The Multiverse in Karch-Randall Braneworld

Table of Links

Abstract

Acknowledgment

PART I

Chapter 1: Introduction

Chapter 2: SU(3) LECs from Type IIA String Theory

Chapter 3: Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate Coupling in the Absence and Presence of Rotation

Chapter 4: Conclusion and Future Outlook

PART II

Chapter 5: Introduction

Chapter 6: Page Curves of Reissner-Nordström Black Hole in HD Gravity

Chapter 7: Entanglement Entropy and Page Curve from the M-Theory Dual of Thermal QCD Above Tc at Intermediate Coupling

Chapter 8: Black Hole Islands in Multi-Event Horizon Space-Times

Chapter 9: Multiverse in Karch-Randall Braneworld

Chapter 10: Conclusion and Future outlook

APPENDIX A

APPENDIX B

APPENDIX C

Bibliography

CHAPTER 9 - MULTIVERSE IN KARCH-RANDALL BRANEWORLD

9.1 Introduction

Wedge holography is constructed by embedding two Karch-Randall (KR) branes in bulk. These branes are joined at the defect via the transparent boundary conditions so that degrees of freedom can be exchanged between them. The wedge holography is very useful for getting the Page curve of black holes. For the paper on wedge holography, see [169, 237–239]. Most of the papers discuss the application of wedge holography to resolve the information paradox. We have already discussed the wedge holography in chapter 5; for more details about the same, please see 5.3.3. As discussed earlier, the wedge holography contains two KR branes. A natural question that came to our mind is whether it is possible to construct the wedge holography with many KR branes instead of only two KR branes. If yes, then what will describe this setup? This is what has been addressed in this chapter based on the paper [13]. In the process of answering these questions, we found that the wedge holography with many KR branes describes a “Multiverse”. We would like to make a remark that we ask the question in reverse order, i.e., can we describe the Multiverse from wedge holography? In any way, we reach the same theoretical model. We explained the Multiverse by constructing the wedge holography in such a way that there are 2n KR branes that are embedded in the (d + 1)-dimensional bulk. The aforementioned branes have Einstein gravity localized on them. Therefore, we have 2n copies of the gravitating system, and all these gravitating systems are connected to each other via the transparent boundary condition at the defect. This model has been used to obtain the Page curve of black holes with multiple horizons and a qualitative idea to resolve the “grandfather paradox”. Let us discuss these concepts in more detail.

9.2 Emerging Multiverse from Wedge Holography

We’re going to discuss how we can construct a multiverse from wedge holography in this section. When discussing the multiverse, α and β will have 2n values, however when discussing wedge holography, α, β = 1, 2 as in 5.3.3.

9.2.1 Anti de-Sitter Background

In this case, we build a multiverse using AdS spacetimes. Let’s start with the most basic scenario covered in 5.3.3. We require many Karch-Randall branes at r = ±nρ such that bulk metric must satisfy the Neumann boundary condition at the locations mentioned above in order to define the multiverse. Extrinsic curvature associated with the Karch-Randall brane and related trace is calculated as:

The Einstein equation of bulk action (9.2) is going to be the identical to (5.19), therefore the solution is as follows:

Figure 9.1: Multiverse constructed from 2n Karch-Randall branes (Q−n,−n+1,...,1,2,...,n−1,n) which are d-dimensional gravitating objects and these branes are embedded in the (d + 1)-dimensional bulk. The dimension of defect P is (d − 1)

The equation mentioned above is obtained from the Einstein-Hilbert term, which includes a negative cosmological constant on the brane:

• Boundary description: d-dimensional boundary conformal field theory with (d−1)- dimensional boundary.

• Intermediate description: All 2n gravitating systems are connected at the interface point by transparent boundary condition.

• Bulk description: Einstein gravity in the (d + 1)-dimensional bulk.

Because there exists a transparent boundary condition at the defect in the intermediate description, the multiverse created in this configuration consists of communicating universes located on Karch-Randall branes. The following is a Wedge holography dictionary for the “multiverse” having 2n AdS branes.

The braneworld holography connects the first and second lines [142,143] and the AdS/CFT correspondence connects the second and third lines [17] since there is gravity on the KR branes. Hence, there exists co-dimensional two duality between the (d + 1)-dimensional classical gravity on AdSd+1 background and the (d−1)-dimensional defect conformal field theory, CF Td−1.

