Authors:
(1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute.
Table of Links
PART I
Chapter 2: SU(3) LECs from Type IIA String Theory
Chapter 4: Conclusion and Future Outlook
PART II
Chapter 6: Page Curves of Reissner-Nordström Black Hole in HD Gravity
Chapter 8: Black Hole Islands in Multi-Event Horizon Space-Times
Chapter 9: Multiverse in Karch-Randall Braneworld
Chapter 10: Conclusion and Future outlook
Part-II (HD) Gravity Islands, and Multiverse
“God does not play dice.” - Albert Einstein
“God not only plays dice, he sometimes throws the dice where they cannot be seen.” - Stephen Hawking
“If God throws dice where they cannot be seen, they cannot affect us.” - Don Page
CHAPTER 5 - INTRODUCTION
In this chapter, we present the introduction of the materials needed to understand the information paradox and its resolution from holography. We start with the discussion on entanglement entropy in section 5.1, we discuss the information paradox and the Page curve in section 5.2 and finally we discuss the resolution of the information paradox in 5.3 from the island proposal, doubly holographic setup and wedge holography in 5.3.1, 5.3.2 and 5.3.3 respectively
5.1 Holographic Entanglement Entropy: Ryu-Takayanagi and Dong’s Proposals
Entanglement Entropy in Quantum Mechanics (QM): Let us first discuss the entanglement entropy in quantum mechanical system. Let us consider a system whose state is denoted by |ψ⟩. The density matrix of the system is defined as:
Entanglement entropy is measured by the von-Neumann entropy. For this, first we have to partition the system into two subsystems A and B. States in the subsystems A and B are denoted by |ψ⟩A and |ψ⟩B such that |ψ⟩ = |ψ⟩AB = |ψ⟩A ⊗ |ψ⟩B . Reduced density matrix of the subsystem A is obtained by tracing over the degrees of freedom of the subsystem B and vice-versa.
Now, the von-Neumann entropy is defined as:
Entanglement Entropy in Quantum Field Theory (QFT): It is not easy to compute the entanglement entropy in quantum field theories (QFTs) by factoring the system into subsystems because factorization is not always possible in QFTs. Entanglement entropy in QFT is calculated using the replica trick. First, let us define the Renyi entropy:
• We need to find out a co-dimension two surface (ϵA) in the bulk Md+1 which is anchored on ∂A.
• There is possibility of many surfaces but we have to consider the one which satisfy the homology constraint, i.e., ϵA is smoothly retractable to the boundary region A.
• Out of those surfaces which satisfy homology constraint, we need to pick the one with the minimal area then the entanglement entropy is defined as:
The Ryu-Takayangi formula has certain limitation, it is applicable to the the timeindependent backgrounds. For the time dependent background, one is required to use the HRT formula [125] where HRT stands for Hubney, Rangamani and Takayanagi. Quantum corrections to all order in ℏ to the Ryu-Takayanagi formula was incorporated in [126] where one is required to extremize the generalised entropy. Surfaces which extremize the generalised entropy are known as quantum extremal surfaces(QES). If there are more than one quantum extremal surfaces then we need to consider the one with minimal area. In [6], authors generalised the QES prescription to island surfaces where we are required to extremize the generalised entropy like functional which includes contribution from the island surfaces. In this case, extremal surfaces are known as quantum extremal islands. Since, in this thesis, we are confining ourselves to the time independent backgrounds and therefore we will not discuss the HRT formula.
• Let us label the each term as αth term after getting the final expression of the second term which is obtained from the differentiation of Lagrangian with respect to Riemann tensor twice.
• We need to perform the following transformations on certain components of the Riemann tensors:
The reason for discussing these proposals is that when we compute the Page curve of black holes in doubly holographic setup and wedge holography then these proposals will be useful. The holographic entanglement entropy has also been computed from the holographic stress tensor and surface terms in [128] and [129] respectively.
