Page curves for accelerating black holes

The island paradigm for the fine-grained entropy of Hawking radiation is applied to eternal charged accelerating black holes. In the absence of the island, the entanglement entropy grows linearly and divergent at late times, while once the island outside the event horizon is taken into account, the unitary Page curve is reproduced naturally. The impact of the charge and the acceleration on Page curves is investigated at late times. For the Page time and the scrambling time, they both increase as the acceleration increases, while decreasing as the charge increases. In particular, neutral black holes have the largest Page time and scrambling time. It is worth noting that the Page time and the scrambling time is divergent at the extremal case, which implies that islands may be related to the causal structure of spacetime.


Introduction
Quantum gravity is one of the most active fields in modern physics.The black hole information paradox, which lasted about half a century, is one of the key signposts to the theory of quantum gravity [1].Hawking calculated in 1974 that black holes have a temperature and can emit Hawking radiation, which resembles black-body radiation [2].As the black hole evaporates, it takes the information that falls into the black hole with it.Thus, Hawking predicted in 1976 that the evaporation of a black hole would lead to information loss, known as the black hole information paradox [1].This paradox leads to persistent debate and studies in the field of theoretical physics.On the one hand, through quantum mechanics, the evolution of physical systems must follow the principle of the unitary.On the other hand, Hawking radiation is produced using quantum fluctuations near the event horizon of a black hole, so the information loss undermines the unitary.More specifically, a black hole formed from a pure state will evolve into a mixed state during evaporation.
The issue has taken a turn for the better until 1993 -The Page curve [3,4], which describes the entanglement entropy of Hawking radiation is depicted in detail by the Page theorem [5] under the unitary hypothesis.The black hole information paradox can be solved if the Page curve can be calculated directly from semi-classical gravity without this hypothesis.Although still difficult, the development of the holographic principle provides confidence in this issue [6].The AdS/CFT duality has been an important and fascinating topic, not only providing a key link between the theory of gravity in AdS (anti-de Sitter) spacetime and the CFT (conformal field theory) that lives at its boundary, but also opening a window into quantum gravity.Many recent advances in holographic descriptions of the evaporation of black holes further support the view of information conservation.The Ryu-Takayanagi (RT) formula for calculating the holographic entanglement entropy is undoubtedly the best of these works [7,8].When one considers higher-order quantum corrections, the RT formula is generalized to the quantum extremal surface (QES) formula of the entropy functional of the sum of the contributions of the bulk field and the area term [9].Once we use the QES formula to calculate the entropy of Hawking radiation based on the semi-classical gravity method, the Page curve is naturally reproduced [10].In particular, the most critical point is that the bulk field contribution contains a disconnected region located in the interior of the black hole, called the "island " [11].The idea of reconstructing the Page curve by correctly tracking the radiation outside black holes and the interior of black holes is summarized as the "entanglement wedge reconstruction" [12].Finally, we can express the formula for calculating the fine-grained entropy of Hawking radiation as the so-called "island formula" [13]: where S gen represents the generalized entropy of radiation, which can be written as where the first term is the area term of islands, in which I refer to the island, ∂I is denoted the boundary of the island.The second term is the entropy that is contributed by matter fields around the black hole.One should note that besides the radiation region R, the region I inside the black hole is also considered.It also provides a glimmer of insight into the interior of black holes.In brief, the island formula (1.1) gives us an instruction: in the explicit calculation, one first extremize the generalized entropy (1.2), these saddle points corresponding to positions of QES, namely the island.Then, the minimal value of the entropy is our candidate.For the whole process of an evaporating black hole, at early times, the QES is just a trivial (vanishing) surface, so there is no island at early times, which leads to a Hawking curve.However, the minimal condition requests that the QES is a non-trivial (non-vanishing) surface at late times, which provides a decreasing curve for the entropy.
Therefore, combining two curves, we finally obtain the Page curve.The island formula (1.1) then can be rewritten as follows: S Rad = Min S gen (without island), S gen (with island) . (1. 3) The first fruits of victory were harvested in the evaporating Jackiw-Teitelboim (JT) black hole in two-dimensional (2D) gravity, where the auxiliary thermal bath is coupled to the asymptotically AdS black hole [10,14].However, the range of adaptation of the island paradigm extends far beyond the framework of AdS black holes, one can refer to  for an inexhaustive list of recent studies.
There is no doubt that 2D gravity, despite the simplicity of calculations it provides, is still only a toy model.The more representative and acceptable Einstein gravity, in which the Kerr-Newmann family is a typical four-dimensional (4D) black hole prototype.They are described by the mass, charge, and angular momentum parameters.However, there is a littleknown exact solution for a 4D black hole: the "C-metric", which describes an accelerating black hole [74][75][76].For this kind of interesting black hole, there is a conical defect angle along the polar axis of the black hole.It is because of the defect angle that it provides the driving force for the black hole to accelerate.The exact solution described by the "C-metric" is ideal.Nevertheless, its applications are widespread beyond the theory of classical general relativity.It can also describe the generation of black hole pairs in electromagnetic fields [77], and the splitting of cosmic strings [78], especially recent work on its thermodynamics [79].More work in this field can be referred to [80][81][82] Based on these remarkable special properties of the C-metric, in this paper we investigate the information issue in the charged version of the C-metric.In order to solve the information issue, we calculate the entanglement entropy of the charged accelerating black hole explicitly using the island formula (1.1) and reconstruct the corresponding Page curve.We also discuss the effects of the charge and the acceleration, in particular, for the neutral black hole, which has the largest Page time.In addition, it is noted that in the extremal case, due to the vanishing of the temperature, some physical quantities become divergent, which strongly implies that quantum information is closely related to the causal structure of spacetime.
The structure of this paper is as follows.In Sec.2, we briefly review some properties of the charged C-metric.In Sec.3, the entanglement entropy of Hawking radiation in the with and without island configuration is calculated by the island paradigm.In Sec.4,we obtain the Page curve satisfying the unitary and the scrambling time, then discuss the impact of the charge and the acceleration parameters.The summary is in Sec. 5.In what follows, we will use the nature units where c = = k B = 1.

