Stephen D. H. Hsu
We investigate isolated white holes surrounded by vacuum, which correspond to the time reversal of eternal black holes that do not evaporate. We show that isolated white holes produce quasi- thermal Hawking radiation. The time reversal of this radiation, incident on a black hole precursor, constitutes a special preparation that will cause the black hole to become eternal.
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We present the gravity description of evaporating black holes that end up with eternal traversable wormholes where every would-be behind horizon degree is available in asymptotic regions. The transition is explicitly realized by a time-dependent bulk solution in the two-dimensional Einstein-dilaton gravity. In this solution, the initial AdS$_2$ black hole is evolved into an eternal traversable wormhole free of any singularity, which may be dubbed as transparentization of black holes to eternal traversable wormholes. The bulk construction completely matches with the boundary description governe…
When the slow-roll parameter $ε_H$ is smaller than $H^2/M_{\rm Pl}^2$, the quantum fluctuations of the inflaton after the horizon crossing are large enough to realize eternal inflation. Whereas they do not generate a sufficient amount of density fluctuation of the inflaton to produce the black hole in quasi-de Sitter space, they can also generate the sizeable density fluctuation of the radiation when the number of degrees of freedom increases rapidly in time, as predicted by the distance conjecture. We argue that the condition that the density fluctuation of the radiation is not large enough t…
The microcanonical functional integral for an eternal black hole system is considered. This requires computing the microcanonical action for a spatially bounded spacetime region when its two disconnected timelike boundary surfaces are located in different wedges of the Kruskal diagram. The path integral is a sum over Lorentzian geometries and is evaluated semiclassically when its boundary data are chosen such that the system is approximated by any Lorentzian, stationary eternal black hole. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical…
We present the Hamiltonian, quasilocal energy, and angular momentum for a spacetime region spatially bounded by two timelike surfaces. The results are applied to the particular case of a spacetime representing an eternal black hole. It is shown that in the case when the boundaries are located in two different wedges of the Kruskal diagram, the Hamiltonian is of the form $H = H_+ - H_-$, where $H_+$ and $H_-$ are the Hamiltonian functions for the right and left wedges respectively. The application of the obtained results to the thermofield dynamics description of quantum effects in black holes …
Both reflected entropy and entanglement negativity provide valid measures of entanglement between subsystems of a mixed state. For general 2D eternal black holes coupled with CFT matters in large $c$ limit, we perform the replica-trick computation and find that both effective Renyi reflected entropy and effective entanglement negativity can be expressed in terms of the combination of modified backreacting cosmic branes in $\mathrm{AdS}_3$ bulk. We then develop a holographic scheme to calculate effective reflected entropy and entanglement negativity for general 2D eternal black holes coupled wi…
We review the relation between AdS spacetime in 1+2 dimensions and the BTZ black hole. Later we show that a ground state in AdS spacetime becomes a thermal state in the BTZ black hole. We show that this is true in the bulk and in the boundary of AdS spacetime. The existence of this thermal state is tantamount to say that the Unruh effect exists in AdS spacetime and becomes the Hawking effect for an eternal BTZ black hole. In order to make this we use the correspondence introduced in Algebraic Holography between algebras of quasi-local observables associated to wedges and doble cones regions in…
We show that, in the AdS/CFT correspondence, states obtained by Hamiltonian evolution of the thermofield doubled state are also dual to an eternal black hole geometry, which is glued to the boundary with a time shift generated by a large diffeomorphism. We describe gauge invariant relational observables that probe the black hole interior in these states and constrain their properties using effective field theory. By adapting recent versions of the information paradox we show that these observables are necessarily described by state-dependent bulk-boundary maps, which we construct explicitly.