5.2 Hawking’s Information Paradox and Page Curve
Hawking’s black hole information paradox is a long-time puzzle that started with his papers [130, 131]. When matter collapses to form a black hole, the whole matter is stored in the singularity. The horizon of the black hole covers the black hole singularity. Initially, the system is in a pure state. Hawking studied the creation of particles in pairs with negative and positive energy in the presence of quantum effects, and he found that a particle with negative energy gets trapped inside the black hole, whereas the particle with positive energy scattered off to infinity is what we receive in Hawking radiation. We can get the radiation from the black hole due to quantum mechanics, which allows the possibility of quantum tunneling through a potential barrier. In the case of a black hole, the horizon acts as the potential barrier. Hawking calculated the spectrum of the particles coming out of the black hole and found that the spectrum behaves as the spectrum of thermal radiation with a temperature known as Hawking temperature, which implies a mixed state. This means that the black hole evolves from the pure state to the mixed state, and hence unitary evolution of quantum mechanics is not preserved. This leads to the famous “information paradox”.
Page suggested that when we include the quantum effects, then the black hole must follow the unitary evolution [132]. If we consider the black hole and radiation region as a single system, then one should get the Page curve to resolve the paradox. For the evaporating black hole, entanglement entropy of the Hawking radiation first increases linearly with time up to the Page time and then falls back to zero [132]. We are interested in eternal black holes, and for these black hoes, instead of falling to zero of the entanglement entropy, one obtains the constant entanglement entropy after the Page time, and this constant value is equal to the twice the thermal entropy of the black holes.
In this part of the thesis, we focus on getting the Page curve of eternal black holes using the recent proposals given in the literature, e.g., island proposal, doubly holographic setup, and wedge holography. Apart from getting the Page curve, we have also obtained other exciting results, which are discussed in the upcoming chapters.
5.3 Resolution of Information Paradox from Holography
Following three proposals are available in the literature which started with the idea of holography to resolve the black hole information paradox.
5.3.1 Island Proposal and its Extension to HD Gravity
Authors in [6] proposed a method to resolve the information paradox which is equivalent to getting the Page curve. Idea is that at early times we get only contribution from the radiation region which gives the divergent part of entanglement entropy at late times because the entanglement entropy of Hawking radiation turns out to be proportional to the time. According to [6] at early times situation remains the same whereas at late times interior of the black holes becomes part of the entanglement wedge and hence at late times entanglement entropy receives the contributions from the radiation as well as interior of the black holes. The part of the interior the black holes which contributes to the entanglement entropy is known as “island”.
Island rule was proposed from a setup where we couple the evaporating JT(Jackiw Teitelboim) black hole plus conformal matter on the Planck brane with the two-dimensional CFT bath [6]. The black hole is contained on the Planck brane, and the Hawking radiation is collected in the 2D conformal bath. This setup has the following three descriptions.
• 2D-Gravity: The Planck brane is coupled to the external CFT bath, which acts as the sink for the Hawking radiation.
• 3D-Gravity: Two-dimensional conformal field theory has the three-dimensional gravity dual with metric AdS3 via AdS/CFT correspondence.
• QM: The boundary of the external CFT bath is one dimensional where quantum mechanics (QM) is present.
The island formula was derived from the gravitational path integral using the replica trick for special JT black holes in [133, 134]. The authors obtained the Page curve from the disconnected and connected saddles. One obtains the linear time dependence in the Page curve from the disconnected saddles, whereas connected saddles produce the finite part of the Page curve. The discussion of [133] holds for the replica wormholes with n boundary as well. The generalised entropy in the presence of island surface is written as follows:
where R, GN and I are representing the radiation region, Newton constant and the island surface. Equation (5.11) contains two terms: island surface’s area and the matter contribution from the radiation and island regions. From (5.11), we can easily see that when island surface is absent then Sgen(r) = Smatter(R). It has been shown in the literature that island surface emerges at late times and hence initially one obtains the linear time dependence in the Page curve and at late times, when island surface’s contribution dominates then one obtains the fall of the entanglement entropy for the evaporating black holes whereas constant part (twice of their thermal entropies) for the eternal black holes. Hence, when we include these contributions, we obtain the Page curve. If there are more than one island surfaces then we have to consider the one with the minimal area. We have followed this proposal to obtain the Page curve of Schwarzschild de-Sitter black hole in [12] and discussed in detail in the chapter 8 of this thesis. See [135–137] for the application of island proposal in the context of JT gravity and other issues [138–140].
