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.
We consider evaporation of the Schwarzschild black hole and note that island configurations do not provide a bounded entanglement entropy. The same remark is valid also for some other static black holes. A proposal for improving the situation is discussed
We study 4-dimensional Schwarzschild de Sitter black holes in the regime where the black hole mass is small compared with the de Sitter scale. Then the de Sitter temperature is very low compared with that of the black hole and we study the black hole, approximating the ambient de Sitter space as a frozen classical background. We consider distant observers in the static diamond, far from the black hole but within the cosmological horizon. Using 2-dimensional tools, we find that the entanglement entropy of radiation exhibits linear growth in time, indicative of the information paradox for the bl…
We reconsider $D\geq4$ dimensional asymptotically flat eternal Schwarzschild black hole, and focus on the situation where the inner boundary of the radiation region is chosen to be near the horizon (i.e. $β\ll1$). The tension between the near-horizon condition and the short-distance approximation emerges in large dimensions in $[JHEP 06 (2020) 085]$. We remove this tension by introducing a more proper near horizon condition, thus the resulting island solution is well-behaved in any $D\geq4$ dimensional spacetime. Interestingly, a novel constraint is obtained in this situation as required by th…
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…
By using the quantum extremal island formula, we perform a simple calculation of the generalized entanglement entropy of Hawking radiation from the two dimensional Liouville black hole. No reasonable island was found when extremizing the generalized entropy. We explain qualitatively the reason why the page curve cannot be reproduced in the present model. This suggests that the islands may not necessarily save the information paradox for the Liouville black holes.
The island rule for the entanglement entropy is applied to an eternal Reissner-Nordström black hole. The key ingredient is that the black hole is assumed to be in thermal equilibrium with a heat bath of an arbitrary temperature and so the generalized entropy is treated as being off-shell. Taking the on-shell condition to the off-shell generalized entropy, we find the generalized entropy and then obtain the entanglement entropy following the island rule. For the non-extremal black hole, the entanglement entropy grows linearly in time and can be saturated after the Page time as expected. The ent…
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 $\partial I$…
In this paper, we have analyzed the information paradox and its resolution using island proposal in Schwarzschild de-Sitter black hole spacetime. First, we study the information paradox for the black hole patch by treating de-Sitter patch on both sides as a frozen background (by inserting thermal opaque membrane) and then we carry out similar study for de-Sitter patch also. In both cases, when there is no island surface then the entanglement entropy has as usual the linear time dependence whereas in the presence of an island surface entanglement entropy become constant (equal to twice of therm…
By applying the island rule proposed recently, we compute the entanglement entropy of Hawking radiation and study the Page curve for the eternal black holes in massive gravity. We investigate for both the neutral and charged black holes which the corresponding results of Schwarzschild and Reissner-Nordström black holes are restored in the limit of massless graviton. We show for the neutral and non-extremal charged black holes that the island is not formed at the early times of the evaporation and hence the entanglement entropy increases linearly in time. However, for the extremal charged black…
Recently it was proposed that the entanglement entropy of the Hawking radiation contains the information of a region including the interior of the event horizon, which is called "island." In studies of the entanglement entropy of the Hawking radiation, the total system in the black hole geometry is separated into the Hawking radiation and black hole. In this paper, we study the entanglement entropy of the black hole in the asymptotically flat Schwarzschild spacetime. Consistency with the island rule for the Hawking radiation implies that the information of the black hole is located in a differ…
We derive the equations of cosmological evolution from an AdS Schwarzschild black hole via holographic renormalization with appropriate boundary conditions.
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 globally embed (3+1)-dimensional Schwarzschild and Schwarzschild-AdS black holes in massive gravity into (5+2)-dimensional flat spacetimes. Making use of embedding coordinates, we directly obtain the generalized Hawking, Unruh and freely falling temperatures in a Schwarzschild and Schwarzschild-AdS black hole due to massive graviton effects.
A quantum extremal island suggests that a region of spacetime is encoded in the quantum state of another system, like the encoding of the black hole interior in Hawking radiation. We study conditions for islands to appear in general spacetimes, with or without black holes. They must violate Bekenstein's area bound in a precise sense, and the boundary of an island must satisfy several other information-theoretic inequalities. These conditions combine to impose very strong restrictions, which we apply to cosmological models. We find several examples of islands in crunching universes. In particul…
Explicit regular coordinates that cover all of the Tangherlini solutions (Schwarzschild black holes of dimension $D>4$) are given. The coordinates reduce to Israel coordinates for D=4.