9.2.2 de-Sitter Background

The third line in the aforementioned duality originated from dS/CFT correspondence [240, 241]. The authors of [238] precisely computed the central charge of dual CFT and found that central charge is imaginary implying that CFT located at the defect is non-unitary. The aforementioned explanation also works for AdS bulk too. In the present scenario, the wedge holographic dictionary is:

To begin discussing the existence of the multiverse, we will use the bulk metric [9]:

By utilizing Neumann boundary condition (5.21) for the de-Sitter branes and substitution of (9.7) into (5.18), one may construct Einstein-Hilbert terms with a positive cosmological constant on the Karch-Randall branes; the resultant action is provided as follows:

• Boundary description: d-dimensional BCFT with (d − 1)-dimensional defect.

• Intermediate description: 2n gravitating systems with de-Sitter geometry connected to each other at the (d − 1)-dimensional defect.

• Bulk description: (d + 1)-dimensional Einstein gravity with negative cosmological constant in the bulk.

The first and third descriptions are connected with one another via AdS/BCFT correspondence, and a (d − 1)-dimensional non-unitary defect CFT arises due to dS/CFT correspondence [240,241]. de-Sitter space persists for a certain period of time and then ceases to exist. Another de-Sitter space formed following the end of the preceding one [242]. As a result, it is feasible to have a “multiverse” (say M1) containing de-Sitter branes if they are all formed at the same “creation time”(the “time” when any universe is born [242] is termed as creation time), but this will only persist for a finite period until M1 ceases to exist. Following the extinction of M1, another multiverse (say, M2) consisting of numerous de-Sitter branes created at exactly the same creation time as all the de-Sitter branes.

9.2.3 Braneworld Consists of Anti de-Sitter and de-Sitter Spacetimes

One could wonder why we are so interested in a configuration that has both anti-de-Sitter and de-Sitter branes. The reason given is that this model helps in studying of the information paradox of the Schwarzschild de-Sitter black hole with two horizons using wedge holography. To accomplish this, AdS branes in M1 must be replaced by flat-space branes with n1 = 1. Overall, we’re left with two flat-space branes and two de-Sitter branes with n1 = n2 = 1 and has been discussed in 9.3.2. The question now is whether the above description makes sense. When d-dimensional AdS spacetimes are embedded in AdSd+1, their branes come together at the time-like surface located at the AdSd+1 boundary, whereas dSd Karch-Randall branes intersect at the space-like surface of the AdSd+1 boundary. Fortunately, there is no issues in Fig. 9.4 when M1 and M2 remain separated from themselves. This is the approach used in refIP-SdS to get the Page curve of a Schwarzschild de-Sitter black hole by considering the Schwarzschild and de-Sitter patches independently. We have explored the embedding of several types of Karch-Randall branes in distinct bulks that are not related to each other.

The authors of [242] looked at the numerous options for embedding several sorts of branes, such as Minkowski, de-Sitter, and anti-de-Sitter branes, in the identical bulk. The existence of multiple branes is defined by the creation time τ∗. Minkowski and de-Sitter branes have been created for a finite period of time, however anti de-Sitter branes have no

creation time. Authors noted out that among the several possibilities stated in [242], one can find Minkowski, de-Sitter, and anti de-Sitter branes at the same moment at creation time τ∗ = −π/2 in a particular bulk. Branes have a time-dependent location in this scenario. We begin by summarizing this conclusion and then remark on its implementation using wedge holography. For more details, see citemismatched-branes. The bulk AdS5 metric is written as follows:

Comment on the Wedge Holographic Realization of Mismatched Branes: Using the AdS/BCFT concept, one may build a double holographic system from (9.11). Let us provide three acceptable descriptions of a double holographic setup made of (9.11).

• Boundary description: 4D quantum field theory (QFT) at conformal boundary of (9.11).

• Intermediate description: Dynamical gravity localized on 4D end-of-the-world brane coupled to 4D boundary QFT.

