice.25 →∞ X T Σ + Σ - Σ r = r r = Figure 14 . Schematic representation of the slice t → ∞ . Notice that when t → ∞ , thered line inside the horizon becomes infinitely long and Σ − and, Σ + lie on the horizons, atnull infinity. For the state defined above we can compute its associated entanglement entropy.We will exemplify this calculation by posing the problem of computing the entagle-ment entropy for the segmets Σ − and Σ + on the silce t < ∞ , see Fig. 5. Note that,at least, mathematically we can pose the problem on these segments for t < ∞ . Forthem we have r ≤ r ≤ ∞ , with r < ≤ r ≤ ∞ , with r > t → ∞ .To address this calculation, we must first define the replica manifold of thisgeometry. We can start by defining the reduced density matrix ˜ ρ [1 (cid:48) , (cid:48) ; 1 ,
2] associatedto Σ − and Σ + . To build this object first, we perform the complex extension on thesegments Σ − and Σ + , Fig. 15. Then we should fill in the geometry for the extensionof the Σ slice. The symbol ! =, in the definition of the density matrix in Fig. 15indicates that ˜ ρ [1 (cid:48) , (cid:48) ; 1 , t does not appearexplicitly on Σ , and this slice does not evolve forward in time, it only grows. Theextension of it is determined only by the metric’s boundaries values on δ − and δ +0 ;and the induced metric on Σ . To fully specify the reduced density matrix, we haveto fill in the geometry in between the two cylinders in Fig. 15, as we did in theprevious section.The geometric representation of the reduced density matrix is depicted in Fig.16. This reduced density matrix can be obtained by taking the partial trace of thedensity matrix defined in the previuos section over the degrees of freedom on Σ (redslice, see for instance Fig. 12), i.e., ˜ ρ = Tr Σ [ ρ ]. In this way the partition functionwould be Z = Tr Σ − ∪ Σ + [ ˜ ρ ].Using ˜ ρ , we can compute the density matrix of the replicated manifold. However,this construction comes with a caveat, and extra care is needed when we apply it26 igure 15 . Complex extension of the segments Σ − and Σ + , and geometric representationof the reduced density matrix. Here the reduced density matrix has not been fully specifiedyet. To fully specify it, we must fill in the geometry in between the two cylinders. Figure 16 . Geometric representation of the reduced density matrix. to construct and associate ˜ ρ n to the replicated manifold. We should remember thatthere is an ambiguity when extending the Σ slice. To see the consequences of suchambiguity, let us construct the manifold associated to ˜ ρ .To compute ˜ ρ [1 (cid:48) , (cid:48) ; 1 , ρ [1 (cid:48) , (cid:48) ; 1 , ! = (cid:88) (3 , (cid:48) ) ˜ ρ [1 (cid:48) , (cid:48) ; 3 , (cid:48) ] ˜ ρ [3 , (cid:48) ; 1 , . (6.1)The symbol ! =, in (6.1) indicates that the matrix ˜ ρ , in Fig. 17 has not been fully Figure 17 . Complex extension of two copies of the segments Σ − and Σ + , and geometricrepresentation of the reduced density matrix for the replicated manifold. Here the reduceddensity matrix associated with the replicated manifold has not been fully specified yet. Tofully specify it, we must fill in the geometry in between the four cylinders. specified yet. To fully specify ˜ ρ , we must fill in the geometry in between, and thentake a trace over the red segments. At this point is where the ambiguity shows up.There are several ways in which we can fill in the geometry. The first and obviouscase is represented in Fig. 18It can be regarded as the genuine ˜ ρ [1 (cid:48) , (cid:48) ; 1 , ρ [1 (cid:48) , (cid:48) ; 1 , ρ [1 ,
2; 1 (cid:48) , (cid:48) ], because it is not the square of the matrix in Fig.16. The geometry is connecting the two copies and can be regarded as a complexwormhole. On the one hand, obviously, it is not the square of the matrix in Fig. 16.28 igure 18 . Geometric representation of the genuine ˜ ρ reduced density matrix associatedto the replicated fully disconnected manifold. On the other hand, we can see how the ambiguity in extending the Σ slice has ledus to the concept of replica wormhole.There are more geometries one could include in the density matrix definition.The one we have considered so far in Fig. 19 is topologically equivalent to an S ,with four punctures. Certainly, we could include the higher genus ones. However,like in the two-dimensional case, we believe that they would be suppressed by sometopological mechanism [21].Other connecting-geometries could be considered. For instance, we could connectthe two cylinders on the left and the two on the right in Fig. 17; or the upper cylinderon the left with the lower on the right and the lower on the left with the upper onthe right. However, these geometries are not allowed because they do not satisfy theboundary conditions on the red slices. In other words, the red slices can not be fullyinscribed in these geometries.At this point we find it convenient to make a distinction among these matrices. Inwhat follows we regard ˜ ρ ( n ), as the most general density matrix can be associated to aparticular, non-fully specified manifold , for instance Fig. 17. To construct ˜ ρ ( n ), andfully specify it we can procced as in [16], and in equation (2.11). We can consider allthe contributions comming from the disconected and connected geometries fulfilling By non-fully specified manifold we mean the manifold before having connected all its internalboundaries, as in Fig. 15 and Fig. 17. igure 19 . Geometric representation of the ˜ ρ reduced density matrix associated withthe replicated connected manifold. This geometry can be regarded as a complex wormholeconnecting the copies. the boundary conditions on the internal boundaries and on the red slices, i.e.,˜ ρ ( n ) = ˜ ρ n disconnected + ˜ ρ n connected . (6.2)Here, ˜ ρ n is genuinely the n th power of the matrix ˜ ρ .Now, we can regard, for instance, ˜ ρ (2)[1 (cid:48) , (cid:48) ; 1 , ρ (2)[1 (cid:48) , (cid:48) ; 1 ,
2] = ˜ ρ [1 (cid:48) , (cid:48) ; 1 , disconnected + ˜ ρ [1 (cid:48) , (cid:48) ; 1 , connected . (6.3)One of the advantage of these distinctions (or definitions) is that we can avoid thefactorization problem [21]. By avoiding this problem, no ensemble average is neededto make the setup consistent.Having ˜ ρ ( n ), we can compute the following quantity S = − lim n → ∂ n Tr (cid:104) ˜ ρ ( n ) (cid:105) = − lim n → ∂ n Tr (cid:104) ˜ ρ n + ˜ ρ n (cid:105) . (6.4)This quantity can not be identified as the entanglement entropy of the segments,in the ordinary QFT sense, see [37] for a discussion about this idetinfication and30ther issues related to the replica wormhole calculus. The reason is the derivative ofTr (cid:104) ˜ ρ ( n ) (cid:105) , does not lead to − Tr (cid:104) ˜ ρ log ˜ ρ (cid:105) , instead it leads to S = − Tr (cid:104) ˜ ρ log ˜ ρ (cid:105) − lim n → ∂ n Tr (cid:104) ˜ ρ n (cid:105) , (6.5)where the connected contribution appears. Of course, if we assume that in QG, thedefinition of entanglement entropy should be generalized to (6.4), which seems to besupported by [20–25], when using the replica trick, then we would be computing theactual entanglement entropy associated to Σ − and Σ + .We want to point out the following fact. We have posed the problem of computingthe entanglement entropy for two segments on a nice slice for t (cid:54) = 0. Instead, if wehad posed the problem for t = 0, where no red slice appears, see Fig. 8, no wormholewould have appeared in the calculation of the entanglement entropy. Of course, afterevolving the state in Lorentzian time, we would have room again for including thereplica wormholes. This paper has combined several ideas to propose a new semiclassical QG stateon a nice slice for a Schwarzschild BH. On these slices, the low energy descriptionremains valid during most of the BH evaporation. For this to happen, a portionof the nice slices inside the BH must be fixed at some r < n copies of that manifold, and glue them together according to the region31e are interested in computing the entanglement entropy. This new manifold definesthe reduced density matrix. After extending n from the Integers to the Reals, we canuse it to compute the entanglement entropy according to the usual rules in QFT.In QG, the fact that the geometry is not fixed affects the density matrix def-inition we can associate to the replicated manifold. In fact, it directly affects thevery concept of replicated manifold. Also, in QG, there is an exact prescription toprepare a semiclassical QG state through complex time evolution [6]. Of course, thisprescription is subjected to the appearance of time on those surfaces where we areinterested in defining the state.On the nice slices, the fact that there are portions that do not evolve forwardin time introduces an ambiguity in associating a replicated manifold to a particulardensity