We derive a novel four-dimensional black hole with planar horizon that asymptotes to the linear dilaton background. The usual growth of its entanglement entropy before Page's time is established. After that, emergent islands modify to a large extent the entropy, which becomes finite and is saturated by its Bekenstein-Hawking value in accordance with the finiteness of the von Neumann entropy of eternal black holes. We demonstrate that viewed from the string frame, our solution is the two-dimensional Witten black hole with two additional free bosons. We generalize our findings by considering a g…
We show that the hidden conformal symmetry of the Schwarzschild black hole is realized from the AdS$_2$ sector of the AdS$_2\times S^2$, but not from the Rindler spacetime which is the genuine near-horizon geometry of the Schwarzschild black hole. This implies that purely imaginary quasinormal frequencies obtained using the hidden conformal symmetry is not suitable for describing the largely damped modes around the Schwarzschild black hole.
The statistical entropy of a Schwarzschild black string in five dimensions is obtained by counting the black string states which form a representation of the near-horizon conformal symmetry with a central charge. The statistical entropy of the string agrees with its Bekenstein-Hawking entropy as well as that of the Schwarzschild black hole in four dimensions. The choice of the string length which gives the Virasoro algebra also reproduces the precise value of the Bekenstein-Hawking entropy and lies inside the stability bound of the string.
Recently an anomalous decay rate of the quasinormal modes of a massive scalar field in Schwarzschild black holes backgrounds was reported in which the longest-lived modes are the ones with higher angular number, for a scalar field mass smaller than a critical value, while that beyond this value the behaviour is inverted. In this work, we extend the study to other asymptotic geometries, such as, Schwarzschild-de Sitter and Schwarzschild-AdS black holes. Mainly, we found that such behaviour and the critical mass are present in the Schwarzschild-de Sitter background. Also, we found that the value…
The string-black hole correspondence is considered in the context of the correspondence principle proposed recently by Horowitz and Polchinski. We demonstrate that the entropy of string states and the entropy of a Schwarzschild black hole can be matched including the subleading terms which depend on mass logarithmically. We argue the necessity to include the string interaction (with coupling $g$) in the consideration and propose the $g^2$-dependent modification of the string entropy. The matching of it with the entropy of Schwarzschild black hole is analyzed.
We study physical properties of a Schwarzschild-type black hole in the framework of the recently proposed Starobinsky-Bel-Robinson (SBR) modified theory of gravity, working perturbatively in the new coupling constant. In particular, we compute the temperature, entropy, pressure and lifetime of a Schwarzschild-type black hole.
We revisit the classical stability of Schwarzschild black hole in fourth-order theories of gravity. On the contrary of the stability of this black hole, it turns out that the linearized perturbations exhibit unstable modes featuring the Gregory-Laflamme instability of five-dimensional black string. This shows clearly an instability of black hole in fourth-order gravity with $α=-β/3$.
Thermodynamic metric usually works only for those black holes with more than one conserved charge, therefore the Schwarzschild black hole was excluded. In this letter, we compute and compare different versions of offshell thermodynamic metric for the Schwarzschild-like black hole by introducing a new degree of freedom. This new degree of freedom could be the running Newton constant, a cutoff scale for regular black hole, a noncommutative deformation, or the deformed parameter in the nonextensive Tsallis-Renyi entropy. The onshell metric of the deformed Schwarzschild solution would correspond t…
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 analytically calculate to second order the correction to the asymptotic form of quasinormal frequencies of four dimensional Schwarzschild black holes based on the monodromy analysis proposed by Motl and Neitzke. Our results are in good agreement with those obtained from numerical calculation.
In this paper, we investigate a tunneling mechanism of massless particles from the Schwarzschild black hole in the framework of polymer quantum mechanics. According to the corresponding invariant Liouville volume, we determine the tunneling rate from Schwarzschild black hole by the polymeric quantization procedure. In this regard, we show that the temperature and tunneling radiation of the black hole receive new corrections in such a way that the exact radiant spectrum is no longer precisely thermal.
The role played by the mutual information of subsystems on the Page curve is explored in this paper. With the total system consisting of the black hole and radiation, together with the inclusion of island, we observe that the vanishing of mutual information between $B_+$ and $B_-$ which in turn means the disconnected phase of the entanglement wedge corresponding to $B_+\cup B_-$, yields a time scale of the order of scrambling time. This results in a time independent expression for the fine grained entropy of Hawking radiation consistent with the correct Page curve. We also find corrections to …
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 …
In this paper, we successfully derive the Bekenstein-Hawking entropy for Schwarzschild black holes in various dimensions by using a non-trivial phase space. It is appealing to notice that the thermodynamics of a Schwarzschild black hole actually behaves like that of a $1$-dimensional quantum mechanical system. Our result suggests that black hole should be viewed as a system with the equation of state $P=ρ$, and it also suggests that a holographic stage should exist in the early universe.