• Bulk description: 4D QFT defined in the first description has 5D gravity dual whose metric is (9.11)

Because of the covariant character of the AdS/CFT duality, the duality stays the same when one is working with altered coordinates of the bulk, i.e. different AdS parametrizations don’t suggest distinct dualities, and thus in the aforementioned doubly holographic setup, it is expected the defect to be 3-dimensional conformal field theory since 4-dimensional gravity serves as simply FRW parametrization of AdS4 spacetime (9.12). This type of duality has been studied by the authors in [243], wherein the bulk represents the de-Sitter parametrization of AdS4 and conformal field theory was QFT on dS3. As addressed thoroughly in appendix A of [242] and summarized within this chapter, de-Sitter and Minkowski branes can also exist in this coordinate system (9.11).

Let us now look at why defining wedge holography using “mismatched branes” is problematic. The “defect CFT” in wedge holography is produced via dynamical gravity on Karch-Randall branes. Assume that there are two Karch-Randall branes with differing geometries, one AdS brane and one de-Sitter brane. The defect CFT must be unitary owing to the AdS brane and non-unitary because of the de-Sitter brane. We appear to have two distinct CFTs at exactly the same defect. This condition will not alter even if four branes or 2n branes are considered. As a result, we may be unable to properly describe the “multiverse” using mismatched branes via wedge holography. Due to the “time-dependent” location of branes, the shared boundary of multiverses M1 and M2 (explained in Fig. 9.4) cannot be identical. All AdS branes in M1 are capable of communicating with one another via transparent boundary conditions across the defect, and all de-Sitter branes in M2 can communicate as well. However, even with the metric (9.11), there does not exist a connection between M1 and M2.

As a result, we come to the conclusion that it is possible to construct a multiverse of identical branes (AdS or de-Sitter) however not as a mixture of both of them. As a result, the issue of mismatched branes does not change from a wedge holography standpoint. The multiverse of AdS branes survives indefinitely, but the multiverse of de-Sitter branes has a finite lifespan.

9.3 Application to Information Paradox

The multiverse is made up of 2n Karch-Randall branes that are embedded in the AdSd+1 bulk. As a result, just one Hartman-Maldacena surface can stretch between the defect CFTs which are thermofield double partners and the total n island surfaces (I1,I2,.....,In) stretched (r = ±nρ) across identical branes of the same positions with opposing sign as shown in Fig. 9.5. Let us give a specific assertion about the wedge holographic dictionary with 2n branes.

CFT is going to be non-unitary if the metric of Karch-Randall branes corresponds to the de-Sitter metric. As a result, this description is identical to the normal wedge holography involving two Karch-Randall branes, with the exception that we are now equipped with 2n Karch-Randall branes.

Let us now construct a mathematical formula to obtain entanglement entropies. We take Q1,2,....,n to be black holes that produce Hawking radiation, which is contained by gravitational baths Q−1,−2,....,−n, see Fig. 9.5. In this configuration, the entanglement entropy of the island surfaces will have the following form:

• Boundary Description: BCFT is living at the AdSd+1 boundary with (d − 1)- dimensional boundary.

• Intermediate Description: 2n gravitating systems interact with each other via transparent boundary conditions at the (d − 1)-dimensional defect.

• Bulk Description: Gravity dual of BCFT is Einstein gravity in the bulk.

Consistency Check: Let’s look at the formula provided in (9.14) when n = 2 whether it is giving the consistent results or not.

9.3.1 Page Curve of Eternal AdS Black Holes in n = 2 Multiverse

We began by calculating the thermal entropy of black hole. The black hole metric in the AdS background has the following structure:

Let us now produce the Page curve of two eternal black holes utilizing the formula presented in (9.14).

The induced metric associated with the Hartman-Maldacena surface which has the parametrization, r ≡ r(z) and v ≡ v(z), is derived as:

The embedding v(z) equation of motion is:

Let us now look into the late-time behavior exhibited by the Hartman-Maldacena surface area:

Since,

Hence,

This amounts to an unlimited quantity of Hawking radiation if t1 → ∞, i.e. at late times, and so results in the information paradox.

