**Launch your smart contract idea in just 4 days**

by Multiverse Theory: as real as the movies make it out to beFebruary 22nd, 2024

**Authors:**

(1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute.

**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

Applying the island proposal described in 5.3.1, we are able to compute the Page curves of eternal black holes [6]. In the papers [172–175], the Page curves of the Reissner-Nordström black hole, charged dilaton black hole, Schwarzschild black hole, and hyperscaling violating black branes were found. In the research referred to as [176], the page curve in the charged linear dilaton model for both the non-extremal black hole and the extremal black hole was analyzed. Islands in Kerr-de Sitter spacetime and generalized dilaton theories were the subject of research in the article [177, 178]. In the research referred to as [141], the effect that mutual knowledge between subsystems on the Page curve was analyzed. Calculations of the page curve used in higher derivative gravity theories could be found in the papers [11,141]. Page curves of Schwarzschild black holes were obtained by the authors of the paper mentioned above [141] when higher derivative terms, known as O(R2) and Gauss-Bonnet terms, were present in the gravitational action.

In the papers [179–185] researchers looked at charged black holes in Einstein-GaussBonnet gravity in four dimensions. See [186] for a review of the Einstein-Gauss-Bonnet theory of gravity when applied to four dimensions. Since we were dealing with higher derivative theories of gravity, we employed [127] to compute the entanglement entropy for those cases where it was necessary. The non-extremal black holes are what we’ll be discussing in this chapter. For extremal black holes, see [176, 187, 188]. *It has come to our attention that research on the effect of the Gauss-Bonnet coupling on the Page curve of a charged EinsteinGauss-Bonnet black hole had not been done. As a result, it would be quite fascinating to investigate how the Page curves of the Reissner-Nordström black hole will change when the Gauss-Bonnet term is present. As a result of this inspiration, we have computed the Page curves of charged black holes in higher derivative gravity with O(R2 ) terms and for charged Einstein-Gauss-Bonnet black holes in this work.*

This section has been broken up into two different subsections. We will provide a concise review of the charged black hole in Einstein-Guass-Bonnet gravity in the vanishing cosmological constant limit [189] and the Page curve calculation of the Reissner-Nordström black hole [172] in four dimensions in 6.2.1 and 6.2.2.

The gravitational action (6.2) has the following black hole solution:

where the black hole function F(*r*) is defined as:

If we do a small expansion in α with a negative sign chosen in (6.5), then we get the following result:

Solving F(*r*) = 0 with negative sign of (6.5) yields the black hole’s horizons, which are presented in the following form:

where the physical horizon of the charged black hole is denoted by the symbol r+ and the Cauchy horizon is denoted by the symbol r_.

Let us now examine the relationship between the Gauss-Bonnet coupling α and the black hole horizon r+. We have depicted it in figure 6.1 for M = 1 and for a variety of different values for the black hole charges. When the Gauss-Bonnet coupling (α) is zero, as shown by the figure 6.1 and the equation (6.7), we get the typical Reissner-Nordström black hole. The decrease of Gauss-Bonnet coupling results in the increase of the black hole horizon and vice-versa. When M = 1, the following values for the Gauss-Bonnet coupling α have been specified by the authors of [179, 182]:

In contrast to the region B, which only contains the physical horizon (r+), the region A contains both horizons (r±). Since we’re looking for the non-extremal black hole, the region A is of interest to us. Using Q = 0.8, we found that for the region A, α ∈ (−6.66, 0.36).