It is generally believed that the eternal AdS black hole is dual to two conformal field theories with compact spatial sections that are together in a thermofield double state. We argue that this proposal is incorrect, and by extension so are the "entanglement=geometry" proposal of Van Raamsdonk and "ER=EPR" proposal of Maldacena and Susskind. We show that in the bulk there is an interaction needed between the two halves of the Hilbert space for connectivity across the horizon; however, there is no such interaction between the CFTs. This rules out the possibility of the dual to the CFTs being t…
We study the entanglement islands and subsystem volume complexity corresponding to the left/ right entanglement of a conformal defect in $d$-dimensions in Randall-Sundrum (RS) braneworld model with subcritical tension brane. The left and right modes of the defect mimic the eternal black hole and radiation system respectively. Hence the entanglement entropy between the two follows an eternal black hole Page curve which is unitarity compatible. We compute the volumes corresponding to the left and right branes with preferred Ryu-Takanayagi (RT) surfaces at different times, which provide a probe o…
We study a higher order conformally coupled scalar tensor theory endowed with a covariant geometric constraint relating the scalar curvature with the Gauss-Bonnet scalar. It is a particular Horndeski theory including a canonical kinetic term but without shift or parity symmetry for the scalar. The theory also stems from a Kaluza-Klein reduction of a well defined higher dimensional metric theory. Properties of an asymptotically flat spherically symmetric black hole are analyzed, and new slowly rotating and radiating extensions are found. Through disformal transformations of the static configura…
Eternal inflation near inflection points: a challenge to primordial black hole models
该研究指出，具有拐点势的膨胀模型中，永恒膨胀的条件为λ₁ ≤ 3，且在特定参数下会导致婴儿宇宙的形成，从而挑战原初黑洞模型与大尺度结构观测的兼容性。
The analogy between black hole radiation and accelerating mirror radiation (the dynamical Casimir effect) is particularly strong for mirror trajectories giving rise to a constant thermal flux of particles. We present new ways to achieve such thermal plateaus, and customize their finite, semi-infinite, and eternal presence, corresponding to forming/collapsing, complete-evaporation/remnants, and eternal black holes. We find simple expressions for the energy flux in terms of the mirror rapidity as a function of proper time and null time.
In this short paper, we conjecture a correspondence between the creation of the universe in entangled pairs and eternal black holes. We shall see that this correspondence will restore the matter-antimatter asymmetry at the beginning of the universe and provide a new cosmological model that can be used to map the physics of the entire universe to the physics of the black hole horizon.
In this article we deduce two new exact solutions of Einstein's equations for eternal black holes, now related to stiff matter, one `static' and another rotating (stationary like the Kerr one), thus the number of these eternal solutions grows from four to six. Dealing with these static and rotating solutions we automatically rehabilitate the practically forgotten Sommerfeld-Lenz approach to the deduction of our results for the satisfaction of Einstein's equations, e.g., in these rotating cases. At the same time, we come to the confirmation of our new relativistic generalization of the inertial…
The one-loop quantum corrections to geometry and thermodynamics of black hole are studied for the two-dimensional RST model. We chose boundary conditions corresponding to the eternal black hole being in the thermal equilibrium with the Hawking radiation. The equations of motion are exactly integrated. The one of the solutions obtained is the constant curvature space-time with dilaton being a constant function. Such a solution is absent in the classical theory. On the other hand, we derive the quantum-corrected metric (\ref{solution}) written in the Schwarzschild like form which is a deformatio…
It is shown that the correspondence principle and the holographic principle are incompatible in the background of an eternal Schwarzschild-anti-de Sitter geometry. The argument is based on the observation that algebraic structures of local quantum field and CFT operators are not equivalent. This implies, in particular, the bulk CFT must be singular near the black-hole horizon. A CFT Hilbert space representation is elaborated which may correspond to the AdS black hole in the dual theory.