Entanglement entropy contribution from Island surfaces: Consider the parametrized island surfaces t = constant and z ≡ z(r). The entanglement entropy associated with two eternal AdS black holes regarding the island surfaces was calculated using (9.14). Since there exist two island surfaces (I1 and I2) that extend between the Karch-Randall branes at r = ±ρ (I1) and r = ±2ρ (I2), we could use (9.14) for the same.

We begin by calculating AI1 . The induced metric for Karch-Randall branes could be derived utilizing (9.15) by parametrizing the island surface as t = constant and z = z(r) and constraining to d = 4 with f(z) = 1 − z 3 (since zh = 1):

The area associated with the island surface I1 resulted from (9.29) is given as:

Only if the first term of the preceding equation disappears will certainly the variational principle become meaningful. The second term represents the EOM associated with the embedding z(r). Let’s have a look at what this means.

(9.32) disappears if we enforce the Dirichlet boundary condition on the branes, i.e., δz(r = ±ρ) = 0 or Neumann boundary condition on the branes, i.e., z ′ (r = ±ρ) = 0. Neumann boundary conditions enable RT surfaces to travel along branes in gravitating baths. Under this scenario, the black hole horizon [162] is the minimum surface. The Euler-Lagrange equation of motion for the action with embedding z(r) becomes:

The second island’s minimum surface area (I2) will end up being identical as that of the first (9.34), with various integration limits resulting from different Karch-Randall brane positions (r = ±2ρ)

We obtained the total entanglement entropy of island surfaces by inserting (9.34) and (9.35) into (9.28), which is written below:

The additional island surface obtained via the thermofield double partner is the source of the factor “2” in (9.36). The Page curve associated with the n = 2 multiverse is obtained using (9.27) and (9.36), which is illustrated in Fig. 9.6.

9.3.2 Page Curve of Schwarzschild de-Sitter Black Hole

Here, we examine the Schwarzschild de-Sitter black hole’s information problem. We cannot join mismatched branes at the identical defect, as stated in 9.2.3. As a result, we divide our analysis of this issue into two parts and compute the Page curve corresponding to the Schwarzschild patch first, followed by the Page curve associated with the de-Sitter patch similar to [12]. The next is how to do this. Two flat space branes embedded in the bulk are taken into consideration when studying the Schwarzschild patch in 9.3.2.1, and two de-Sitter branes are taken into consideration when studying the de-Sitter patch in 9.3.2.2. The set-up has been shown in Fig. 9.7. With flat space and de-Sitter branes in Schwarzschild and de-Sitter patches, respectively, the setup consists of two copies of wedge holography.

9.3.2.1 Schwarzschild patch

Given that Λ = 0 for the Schwarzschild black hole, we must take into account flat space branes in order to understand the Schwarzschild black hole on the Karch-Randall brane. It has been shown in [9] that one could acquire Karch-Randall branes with flat space black holes under the condition that bulk metric possess a particular structure:

(9.38) represents the equation of motion containing the brane Einstein-Hilbert term:

The induced metric associated with the Hartman-Maldacena surface is given as follows for the parametrization r = r(z) and v = v(z).

The area associated with the Hartman-Maldacena surface was calculated using (9.42) as:

In late time approximation, i.e., t → ∞, r(z) → 0

Island Surface: t = constant and z = z(r) define the island surface. The area associated with the island surface could possibly be calculated using the induced metric in terms of embedding(z(r)) and its derivative obtained from the bulk metric (9.37):

For simplicity, we assign zh = 1 in the preceding equation, therefore f(z) ≥ 0 needs z > 1. Substituting (9.47)’s Lagrangian in (9.31), the first term that appears on the last line of (9.31) for (9.47), implies:

As a result, we are guided by the well-defined variational principle of (9.47) assuming the embedding function meets the Neumann boundary condition on the branes, i.e., z(r = ±a1) = 0, and therefore the minimal surface is going to be the black hole horizon, i.e., z(r) = 1, as in [162,166]. The following equation of motion of z(r) can be used to reach the same result.

The black hole horizon is the solution of (9.49), i.e. z(r) = 1, which is compatible with the Neumann boundary condition on the branes [162]. As a consequence of inserting z(r) = 1 in (9.47), the smallest area of the island surface is produced, and the final outcome is:

As a result, the entanglement entropy associated with the Schwarzschild patch’s island surface becomes:

As a result, we obtained the Page curve by plotting (9.45) and (9.51) for the Schwarzschild patch depicted in Fig. 9.8.