Here, we take a look back at [172]’s computation of the Reisnner-Nordström black hole’s Page curve. The action that describes the Einstein-Maxwell gravity is given as:

where R represents the Ricci scalar, Fµν denotes the field strength tensor associated with the Aµ gauge field, and I*matter* represents the matter part of the action. The Reissner-Nordstörm black hole has the following metric:

The following are the black hole horizons that result from solving F(*r*) = 0:

Below is a version of metric (6.9) written in Kruskal coordinates:

where,

and the tortoise coordinate r∗ is given by:

For both r+ and r_, we have the following expressions for the surface gravities κ±:

The following are definitions of the charged black hole’s Hawking temperature and Bekenstein-Hawking entropy:

The conformal factor in the metric, expressed in Kruskal coordinates, is defined as follows:

As a result, the metric can be expressed as follows:

The Penrose diagrams of an eternal Reissner-Nordström black hole are shown in figures 6.2 and 6.3. In figures 6.2 and 6.3, the R_ and R+ denote the left and right wedges of radiation regions, the b_ and b+ denote the boundaries of R_ and R+, and the a_ and a+ denote the boundaries of island surface in the left and right wedges, respectively.

Since there is no island surface to begin with, the entanglement entropy of the Hawking radiation will be obtained by applying the formula shown below [173, 192, 193]:

Now, utilizing equations (6.13), (6.14), (6.17), (6.19) and (6.20), the entanglement entropy of Hawking radiation in the absence of an island surface has been simplified to [172]:

Therefore, we can observe that the entanglement entropy of the Hawking radiation in without an island surface is increasing linearly with time and approaches infinite at later times via the equation (6.22). This is what causes the information paradox for the Reissner-Nordström black hole.

Here, we will go through the computation of the entanglement entropy of the Hawking radiation when an island surface is present. We are able to compute it by utilizing the formula [173, 192, 193], which is as follows:

Therefore, generalized entropy with inclusion of an island surface at late times is as follows [172]:

Substitution of a using (6.27) in (6.25), results in Hawking radiation’s total entanglement entropy at late times:

We may deduce from the preceding equation that the total entanglement entropy remains the same, which means that it takes precedence after the Page time, and as a result, we have the Page curve of an eternal Reissner-Nordström black hole.

We showed that the entanglement entropy without an island surface increases linearly with time by using (6.22), and we found that the entanglement entropy remains constant at late times by using (6.28). Island surface appears after the Page time, which saturates the linear growth of entanglement entropy and attains a fix value, which equals twice of the Bekenstein-Hawking entropy of the Reissner-Nordström black hole, we are able to get the Page curve of the Reissner-Nordström black hole.

We used only the leading order term in the GN in equation (6.28) when we drew the Page curve of the Reissner-Nordström black hole (shown in figure 6.4) with the value M = 1, Q = 0.8, G*N* = 1. The blue line in figure 6.4 refers to the linear time growth of the entanglement entropy equation (6.22)), and the orange line refers to the entanglement entropy with island surface (equation (6.28)), which takes over after the Page time and results in the Page curve.

**Page time:** Page time is described as the point in time at which the entanglement entropy of the Hawking radiation begins to approach zero for a black hole that is evaporating and reaches a value that is constant for a black hole that is eternal. From matching the equations (6.22) and (6.28), we are able to obtain the Page time, which is represented by the following:

**Scrambling time:** The term “scrambling time” refers to the time when we are able

to retrieve the information that was lost when it was sucked into the black hole in the form of Hawking radiation [194, 195]. Information from a black hole that has evaporated halfway can be quickly retrieved as Hawking radiation, as detailed in [194]. If the black hole hasn’t evaporated halfway, we’ll have to wait until it does so that we can swiftly retrieve the information we’ve lost. Scrambling time, as used in the terminology of entanglement wedge reconstruction [196], is the amount of time it takes the information from the cut-off surface (r = b+) to arrive at the island surface’s boundary (r = a+). Once the information thrown into the black hole has traveled to the boundary of the island surface (r = a+), the entanglement entropy of Hawking radiation begins to include contribution from the island surface, as the degrees of freedom of the island become part of the entanglement wedge of the Hawking radiation. Time it takes the information to travel from the cut-off surface (r = b) to the boundary of the island surface (r = a) when sent into a black hole is provided by:

Scrambling time of the Reissner-Nordström black hole is found to be when the value of a from equation (6.27) is substituted into the previous equation and the obtained expression is written as:

This result is comparable to [195], which suggests that black holes are the most efficient scramblers. We can observe this by looking at the equation (6.31), which shows that the scrambling time is logarithmic of the thermal entropy of charged black hole analogous to [195].