Factorization of the Hilbert space of eternal black holes in general relativity
本文通过构造视界后含重物质壳层的黑洞微观态，证明任意维度负宇宙学常数量子引力中永恒黑洞希尔伯特空间在半经典极限下可因子化。
The entropy of a quantum-statistical system which is classically approximated by a general stationary eternal black hole is studied by means of a microcanonical functional integral. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical black hole in path integral descriptions of its thermodynamical properties. If the functional integral is interpreted as the density of states of the system, the corresponding entropy equals ${cal S} = A_H/4 - A_H/4 =0$ in the semiclassical approximation, where $A_H$ is the area of the black hole horizon. The f…
Mass and entropy of asymptotically flat eternal quantum black holes in 2D
研究二维渐近平坦永恒量子黑洞的质量和熵，通过解析和数值方法分析其热力学性质及稳定性。
It has been argued i) that Lorentz-signature solutions with wormholes connecting n asymptotically AdS regions describe bulk quantum states dual to n entangled but non-interacting CFTs and ii) that such bulk wormhole states should be identified with similar entangled but non-interacting bulk systems, each describing quantum geometries with only a single asymptotic region. But if the wormhole is to behave semiclassically, we show that conjecture (ii) cannot hold. Instead, the theory of asymptotically AdS bulk quantum gravity must admit superselection sectors with respect to the CFT observables t…
We present the gravitational action and Hamiltonian for a spatially bounded region of an eternal black hole. The Hamiltonian is of the general form $H=H_{+} - H_{-}$, where $H_{+}$ and $H_{-}$ are respectively the Hamiltonians for the regions $M_+$ and $M_-$ located in the left and right wedges of the spacetime. We construct explicitly the quasilocal energy for the system and discuss its dependence on the time direction induced at the boundaries of the manifold. This paper extends the analysis of Ref.~[1] to spacetimes possesing a bifurcation surface and two timelike boundaries. The constructi…
We study the Page curve for asymptotically flat eternal Schwarzschild black holes in four and higher spacetime dimensions. Before the Page time, the entanglement entropy grows linearly in time. After the Page time, the entanglement entropy of a given region outside the black hole is largely modified by the emergence of an island, which extends to the outer vicinity of the event horizon. As a result, it remains a constant value which reproduces the Bekenstein-Hawking entropy, consistent with the finiteness of the von Neumann entropy for an eternal black hole.
The glow of eternal black holes
该研究计算被吸积盘环绕的永恒史瓦西黑洞的光学外观，通过替换过去奇点为最后散射面得到中心亮斑，推导其干涉特征，结合M87*和Sgr A*观测约束相关参数，为检验黑洞是否为永恒提供新途径。
We point out how to construct the Hartle-Hawking-Israel state for the minimaly coupled massless quantum real scalar field in the two dimensional BTZ black hole. We also calculate the renormalized energy-momentum tensor for the same field in the eternal CGHS black hole, AdS, Robertson-Walker and Rindler spacetime in two dimensions. We also discuss the Boulware, the Hartle-Hawking-Israel and the Unruh state for the eternal CGHS black hole.
The work is an attempt to model a scenario of inflation in the framework of Anti de Sit- ter/Conformal Field theory (AdS/CFT) duality, a potentially complete nonperturbative description of quantum gravity via string theory. We look at bubble geometries with de Sitter interiors within an ambient Schwarzschild anti-de Sitter black hole spacetime and obtain a characterization for the states in the dual CFT on boundary of the asymptotic AdS which code the expanding dS bubble. These can then in turn be used to specify initial conditions for cosmology. Our scenario naturally interprets the entropy o…
The non-orientable spacetime of the eternal black hole
本文通过交换时空坐标特性推导BTZ黑洞内部解度规并最大延拓其几何，构造出分别对偶于外部解和内部解的两个独立热场双重态，进而计算体空间配分函数，揭示其对应不可定向时空，并提出连接相反时空取向区域的类热场双重态。
We apply the recently proposed quantum extremal surface construction to calculate the Page curve of the eternal Reissner-Nordström black holes in four dimensions ignoring the backreaction and the greybody factor. Without the island, the entropy of Hawking radiation grows linearly with time, which results in the information paradox for the eternal black holes. By extremizing the generalized entropy that allows the contributions from the island, we find that the island extends to the outside the horizon of the Reissner-Nordström black hole. When taking the effect of the islands into account, it …
New approach to exact solvability of dilaton gravity theories is suggested which appeals directly to structure of field equations. It is shown that black holes regular at the horizon are static and their metric is found explicitly. If a metric possesses singularities the whole spacetime can be divided into different sheets with one horizon on each sheet between neighboring singularities with a finite value of dilaton field (addition horizons may arise at infinite value of it), neighboring sheets being glued along the singularity. The position of singularities coincide with the values of dilato…
In this letter we study the process of Hawking radiation of a black hole assuming the existence of a limiting physical curvature scale. The particular model is constructed using the Limiting Curvature Hypothesis (LCH) and in the context of two-dimensional dilaton gravity. The black hole solution exhibits properties of the standard Schwarzschild solution at large values of the radial coordinate. However, near the center, the black hole is nonsingular and the metric becomes that of de Sitter spacetime. The Hawking temperature is calculated using the method of complex paths. We find that such bla…
We calculate perturbative quantum gravity corrections to eternal two-dimensional black holes. We estimate the leading corrections to the AdS_2 black hole entropy and determine the quantum modification of N-dimensional Schwarzschild spacetime.