Review for Accelerating Black Holes
In this section, we give a brief review of accelerating black holes.Physically, the accelerating black holes can be produced by a cosmic string collision with a black hole, so that a gravitational interaction could result in an acceleration of the black holes.The accelerating black hole can be described by the C-metric, which represents a pair of black holes that accelerating away from each other in an opposite direction.For the charged version of the C-metric, it can be regarded simply as a generalization of Reissner-Nordström (RN) black holes.Besides the parameters of mass M and electric charge Q, it also includes an extra parameter which associated with the acceleration α.This charged C-metric can be written in a spherical coordinate as follows [83] where ) The gauge potential is given by One can easily finds that this metric is asymptotically flat.In the limit α → 0, the charged C-metric approaches the RN black hole.As for the main features of accelerating black holes, one of the key features is that they would be moving through space at a constant velocity rather than remaining stationary.Another important property of black holes is that these accelerating black holes would emit gravitational waves.Further, in the vanishing charge case, one can obtain Schwarzschild black holes.There is an intrinsic singularity at r = 0, but there exist three coordinate singularities at the root of the metric function f (r), then we define the following constraints: ) which corresponds to the acceleration horizon, the event horizon, and the Cauchy horizon, respectively.In order to avoid the naked singularity, one must insist on the following relation: Thus, for non-extremal case, namely r + < r − < r α , the metric function f (r) is always positive, which also imply that the angular function P (θ) is also positive for the range [0, π].
Different from RN black holes, in addition to become extremal when M = Q, i.e. the event and Cauchy horizons are coincides.The accelerating black hole can also becomes extremal for α = 1 r + , which is called the Nariai limit.In this paper, we mainly consider the non-extremal case for simplicity.
The other interesting property of accelerating black holes is that there are two conical singularities at the axis at θ = 0 and θ = π 2 .Therefore, the ratio of the circumference to the radius there is not to equal to 2π.Nevertheless, one can still remove one of two conical singularities by resetting the coordinate φ.More specifically, we first consider a range for φ ∈ [0, 2πa), where a is a constant.Then, if there is regular at θ = 0 with a constant t and r, we obtain where C m and R stand for "circumference" and "radius", respectively.Similarity, if the regularity is insisted at θ = π, then Thus, the equations (2.6) and (2.7) indicate that there are different conical singularities for different conicities.One of the two conical singularities can be eliminated, but not both when we choose a reasonable constant a.For example, if one selects that a = P −1 (0), then the deficit angles at the poles θ = 0 and θ = π are [83] In this way, we remove the conical singularity at θ = 0 and left a deficit angle at θ = π.
Likewise, if we choose a = P −1 (π), which leads to a deficit angle at θ = 0 and the regularity at θ = π is maintained, i.e.
Henceforth, we choose a = P −1 (0) and preserve the equation (2.8) for simplicity.Now, we introduce some thermodynamic qualities.For the Hawking temperature, we can use the standard Euclidean method: we analytically continue to the Euclidean signature, namely with β is the inverse temperature.Then the temperature of black holes is determined by Meanwhile, one can determine the Bekenstein-Hawking entropy, which is a quarter of the horizon area A [79] (2.12) This clearly differs from the RN black hole in which the extremal case corresponds to the finite entropy.This means that entropy is not well-defined in the Nariai limit.However, the entropy is finite in the charged limit.Finally, we perform the Kruskal transformation to obtain the maximum extended spacetime.For the charged C-metric (2.1), we define the tortoise coordinate as .
(2.13)So the null Kruskal coordinates are 1 where κ + = 2πT H is the surface gravity on the event horizon.Thus, the metric (2.1) can be recast as with the conformal factor . (2.16) Here, we have rescaled the metric (2.1) conformally by Ω (2.2c): ds 2 = Ω 2 ds 2 .Such rescaling would provide a simpler form but not affect the physical results in the following section.