Island proposal was extended for higher derivative gravity in [141]. The proposal is exactly similar to [6] but we have to replace the first term of (5.11) by the term which can give the information about the entanglement entropy of higher derivative gravity and the formula for the same was proposed by X. Dong in [127] and hence the island proposal in the presence of higher derivative terms in the gravitational action is written as [141]
where Smatter is the same as the Smatter(R ∪ I) of (5.11) and Sgravity will be calculated using the Dong’s formula [127]. For the AdSd+1/CF Td correspondence the Dong’s formulas is given below[1].
where
Figure 5.1: Description of doubly holographic setup. Blue curves are the island surfaces and red curve is the Hartman-Maldacena surface. δM is the conformal boundary, z∗ and zT are the turning points of Hartman-Maldacena and island surfaces.
5.3.2 Doubly Holographic Setup
The doubly holographic setup is a nice setup to calculate the Page curve of black holes. As its name suggests, it is the double copy of the usual holography proposed by J. Maldacena. First, we need to take the bulk and truncate the geometry along one of the spatial coordinates [142, 143]. By doing so, one generates d-dimensional geometry embedded in the (d + 1)- dimensional bulk. The d-dimensional geometry is known as end-of-the-world brane or KarchRandall brane in the literature, and this holography is called “braneworld holography”. The doubly holographic setup is obtained by joining the two copies of the Karch-Randall model. The setup consists of an eternal black hole living on the brane and two baths where we can collect the Hawking radiation. These two baths behave as thermofield double states because these are like two copies of the boundary conformal field theory (BCFT). Let us discuss the double holography in the context of AdSd+1/BCFTd correspondence using a bottom-up approach, and the setup is shown in figure 5.1.
The doubly holographic setup has the three descriptions summarised below.
• Boundary description: d-dimensional BCFT at the conformal boundary of the bulk AdSd+1. The boundary of BCFTd is the (d − 1) dimensional defect.
• Intermediate description: Gravity on the end-of-the-world brane is coupled to BCFT via transparent boundary condition at the defect.
• Bulk description: The holographic dual of BCFTd is AdSd+1 spacetime.
The intermediate description is very crucial to resolving the information paradox. Because in this description black hole living on the end-of-the-world brane directly couples with the external CFT bath. Define S(R) as the von Neumann entropy of the subregion R on a constant time slice in description 1. One can obtain the S(R) in second description from the island rule [6]:
where generalised entropy functional (Sgen(R ∪ I)) is [126]:
A doubly holographic setup is advantageous in the sense that we can obtain S(R) very easily using the classical Ryu-Takayanagi formula [107]. When bulk is (d + 1) dimensional then [107]:
where γ is the minimal co-dimension of two surface in bulk.
In figure 5.1, there are two BCFTs on the conformal boundary of the bulk. The vertical line is the end-of-the-world brane which contains the black hole. The CFT bath collects the Hawking radiation emitted by the black hole. This setup has two possible extremal surfaces: Hartman-Maldacena [144] and island surfaces. The Hartman-Maldacena surface connects the two BCFTs; it starts at the CFT bath, crosses the horizons, reaches up to the turning point, and then meets the thermofield double partner of BCFT. The entanglement entropy is divergent at late times for the Hartman-Maldacena surface, which implies Hawking’s information paradox. The island surface starts at the external CFT bath and lands on the end-of-the-world brane. The island surface’s entanglement entropy turns out to be a constant value (twice of thermal entropy of the black hole). Therefore, one recovers the Page curve by combining the contributions of the entanglement entropies of both these extremal surfaces. See [7, 145–161] for the extensive literature on the doubly holographic setup.