Eternally inflating universes lead to an infinite number of Boltzmann brains but also an infinite number of ordinary observers. If we use the scale factor measure to regularize these infinities, the ordinary observers dominate the Boltzmann brains if the vacuum decay rate of each vacuum is larger than its Boltzmann brain nucleation rate. Here we point out that nucleation of small black holes should be counted in the vacuum decay rate, and this rate is always larger than the Boltzmann brain rate, if the minimum Boltzmann brain mass is more than the Planck mass. We also discuss nucleation of sma…
The newly proposed island formula for entanglement entropy of Hawking radiation is applied to spherically symmetric 4-dimensional eternal Kaluza-Klein (KK) black hole. The "charge" $Q$ of KK black holes quantifies its deviation from Schwarzschild black holes. The impact of $Q$ on the island is studied at late times. The late-time island, whose boundary is located outside but within a Planckian distance of the horizon, is slightly extended by $Q$. While the no-island entropy grows linearly, the late-time entanglement entropy is given by island configuration with twice the Bekenstein-Hawking ent…
2+1 dimensional anti-de Sitter space has been the subject of much recent investigation. Studies of the behaviour of point particles in this space have given us a greater understanding of the BTZ black hole solutions produced by topological identification of adS isometries. In this paper, we present a new configuration of two orbiting massive point particles that leads to an ``eternal'' time machine, where closed timelike curves fill the entire space. In contrast to previous solutions, this configuration has no event or chronology horizons. Another interesting feature is that there is no lower …
We discuss the large-scale structure of the universe in inflationary cosmology and the implications that it may have for the long-term future of civilizations. Although each civilization is doomed to perish, it may be possible to transmit its accumulated knowledge to future civilizations. We consider several scenarios of this sort. If the cosmological constant is positive, it eventually dominates the universe and bubbles of inflationary phase begin to nucleate at a constant rate. Thermalized regions inside these inflating bubbles will give rise to new galaxies and civilizations. It is possible…
The spacetime inside the white hole is like an anisotropic cosmological background with the past singularity playing the role of a big bang singularity. The scale factor along the extended spatial direction is contracting while the scale factor along the two-sphere is expanding. We consider an eternal Schwarzschild manifold and study quantum cosmological perturbations generated near the white hole singularity which propagate towards the past event horizon and to exterior of the black hole. It is shown that an observer deep inside the white hole and an observer far outside the black hole both s…
It has been conjectured that the dual of the eternal black hole in AdS is two entangled but disconnected CFTs. We show that the entanglement created by the process of Hawking radiation creates several challenges for this conjecture. The nature of fuzzball states suggests a different picture, where the dual to two entangled CFTs is two entangled but disconnected spacetimes. We argue for a process of `quick tunneling' where the Einstein-Rosen bridge of the eternal hole tunnels rapidly into fuzzball states, preventing the existence of the eternal hole as a semiclassical spacetime. The regions beh…
Recent work by Shenker, Stanford, and Kitaev has related the black hole horizon geometry to chaotic behavior. We extend this from eternal black holes to black holes that form and then evaporate. This leads to an identity for the change in the black hole S-matrix (over times shorter than the scrambling time) due an addition infalling particle, elaborating an idea of 't Hooft.
It has been suggested in recent work that the Page curve of Hawking radiation can be recovered using computations in semi-classical gravity provided one allows for "islands" in the gravity region of quantum systems coupled to gravity. The explicit computations so far have been restricted to black holes in two-dimensional Jackiw-Teitelboim gravity. In this note, we numerically construct a five-dimensional asymptotically AdS geometry whose boundary realizes a four-dimensional Hartle-Hawking state on an eternal AdS black hole in equilibrium with a bath. We also numerically find two types of extre…
We illustrate scenarios in which Hawking radiation collected in finite regions of a reservoir provides temporary access to the interior of black holes through transient entanglement "islands". Whether these islands appear and the amount of time for which they dominate - sometimes giving way to a thermalization transition - is controlled by the amount of radiation we probe. In the first scenario, two reservoirs are coupled to an eternal black hole. The second scenario involves two holographic quantum gravitating systems at different temperatures interacting through a Rindler-like reservoir, whi…
Creation of thermal distribution of particles by a black hole is independent of the detail of gravitational collapse, making the construction of the eternal horizons suffice to address the problem in asymptotically flat spacetimes. For eternal de Sitter black holes however, earlier studies have shown the existence of both thermal and non-thermal particle creation, originating from the non-trivial causal structure of these spacetimes. Keeping this in mind we consider this problem in the context of a quasistationary gravitational collapse occurring in a $(3+1)$-dimensional eternal de Sitter, set…
We extend the Gross-Perry-Yaffe approach of hot flat space instability to Minkowski space. This is done by a saddle point approximation of the partition function in a Schwarzschild wormhole background which is coincident with an eternal black hole. The appearance of an instability in the whole manifold is here interpreted as a black hole pair creation.