We study the scalar radiation emitted by a source orbiting a Schwarzschild--anti--de Sitter black hole in circular geodesics. We obtain the emitted power using quantum field theory in curved spacetimes at tree level with reflective boundary conditions at infinity. We compare our results with the ones obtained in Schwarzschild and Schwarzschild--de Sitter ($Λ> 0$) spacetimes. In the Schwarzschild-de Sitter spacetime we observe an enhancement of the emitted power associated to the lower multipole modes, when compared with the Schwarzschild case. In contrast, for the Schwarzschild-anti-de Sitter …
Intersecting M-branes are known to describe multi charged black holes. Using a configuration of such intersecting branes and antibranes, together with massless excitations living on them, we give a description of Schwarzschild black holes following Danielsson, Guijosa, and Kruczenski. We calculate the entropy of these black holes and find that it agrees, upto a numerical factor, with the entropy of the corresponding Schwarzschild black holes in supergravity approximation. We give an empirical interpretation of this factor.
The analytic expression obtained in the preceding project for the massless conformal scalar propagator in the Hartle-Hawking vacuum state for small values of the Schwarzschild radial coordinate above $ r = 2M$ is analytically extended into the interior of the Schwarzschild black hole. The result of the analytical extension coincides with the exact propagator for a small range of values of the Schwarzschild radial coordinate below $ r = 2M$ and is an analytic expression which manifestly features its dependence on the background space-time geometry. This feature as well as the absence of any ass…
We study the islands and the Page curve in the 1+1-dimensional eternal dilaton black hole models. Without islands, the entanglement entropy of the radiation grows linearly at late time. However with an island, its growth stops at the value of almost twice of the black hole entropy. Therefore an island emerges at the late time, and the entanglement entropy of the radiation shows the Page curve.
In this paper, we investigate the phase transitions of Reissner-Nordström (RN) and Schwarzschild black holes for the extended uncertainty principle (EUP) framework. Considering temperature $T$, charge $Q$ and electric potential $Φ$ as the state parameters, we show the van der Waals (vdW) like phase transition of RN black hole in $Q-Φ$ diagrams and find the critical points depending on EUP parameter $α$. Furthermore, we find Hawking-Page like phase transition for Schwarzschild black hole. The results imply that the black holes in asymptotically flat space have the similar phase structure with t…
We study the thermodynamic phase transition of a quantum-corrected Schwarzschild black hole. The modified metric affects the critical temperature which is slightly less than the conventional one. The space without black holes is not the hot flat space but the hot curved space due to vacuum fluctuations so that there appears a type of Gross-Perry-Yaffe phase transition even for the very small size of black hole, which is impossible for the thermodynamics of the conventional Schwarzschild black hole. We discuss physical consequences of the new phase transition in this framework.
Following the monodromy technique performed by Motl and Neitzke, we consider the analytic determination of the highly damped (asymptotic) quasi-normal modes of small Schwarzschild-de Sitter (SdS) black holes. We comment the result as compared to the recent numerical data of Konoplya and Zhidenko.
The entropy of the four dimensional Schwarzschild black hole is derived by mapping it onto a configuration of intersecting branes with four charges. This configuration is obtained by performing several boosts and dualities on a neutral black brane of M-theory to which the Schwarzschild black hole is related by trivial compactification. The infinite boost limit is well-defined and corresponds to extremality where the intersecting brane configuration is a marginal one on which a standard microscopic counting of the entropy can be safely performed. The result reproduces exactly the Bekenstein-Haw…
The scalar scattering cross section is computed for Schwarzschild-Tangherlini black holes on the brane. The cross sections are computed numerically via the partial-wave method. The phase shifts are found numerically and also via the Born approximation. It is shown that spacetimes with more than six dimensions present a finite scattering cross section in all directions.
Recent work has established a route towards the semiclassical validity of the Page curve, and so provided evidence that information escapes an evaporating black hole. However, a protocol to explicitly recover and make practical use of that information in the classical limit has not yet been given. In this paper, we describe such a protocol, showing that an observer may reconstruct the phase space of the black hole interior by measuring the Uhlmann phase of the Hawking radiation. The process of black hole formation and evaporation provides an invertible map between this phase space and the spac…
We investigate the effect of quantum correction on Hawking radiation for Schwarzschild black holes. We consider Hawking temperature and entropy to order G^2 and find that the area law of black holes should be modified. We think of Hawking radiation as tunneling and find that at certain frequency ω the radiation spectrum of black holes can be pure blackbody spectrum. Therefore, it is possible to break down information conservation. In order to ensure information conservation at any energy scales, we suggest that black holes cannot release all information they have. We briefly discuss the bound …
The holographic conjecture for black holes localized on a 3-brane in Randall-Sundrum braneworld models RS2 predicts the existence of a classical 5D time dependent solution dual to a 4D evaporating black hole. After briefly reviewing recent criticism and presenting some difficulties in the holographic interpretation of the Gregory-Laflamme instability, we simulate some basic features of such a solution by studying null geodesics of the Schwarzschild black string, in particular those propagating nontrivially in the bulk, and using holographic arguments.