Some authors found that gravity is massive on the end-of-the-world brane [162–165] when we couple the brane to the external CFT bath. In some papers, it was shown by the authors that we could construct the doubly holographic setup with massless gravity on the brane [11, 154, 166, 167]. We constructed the doubly holographic setup from a top-down approach in [11] and details are given in chapter 7. We have a non-conformal bath (QCD2+1) and the holographic dual is M theory inclusive of O(R4 ) corrections [1]. The reason for the existence of massless graviton in our setup is that we required the wave function of the graviton to be normalized, the second reason is due to the Dirichlet boundary condition on the wave function of the graviton, and the third reason is that end-of-the-world brane had non-zero tension and hence localization of graviton is possible on the brane in a “volcano”- like potential. We obtained the Page curve with massless gravity in our setup, which was impossible in other doubly holographic setups without the DGP term. One alternate method to deduce the massless gravity on the brane is to include the Dvali-Gabadadze-Porrati (DGP) term [168] on the brane [166].5.3.3 Wedge Holography
5.3.3 Wedge Holography
In the doubly holographic setup, the external bath is a fixed CFT bath. In some of the papers, it was found that gravity is massive on the end-of-the-world brane and island prescription is not valid in the masssless gravity. Some of the authors considered the bath as gravitating too [8, 9, 162, 169]. This setup is known as wedge holography in the literature. It was also argued that in wedge holography, Hartman-Maldacena surface does not exist and hence no Page curve in wedge holography. In [13], we showed that entanglement entropy of HartmanMaldacena surface is non-zero for the AdS and Schwarzschild black hole and it is zero for the de-Sitter black hole. This implies that one could get the Page curve for the AdS and Schwarzschild black hole but not for the de-Sitter space using wedge holography. See Fig.5.2 for the pictorial description of the wedge holography. One can get the Page curve or not in wedge holography is a debatable topic. Some progress in this direction has been made in [166]. It was shown by the author that we can get the Page curve with massless gravity localize on the Karch-Randall brane provided we have to include the DGP term on the Karch-Randall brane, see [170, 171] for the detailed analysis with examples.
Figure 5.2: Description of wedge holography. Two d-dimensional Karch-Randall branes joined at the (d − 1) dimensional defect, Karch-Randall branes are embedded in (d + 1)-dimensional bulk.
Take into consideration the following action, [8, 9, 169], to describe the mathematical description of wedge holography:
The above equation has the following solution [9]:
Similar to the double holography, wedge holography has also the three descriptions:
• Boundary description: BCF Td on the conformal boundary of the bulk AdSd+1 with the (d − 1) dimensional defect.
• Intermediate description: two gravitating systems are connected with each other via the transparent boundary condition at the defect.
• Bulk description: the holographic dual of BCFTd is classical gravity AdSd+1 spacetime.
The wedge holographic dictionary for the (d + 1)- dimensional bulk is stated as: holographic dual of the (d−1)-dimensional defect conformal field theory is the classical gravity in (d+1)- dimensions. Therefore it is a co-dimension two holography. Now let us understand the how this duality exists.
Braneworld holography [142,143] relates first and second line whereas AdS/CFT correspondence [17] between the dynamical gravity on the Karch-Randall brane and defect CFT connects second and third line. Therefore, classical gravity in (d+ 1) bulk is dual to CFTd−1 at the defect. Wedge holography helps us in getting the Page curve of the black holes similar to doubly holographic setup discussed in 5.3.2. One is required to compute the entanglement entropies of Hartman-Maldacena and island surfaces and plot of these entropies with time will give the Page curve.
[1] We have already written the formula in (5.1), here we are writing the covariant form of (5.1). In this formula, a and i, j represents the tangential and normal directions.
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