Great breakthrough in solving black hole information paradox took place when semiclassical island rule for entanglement entropy of Hawking radiation was proposed in recent years. Up to now, most papers which discussed island rule of asymptotic flat black hole with $D \ge 4$ focus on eternal black hole. In this paper, we take one more step further by discussing island of "in" vacuum state which describes one-sided asymptotically flat black hole formed by gravitational collapse in $D \ge 4$. We find that island $I$ emerges at late time and saves entropy bound. And boundary of island $\partial I$…
It is often argued that quantum gravitational correction to the Heisenberg's uncertainty principle leads to, among other things, a black hole remnant with finite temperature. However, such a generalized uncertainty principle also seemingly removes the Chandrasekhar limit, i.e., it permits white dwarfs to be arbitrarily large, which is at odds with astrophysical observations. We show that this problem can be resolved if the parameter in the generalized uncertainty principle is negative. We also discuss the Planck scale physics of such a model.
We study reflected entropy as a correlation measure in black hole evaporation. As a measure for bipartite mixed states, reflected entropy can be computed between black hole and radiation, radiation and radiation. We compute reflected entropy curves in three different models: 3-side wormhole model, End-of-the-World (EOW) brane model in three dimensions and two-dimensional eternal black hole plus CFT model. For 3-side wormhole model, we find that reflected entropy is dual to island cross sections. The reflected entropy between radiation and black hole increases at early time and then decreases t…
Understanding the end state of black hole evaporation, the microscopic origin of black hole entropy, the information loss paradox, and the nature of the singularity arising in gravitational collapse - these are outstanding challenges for any candidate quantum theory of gravity. Recently, a midisuperspace model of quantum gravitational collapse has been solved using a lattice regularization scheme. It is shown that the mass of an eternal black hole follows the Bekenstein spectrum, and a related argument provides a fairly accurate estimate of the entropy. The solution also describes a quantized …
We consider a massless quantum scalar field on a two-dimensional space-time describing a thin shell of matter collapsing to form a Schwarzschild-anti-de Sitter black hole. At early times, before the shell starts to collapse, the quantum field is in the vacuum state, corresponding to the Boulware vacuum on an eternal black hole space-time. The scalar field satisfies reflecting boundary conditions on the anti-de Sitter boundary. Using the Davies-Fulling-Unruh prescription for computing the renormalized expectation value of the stress-energy tensor, we find that at late times the black hole is in…
A self-consistent exact solution for a Reissner-Nordström black-and-white hole formed as a result of accretion has been considered. Prior to the formation of a black-and-white hole, there is a bulk charged sphere at the center of the system. The occurrence of a turning (bounce) point in the newly formed black-and-white hole is investigated. The occurrence of a turning point was investigated by an example of solution for the accretion of a neutral spherical dust shell. The model equations are written with allowance for the cosmological $Λ$-term. Within the model under consideration, both the bl…
We study a thermo-field double type entangle state on two disjoint gravitating universes, say A and B, with an eternal black hole on each. As was shown previously, its entanglement entropy of the universe A is computed by the generalized entropy on a new spacetime constructed by suitably gluing the black holes on A and B. We study such spacetime gluings when universes are asymptotically flat and AdS cases, especially when the masses of these black holes are different. We also clarify the rule to construct such a glued spacetime in more general settings from the gravitational path integral view…
Dark Matter from Eternity
该研究提出暗物质可能源于永恒膨胀，通过引力坍缩形成原始黑洞，并预测了引力波背景的特性。
Non-Locality induces Isometry and Factorisation in Holography
针对黑洞信息悖论中的非等距与因子化难题，本文通过态平均产生的虫洞非定域量子修正，利用预解式技术证明Anti-de Sitter空间永恒黑洞Hilbert空间有限维且离散能谱，实现向type I von Neumann代数的转变以同时解决两难题。