9.3.2.2 de-Sitter patch

The de-Sitter black hole and associated bath could possibly be placed at r = ±ρ. The metric of bulk containing de-Sitter branes has emerged as:

The Hartman-Maldacena surface is parametrized as r = r(z) and v = v(z), and therefore the area has been calculated using (9.54) with the above-mentioned parametrization and expressed as follows:

The Euler-Lagrange equation of motion of r(z) given from (9.55) is:

The equation above is tough to solve. One simple solution to (9.58) is obtain as:

Cosmological Island Surface Entanglement Entropy: The induced metric obtained in terms of embedding (z = z(r)) and its derivative (9.52) has been used to calculate the area of the island surface parametrized by t = constant, z = z(r), and the final result is:

As a result, f(z) ≥ 0 when 0 < z < 1.

In general, solving the given problem is difficult. remarkably, there exists a z(r) = 1 solution for the aforementioned differential equation, that is just that the originally stated de-Sitter horizon (zs = 1) similar to earlier discussions on the EOM for the island surfaces. Further, this solution consistent with the Neumann boundary condition on the branes, and so the cosmological island surface embedding’s EOM solution is given as:

z(r) = 1.

One could achieve an identical result by demanding the well-defined variational principle of (9.61) and enforcing Neumann boundary conditions on the branes, as discussed in 9.3.1

If we enforce z ′ (r = ±ρ) = 0, then we obtain that the horizon as the minimal surface, hence z(r) = 1 [162]. We derive the smallest area corresponding to the cosmological island surface associated with the de-Sitter patch by inserting z(r) = 1 in (9.61) as given below:

Therefore, cosmological island surface has the following entanglement entropy:

Due to a second cosmological island surface on the thermofield double partner side (seen in Fig. 9.7), an additional numerical factor “2” is present. We could get the Page curve of de-Sitter patch by plotting (9.60) and (9.66). We obtained a flat Page curve for the de-Sitter patch as in [162].

Comment on the Wedge Holographic Realization of Schwarzschild de-Sitter Black Hole with Two Karch-Randall Branes: We produced the Page curves of Schwarzschild and de-Sitter patches independently in 9.3.2. There is yet another technique for us to obtain the Schwarzschild de-Sitter black hole’s Page curve. Below is a summary of the concept:

• Take two Karch-Randall branes Q1 and Q2, where Q1 is a Schwarzschild de-Sitter black hole and Q2 is a radiation-collecting bath.

• Let’s say the structure of the bulk metric is as follows:

• The next thing to do is to solve the Einstein equation (5.19) to determine g(r).

• The bulk metric (9.67) has to fulfill the Neumann boundary condition (5.21) at r = ±ρ after finding the solution.

• To use the Ryu-Takayanagi formula, one must additionally determine if a CFT or non-CFT theory exists at the defect.

• If the aforementioned points are correctly verified, we may compute the areas of the Hartman-Maldacena and island surfaces to derive the Page curve of the Schwarzschild de-Sitter black hole.

“Mathematical concept” is essentially what the debate above is. We could be referring to [242] since there are three potential branes: Minkowski, de-Sitter, and anti de-Sitter. Within the open bracket of (9.67), there is no brane specified with the induced metric. Additionally, there is flat space holography, dS/CFT duality, or AdS/CFT correspondence. The duality among CFT and bulk, which takes the structure of a Schwarzschild de-Sitter metric, does not exist. Owing to the abovementioned reason, there won’t be any defect descriptions and no “intermediate description” of the wedge holography. As a result, we get to the conclusion that one could represent a Schwarzschild de-Sitter black hole using wedge holography utilizing two copies of the wedge holography that define the Schwarzschild patch and the de-Sitter patch, respectively.

9.4 Application to Grandfather Paradox

The “grandfather paradox” is described in this section along with how our model resolves it.

According to the “Grandfather paradox”, Bob is unable to travel back in time. Because he could murder his grandfather in a different universe if he could go back in time. Bob’s will cease to be in the present if his grandfather is dead in another universe [244].