We studied generic higher derivative terms of O(R2 ) for the gravitational action as that of considered in [141], and we obtained the Page curves of an eternal Reissner-Nordström black hole. The gravitational action of the Einstein-Maxwell theory with inclusion of O(R2 ) is given as below:

where R[g] denotes the Ricci scalar, Rµν[g] represents the Ricci tensor, and L*GB* denotes the Gauss-Bonnet term defined in (6.3). The formula to calculate the Wald entropy in higher derivative theories of gravity is given as [102, 103]:

where *h* denotes the determinant of the induced metric of the co-dimension two surface. The metric (6.9) is rewritten as follows:

We found the following for the action (6.32):

For the metric (referred to as (6.34)), the Wald entropy that corresponds to the action (referred to as (6.32)) can be calculated utilizing the equation (referred to as (6.35)) as follows:

Given that r+ ≫ r_, the preceding equation can be simplified to read as follows:

The formula which was given in [127] and is described in chapter 5, can be utilized to determine the holographic entanglement entropy that is associated with higher derivative gravity theories. According to what was discussed in chapter 5, we are able to write the total entanglement entropy in the presence of higher derivative terms as follows [141]:

The result of calculating gravity’s contribution to the entanglement entropy corresponding to the action (6.32) is as follows [141]:

Because of this, the generalized entropy of the action (6.32) is as follows [141]:

The gravitational component of the entanglement entropy of the Hawking radiation (6.39) can therefore be simplified as follows:

Thus, the total generalized entropy is going to be calculated as the sum of the contributions to the entanglement entropy from the gravity component (6.44) and the matter part (6.41), which is given as follows:

where r∗(b), g(a), and g(b) can be replaced from the equations (6.14) and (6.17) in the equation (6.45). The small tb dependent contribution is ignorable in (6.41). The position of the island surface can be determined by extremizing the equation shown above with respect to the variable a, which can be written as:

solving above equation, we obtained:

Because the island is located beyond the event horizon of the black hole, this creates a causality paradox. It was demonstrated in [197] that this conclusion occurs in all two-sided eternal black holes or black holes in the Hartle-Hawking state, and it is possible to restore causality using quantum focusing conjecture (QFC) [198]. When we separate the black hole from the bath, a finite quantity of energy flux is generated. This energy flux pulls the black hole horizon outwards, and as a result, the island always stays behind the horizon. It was pointed out in [199] that a limited quantity of energy is also generated even when we couple the black hole to bath, so such energy flux drives the horizon outwards, that suggests that island lies behind the horizon similar to the way that the decoupling process works, and that we are able to get rid of the causality paradox.

When the value of “a” from the equation (6.47) is substituted into the equation (6.45), the generalized entropy equation is simplified to the following form:

If we only look at the leading order term in G*N*, we are able to see that the total entanglement entropy of an eternal Reissner-Nordström black hole with inclusion of O(R2) terms in the gravitational action approaches a constant value that is compatible with the literature. In the case where α → 0, the equation (6.48) can be simplified to:

The preceding equation makes it abundantly evident that in the case when α → 0 is being considered, the Page time of the Reissner-Nordström black hole with the presence of O(R2 ) terms simplifies to the Page time of the Reissner-Nordström black hole (6.29) when higher derivative terms are removed.

As a result of the fact that the Gauss-Bonnet coupling (α) appears with a positive sign in (6.48) and (6.51), we receive the Page curves at a later or earlier times when α increases or decreases.