We use the identification between null singularities of correlators in the bulk with time singularities in the boundary correlators to study the analytic structure of time-dependent thermal Green functions using the eikonal approximation for classical solutions in the AdS black hole background. We show that the location of singularities in complex time can be understood in terms of null rays bouncing on the boundaries and singularities of the eternal black hole, giving the picture of a `reflecting cavity'. We can then extract the general analytic expression for the asymptotic values of the fre…
Wald-like entropy and Islands in Dimensionally Reduced Einstein-Hilbert Gravity
研究维度约化爱因斯坦 - 希尔伯特模型中永恒及蒸发黑洞的孤岛公式与佩奇曲线，通过洛伦兹规范确定广义熵，其与类瓦尔德诺特定理电荷一致，据此得出量子极值面、佩奇时间和佩奇曲线，并探讨向完整四维几何扩展的可能。
Vacuum bubbles may nucleate and expand during the inflationary epoch in the early universe. After inflation ends, the bubbles quickly dissipate their kinetic energy; they come to rest with respect to the Hubble flow and eventually form black holes. The fate of the bubble itself depends on the resulting black hole mass. If the mass is smaller than a certain critical value, the bubble collapses to a singularity. Otherwise, the bubble interior inflates, forming a baby universe, which is connected to the exterior FRW region by a wormhole. A similar black hole formation mechanism operates for spher…
It was argued recently that there exists an unexpected phenomenon, the reflection of incoming particles on the event horizon of black holes (Kuchiev(2003)). This means that a particle approaching the black hole can bounce back into the outside world due to those events that take part strictly on the horizon. Previously the effect was discussed in relation to eternal black holes. The present work shows that the effect exists for collapsing black holes as well.
Hawking radiation is nowadays being understood as tunnelling through black hole horizons. Here, the extension of the Hamilton-Jacobi approach to tunnelling for non-rotating and rotating black holes in different non-singular coordinate systems not only confirms this quantum emission from black holes but also reveals the new phenomenon of absorption into white holes by quantum mechanical tunnelling. The role of a boundary condition of total absorption or emission is also clarified.
The vivid debate concerning the paradox of information being lost when objects are swallowed by a black hole is shown to be void. We argue that no information is ever missing for any observer neither located above, nor falling beneath the event horizon. The information is preserved in a classical scenario of eternal black holes and semi-classical one allowing Hawking radiation.
The generalized second law of black hole thermodynamics was proved by Frolov and Page for a quasi-stationary eternal black hole. However, realistic black holes arise from a gravitational collapse, and in this case their proof does not hold. In this paper we prove the generalized second law for a quasi-stationary black hole which arises from a gravitational collapse.
This is the first of two papers presenting different versions of quasi-classical toy models for the non-singular evolution of the geometry and the associated effective stress-energy tensor for a spherically symmetric black hole that evolves into a white hole and eventually disappears after evaporating down to the Planck scale. The ansatz for the geometry is inspired by calculations of the semi-classical stress-energy tensor in the Schwarzschild background and ideas from loop quantum gravity for a nonsingular transition to the white hole. In this paper the main emphasis is on the evolution of t…
Recently, various authors have proposed that the first ultraviolet effect on the gravitational collapse of massive stars to black holes is the transition between a black-hole geometry and a white-hole geometry, though their proposals are radically different in terms of their physical interpretation and characteristic time scales [1,2]. Several decades ago, it was shown by Eardley that white holes are highly unstable to the accretion of small amounts of matter, being rapidly turned into black holes [3]. Studying the crossing of null shells on geometries describing the black-hole to white-hole t…
The possibility that a black hole could tunnel into a white hole has recently received attention. Here we present a metric that improves the 'firework' metric: it describes the entire process and solves the Einstein equations everywhere except on a small transition surface that corresponds to the quantum tunnelling. We compute the corresponding ray-tracing map from past infinity to future infinity explicitly.
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