Island Paradigm for Black Holes
In this section, we investigate the entanglement entropy between the radiation and black holes, together with the inclusion of the island.We start with the no-island case, then consider the effect of islands.

Early times: Without Island
The total system is a thermofield double state, or equivalently, the Hartle-Hawking state, since we consider an eternal black hole.We first assume that the black hole is formed by a pure state and then make an S-wave approximation.Under this assumption, we can ignore the angular directions θ and φ, and only focus on the contribution of the radial direction.
Thus, the metric (2.15) is reduced to a 2D form: The corresponding Penrose diagram is plotted in Figure .1.In the absence of island, there is only the radiation part left in the island formula (1.1).For a thermofield double state We substitute the coordinate of b ± , then obtain the result as (3.5) At late times limit, t κ + , we can use the approximation: cosh(κ + t b ) 1 2 e κ + t b , then . (3.6) One can obviously see from this result that the entanglement entropy of radiation is proportional to the time t b for the observer at infinity and the temperature of black holes.
Besides, we also find that increasing acceleration will decrease the rate of growth.Finally, the entanglement entropy behaves divergent at late times.In particular, for an eternal black hole, the amount of radiation is infinite at late times, which leads to the entanglement entropy far exceeding the Bekenstein-Hawking entropy bound [85].Therefore, there exists an intuitive paradox here: on the one hand, our result provides a linear increase in the entanglement entropy; On the other hand, the fine-grained (entanglement) entropy must be small than the coarse-grained (Bekenstein-Hawking) entropy.We expect to solve this thorny issue through the island paradigm.

Late times: With Island
Now we introduce the disconnected island region in our system by following the island paradigm.The Penrose diagram is shown in Figure .2.At this time, the von Neuman entropy of the matter part is given by [84] S However, we still assume that the late times and the large distance are allowed.This approximation allows us to rewrite the above equation into a simpler form and (3.9) Accordingly, the generalized entropy (1.2) read as (3.10) Extremizing the above expression with respect to t a firstly Thus, the only solution for the equation is t a = t b .Substituting this relation and then extremizing it with respect to a: Where x ≡ r (b) − r (a), and f (r) is the derivative of f (r).Now, we make the near horizon limit, where a r + .By taking this approximation, the above equation is recast as Thus, we finally obtain the location of the quantum extremal island is Therefore, at late times, the entanglement entropy of Hawking radiation with the island is given by Therefore, we find that once we consider the contribution of an island, the fine-grained entropy of radiation approaches a saturated value eventually.This is a surprising result which is the opposite of Hawking's calculation: at late times, the entanglement entropy of radiation stops increasing and tends to be about twice of the Bekenstein-Hawking entropy.
This conclusion also respects the principle of unitary.Now the black hole has enough degree of freedom to entangle with the external radiation, since now the fine-grained entropy of black holes is smaller than the coarse-grained/Bekenstein-Hawking entropy.Therefore, we obtain the expression of the fine-grained entropy of Hawking radiation for the whole process of the evaporation where the effect of charges and acceleration is absorbed in the temperature T H and the Bekenstein-Hawking entropy S BH , which we will discuss in detail below.