matrix. It has been well illustrated in section 6. Now, the association is notunique and, to a particular configuration, for instance, in Fig. 17, we can associatemany (perhaps infinitely many, the higher genus geometries) manifolds. In factadding all possible contributions together would lead to a good density matrix too,as in (6.2), in the same spirit of [16]. This ambiguity has led us to the concept ofreplica wormhole connecting different replicas [20, 21].The next step in this construction would be to add matter in it and study theevaporating BH, which is a time-dependent system. The inclusion of matter fordimensions higher than two is not straightforward, mainly because we must considerthe backreaction on the metric for the evaporating BH. Although, in principle, onecould use the approximation in [24].A more delicate point when adding matter in this setup would be to definethe radiation’s information flux properly. Usually, it is defined on I + in the Penrosediagram. Here, however, we have possed the problem of computing the entanglemententropy on segments that extend from a finite r = r < > I + , as in [25]. The key point to properlyaddress this calculation is to note that we can take the limit t → ∞ . As we haveshown, the semiclassical geometry does not dependent on the particular choice of t .When t → ∞ , the segments Σ − and Σ + sit completely on null infinity, i.e., on I + in the Penrose diagram, see Fig. 14.In reference [37], some criticism related to the connection of the replica wormholecalculation and the amplitudes computed according to the usual rules of QFT wasraised. Here we have presented some arguments that partially answer the questions in[37]. For instance, in section 5, we have presented the density matrix interpretation ofthe geometry we have built here, together with a prescription on how the amplitudesmust be assembled to give rise to the density matrix. In Fig 13, we have presentedthe building blocks of this density matrix. It turns out that the building blocksare density matrices too. Each of them, in turn, would be constituted by wavefunctionals. 32fter finishing, we would like to speculate, as mencioned in footnote 5, aboutan intriguing possibility related to the steps we have followed here to define the QGstate. Suppose we want to define a global state on the slice T = 0. As usual, onemight think this state is the one leading to the HH state. To get this state we evolvein complex time the portion of the slice T = 0, with X ≥
0, from τ = 0 to τ = 8 π M.For this we use only the right wedge, or for instance (2.18) with t = 0. Leavingthese two boundaries free (a Pacman figure), we can define a (reduced) density matrixassociate to the segment T = 0, X ≥
0. As it is well known, this state is not pureon this segment, and it leads to the known thermodynamic entropy for a BH.An awkward feature of the geometry representing the partition function, aftertracing the degrees of freedom over the mouth of the Pacman (the disk, see Fig. 4)is that the thermal circle is homotopically equivalent to any circle on the disk. Thisis not what is expected to happen in the statistical interpretation of QFT.Now suppose we follow similar steps in defining the state on the slice T = 0,but this time we evolve in complex time both segments on the left and right wedges,similarly to what we have done on a nice slice. The geometry, in this case, wouldnot be a packman figure. Instead, it would be a double Pacman figure with oppositeorientation overlapping each other and sharing a single point at the horizon. Thisgeometry would be similar to the one we have found here. Hence it would lead to aglobal mixed state. Moreover, the thermal circle (after tracing the degrees of freedomover the two mouths of the double Pacman) would not be homotopically equivalentto any circle on this geometry due to the shared point.It raises the question, whether we can define a pure global state for the BHgeometry on any slice. Notice, for instance, that even at I − , i.e., at t → −∞ ,the state is not a global pure. If our speculations turn out to be correct and wecan extend it to other foliations, for instance, the ordinary one for a SchwarzschildBH (2.17), it would have repercussions on the information paradox because of theintrinsic impossibility for defining pure global states. These repercussions will bestudied elsewhere. Acknowledgments