Let’s now examine how our setup might avoid this issue. In 9.2.1 and 9.2.2, we explained how a multiverse is made up of 2n universes that are Karch-Randall branes. These branes have AdS and de-Sitter spacetimes in 9.2.1 and 9.2.2 as their geometries. All “universes” have a connection at the “defect” in each configuration by a transparent boundary condition. Transparent boundary conditions ensure that communication exists between all of these universes. Assume that Bob resides on Q1 and his grandfather resides on Q2. Then, in order to get around the dilemma, Bob is able to go to places like Q−2, Q−3, etc. where he is able to come across Robert and Alice (see Fig. 9.9). Therefore, the “grandfather paradox” could be resolved in this situation. The “grandfather paradox” has been addressed using a concept similar to that discussed in this debate, which is compatible with the “many world theory”.

9.5 Conclusion

Using the concept of wedge holography, we proposed in this chapter that the Karch-Randall braneworld contains a multiverse. If we consider the 2n universes, then those are going to be represented as Karch-Randall branes embedded within the bulk, this is how the multiverse has been described. These branes could either contain black holes or they won’t, that will be decided by the gravitational action. We looked at three distinct scenarios.

• In 9.2.1, we generated the multiverse using d-dimensional Karch-Randall branes that have been incorporated in AdSd+1 bulk. The aforementioned branes are defined by the AdSd geometry. Transparent boundary conditions at the defect interconnect the 2n anti de-Sitter branes that make up the multiverse in this particular scenario to one another. As soon as constructed, the multiverse made of AdS branes is eternal.

• In 9.2.2, we created a multiverse using d-dimensional de-Sitter spaces on the KarchRandall branes embedded within the (d+1)-dimensional bulk AdSd+1. The 2n de-Sitter branes that makes the multiverse possesses a short lifespan. In such a case, all of deSitter branes must be formed and destroyed at the identical time. As a result of the dS/CFT duality, defect CFT corresponds to non-unitary conformal field theory.

• In 9.2.3, we additionally explored why it wouldn’t be conceivable to define the multiverse as a combination of d-dimensional de-Sitter and anti-de-Sitter spacetimes within the identical bulk. We are able to possess the multiverse with either anti de-Sitter branes (M1) or de-Sitter branes (M2), but it does not have both. Since AdS branes intersect on the “time-like” boundary of AdSd+1 bulk and the de-Sitter branes intersect at the “space-like” boundary. Universes within M1 are able to interact with each other; similarly, the universes in M2 are able to communicate with each other, but M1 is unable to communicate with M2.

We obtained the Page curves associated with two black holes for the n = 2 multiverse as a consistency check. We figured out that the black hole and bath systems are located at −2ρ ≤ r ≤ 2ρ and −ρ ≤ r ≤ ρ. In such a scenario, we showed that the entanglement entropy contribution generated by the Hartman-Maldacena surfaces exhibits a linear time dependence for both AdS and Schwarzschild black holes and becomes zero for the de-Sitter black hole, but the contributions that come via the island surfaces remain constant. As a result, this mimics the Page curve. Utilizing this concept, we obtained the Page curve of the Schwarzschild de-Sitter black hole. This idea could be used to compute the Page curve of black holes that have numerous horizons using wedge holography. We also addressed how we might generate a Page curve for these black holes by employing two Karch-Randall branes, one as a black hole and another as a bath. Within this situation, identifying the island surface and figuring out what type of radiation we are receiving will be difficult. As an example, if a Karch-Randall brane contains a black hole and cosmic event horizons, such as a Schwarzschild de-Sitter black hole, the observer receiving the radiation will be unable to tell whether this is Hawking radiation or Gibbons-Hawking radiation.

We tested our approach for very basic cases without the DGP term placed on the KarchRandall branes, however it is also possible to discuss massless gravity by including the DGP term on the Karch-Randall branes [166]. Tensions associated with branes will be corrected by the additional term in (9.3) in this situation. Furthermore, we proposed that with this system, where all universes interact via transparent boundary conditions at the junction point, one could circumvent the “grandfather paradox”. In order to resolve the paradox, one may travel to another universe where his grandpa is not living, so preventing him from killing his grandfather. We provided a qualitative solution to the “grandfather paradox”, although further study in this area employing wedge holography is required.