Page Curve and Scrambling Time
At present, we have calculated the behavior of the entanglement entropy of radiation from the discussion of applying the island paradigm in the previous section.Therefore, the natural thing to do in this section is to plot the corresponding Page curve and obtain some byproducts, such as the scrambling time of black holes.
At first, we derive the Page time that defined as the moment when the entanglement entropy approaches the peak.In the language of the entanglement island, the entropy without island (3.6) is roughly equal to the entropy with island (3.15) at the Page time.
Accordingly, we obtain the Page time is The corresponding graphs as a function for the charge Q and the acceleration α are shown in which the acceleration α is fixed (in the unit of 6π cG N and set M = 1).On the left, it is the non-extremal case with a small amount of charges, while on the right is the near extremal case.
Through these figures, one can easily find that in the α-fixed case, the Page time increases as charges increase.But in the Q-fixed case, the Page time decreases first and then increases with the acceleration increasing.In particular, the Page time reaches its maximum for neutral black holes (Q = 0).Note that these conclusions are applied only to non-extremal black holes.However, once the acceleration or charges makes the black hole approaches an    We obtain the expression of the scrambling time by invoking the location of the island (3.14) In the last line, we assume that b has the order O(r + ) and the acceleration is not large (α 1 r + ).This result follows the conclusion of the initial Hayden-Preskill experiment [87,88].Then, the scrambling time as a function of the acceleration α and the charge Q is plotted in Figure .6.
In the same way, the scrambling time decreases when the charge increases, while as the acceleration increases, it decreases and then increases.The neutral black hole also has the greatest scrambling time.However, in the extremal case, the scrambling time is also divergent.This is for the same reason that the Page time diverges in extremal cases.We will discuss some remarks in the next section.

Conclusion and Discussion
In this paper, we study the information paradox in the charged "C-metric" that describes an accelerating black hole.Different for the RN black holes, the existence of acceleration parameters α results in accelerating black holes existing three horizons.Accordingly, the corresponding geometry is richer.Therefore, it is very meaningful to study the page curve under

Figure 1 :
Figure 1: The Penrose diagram for the charged C-metric.The solid lines represents null hypersurfaces.The dashed lines are identity.H ± , H ± c , and H ± a represents the future/past event horizons, acceleration horizons, and Cauchy horizons, respectively.Blue lines are labeled as radiation regions R ± , where the boundaries are noted by b ± = (±t b , b) for the left and right wedge.The singularity is located at r = 0.

Figure 2 :
Figure 2: The Penrose diagram for charged accelerating black holes with an island.The disconnected island region is labeled as I, its boundary is a ± = (±t a , a).

Figure 3 :
Figure 3: The Page time by charged accelerating black holes as the function for charges Q, extremal black hole, the Page time behaves divergent.Finally, the Page curves for nonextremal charged accelerating black holes are given in Figure.5.

Figure 4 :
Figure 4: The Page time by charged accelerating black holes as the function for the acceleration α (in the unit of 6πcG N and set M = 1).On the left is the non-extremal case, while on the right is the near extremal case.

Figure 5 :
Figure 5: The Page curve for eternal charged accelerating black holes, where setting M = c = G N = 1.On the left, we fix the acceleration α = 1 2r + .On the right, we fix the charge Q = 0.5M .

Figure 6 :
Figure 6: The scrambling time by black holes as functions of charges Q and accelerations α.Here we set the mass M = G N = 1.