We are grateful to A. Banerjee, E. ´O. Colg´ain, H. Casini, T. Hartman, J. M. Mal-dacena and Kanghoon Lee for discussions, useful comments and suggestions. Wewould especially like to thank the members of APCTP for stimulating questions anddiscussions during the presentation of this work at APCTP.33 eferences [1] D. A. Lowe, J. Polchinski, L. Susskind, L. Thorlacius and J. Uglum, “Black holecomplementarity versus locality,” Phys. Rev. D , 6997-7010 (1995).[2] S. D. Mathur, “The Information paradox: A Pedagogical introduction,” Class.Quant. Grav. , 224001 (2009)[3] J. Polchinski, “The Black Hole Information Problem,” [arXiv:1609.04036 [hep-th]].[4] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. ,199-220 (1975).[5] S. W. Hawking, “Breakdown of Predictability in Gravitational Collapse,” Phys. Rev.D , 2460-2473 (1976).[6] J. B. Hartle and S. W. Hawking, “Wave Function of the Universe,” Adv. Ser.Astrophys. Cosmol. , 174-189 (1987), PhysRevD.28.2960.[7] S. B. Giddings, “Schr¨odinger evolution of the Hawking state,” [arXiv:2006.10834[hep-th]].[8] S. B. Giddings, “Nonviolent unitarization: basic postulates to soft quantumstructure of black holes,” JHEP , 047 (2017).[9] S. B. Giddings, “Black holes, quantum information, and unitary evolution,” Phys.Rev. D , 124063 (2012).[10] S. B. Giddings, “Quantization in black hole backgrounds,” Phys. Rev. D , 064027(2007).[11] S. B. Giddings, “(Non)perturbative gravity, nonlocality, and nice slices,” Phys. Rev.D , 106009 (2006).[12] S. W. Hawking, “The Quantum State of the Universe,” Adv. Ser. Astrophys.Cosmol. , 236-255 (1987).[13] G. W. Gibbons and S. W. Hawking, “Action Integrals and Partition Functions inQuantum Gravity,” Phys. Rev. D , 2752-2756 (1977).[14] S. W. Hawking, “Quantum Gravity and Path Integrals,” Phys. Rev. D ,1747-1753 (1978).[15] S. W. Hawking, “The Density Matrix of the Universe,” Phys. Scripta T , 151(1987).[16] D. N. Page, “Density Matrix of the Universe,” Phys. Rev. D , 2267 (1986).[17] T. Hartman and J. Maldacena, “Time Evolution of Entanglement Entropy fromBlack Hole Interiors,” JHEP , 014 (2013).[18] D. N. Page, “Information in black hole radiation,” Phys. Rev. Lett. , 3743-3746(1993).[19] D. N. Page, “Time Dependence of Hawking Radiation Entropy,” JCAP , 028(2013).
20] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, “ReplicaWormholes and the Entropy of Hawking Radiation,” JHEP , 013 (2020).[21] G. Penington, S. H. Shenker, D. Stanford and Z. Yang, “Replica wormholes and theblack hole interior,” [arXiv:1911.11977 [hep-th]].[22] F. F. Gautason, L. Schneiderbauer, W. Sybesma and L. Thorlacius, “Page Curve foran Evaporating Black Hole,” JHEP , 091 (2020).[23] T. Anegawa and N. Iizuka, “Notes on islands in asymptotically flat 2d dilaton blackholes,” JHEP , 036 (2020).[24] K. Hashimoto, N. Iizuka and Y. Matsuo, “Islands in Schwarzschild black holes,”JHEP , 085 (2020).[25] T. Hartman, E. Shaghoulian and A. Strominger, “Islands in Asymptotically Flat 2DGravity,” JHEP , no.4, 045 (2020), [arXiv:2006.17000 [hep-th]].[28] Y. Chen, V. Gorbenko and J. Maldacena, “Bra-ket wormholes in gravitationallyprepared states,” [arXiv:2007.16091 [hep-th]].[29] B. S. DeWitt, “Quantum Theory of Gravity. 1. The Canonical Theory,” Phys. Rev. , 1113-1148 (1967).[30] J. J. Halliwell and J. B. Hartle, “Integration Contours for the No Boundary WaveFunction of the Universe,” Phys. Rev. D , 1815 (1990).[31] J. B. Hartle and S. W. Hawking, “Path Integral Derivation of Black Hole Radiance,”Phys. Rev. D , 2188-2203 (1976).[32] R. Arnowitt and S. Deser, “Quantum Theory of Gravitation: General Formulationand Linearized Theory,” Phys. Rev. , 745-750 (1959).[33] R. L. Arnowitt, S. Deser and C. W. Misner, “Dynamical Structure and Definition ofEnergy in General Relativity,” Phys. Rev. , 1322-1330 (1959).[34] R. L. Arnowitt, S. Deser and C. W. Misner, “Canonical variables for generalrelativity,” Phys. Rev. , 1595-1602 (1960).[35] J. D. Brown, E. A. Martinez and J. W. York, Jr., “Complex Kerr-Newman geometryand black hole thermodynamics,” Phys. Rev. Lett. , 2281-2284 (1991).[36] J. M. Maldacena, “Eternal black holes in anti-de Sitter,” JHEP , 021 (2003).[37] S. B. Giddings and G. J. Turiaci, “Wormhole calculus, replicas, and entropies,”[arXiv:2004.02900 [hep-th]]., 021 (2003).[37] S. B. Giddings and G. J. Turiaci, “Wormhole calculus, replicas, and entropies,”[arXiv:2004.02900 [hep-th]].