**Scrambling time:** The scrambling time of the charged black hole with the presence of O(R*2*) terms can be found by putting a from (6.47) into (6.30). The resulting equation is as follows:

Hence, scrambling time of the charged black hole with inclusion of O(R*2*) terms may increase or decrease based on the sign of second term of (6.54).

We now obtain the Page curves of charged black hole in Einstein-Gauss-Bonnet gravity. The gravitational action of Einstein-Maxwell-Gauss-Bonnet gravity without cosmological constant has the following form [189]:

The Wald entropy of the charged black hole in Einstein-Gauss-Bonnet gravity has been obtained utilizing (6.33),(6.34),(6.35) and (6.55) and the obtained results is written below:

where *h* denotes the determinant of a given induced metric on a surface with constant coordinates (*t, r = a*):

Now equation (6.57) was simplified by the use of Gauss-Codazzi equation [141, 201]:

Holographic entanglement entropy in Einstein-Gauss-Bonnet gravity can be simplified to the following form through the use of the equations (6.57) and (6.59), which represent the gravitational contribution to the generalized entropy:

Hence, similar to 6.3, the generalised entropy for this case is obtained as:

The variation of (6.62) with respect to a leads to the following equation:

from the previous equation, we found the location of the island surface as written below:

Using (6.64), the total entanglement entropy of the charged Einstein-Gauss-Bonnet black hole (6.62) in late time approximation is obtained as:

Since the total entanglement entropy of the charged black hole in Einstein-Gauss-Bonnet gravity at late times (6.65) is the same as the total entanglement entropy of the charged black hole in higher derivative gravity with the O(R2 ) terms specified in the equation (6.48), it may be concluded that the two quantities are identical. Because of this, the Page curves and Page times of the charged black hole in Einstein-Gauss-Bonnet gravity will turn out to be the same as the Page curves and Page times of the charged black hole in higher derivative gravity with O(R2 ) terms , figure 6.5.

**Scrambling time:** In this particular instance, the scrambling time is not influenced by the higher derivative terms and is identical to the scrambling time of the Reissner-Nordström black hole [172]:

We obtained the Page curves of an eternal Reissner-Nordstrom black hole in this chapter. We concentrated on the non-holographic model and made the assumption that the observer is located a large distance from the black hole. As a result, we were able to employ the s-wave approximation to compute the entanglement entropy of Hawking radiation utilizing the formula associated with two-dimensional conformal field theory. The affect of higher derivative terms on the Page curve was studied in [141] for the neutral black hole. The researchers in that paper took into account the eternal black hole and the evaporating black hole both. They were solely concerned with the more generic O(R2 ) terms. We considered the general O(R2 ) terms as well as Gauss-Bonnet term both as the higher derivative terms for the nonextremal eternal Reissner-Nordström black hole and found that Page time, scrambling time, and the Page curves of eternal black holes are all affected when higher derivative terms are present in the gravitational action. The key results that have been found are summarized below.

1. In the first scenario, we took into account the general terms (O(R2 )) in the gravitational action described in [141] and calculated the Page curves. We came to the conclusion that, because there is initially no island surface, the entanglement entropy of the Hawking radiation will continue to increase in a linear fashion with the passage of time indefinitely. As a result, we ended up at an information paradox for the charged black hole. At later times, an island will appear, and the entanglement entropy of the Hawking radiation will attain a constant value. This value will be equal to double the Bekenstein-Hawking entropy of the black hole, and it will allow us to derive the Page curves for fixed values of the Gauss-Bonnet coupling (α). The higher derivative terms shifts the Page curve and hence Page time. *Page curves move towards later times as the Gauss-Bonnet coupling (α) grows, and Page curves shift towards earlier times as the Gauss-Bonnet coupling (α) drops.*

2. In the next scenario, we solely included the Gauss-Bonnet term as the higher derivative term because it is crucial for the investigation of the charged black hole in Einstein-Gauss-Bonnet gravity [189]. In a manner analogous to the first scenario, we observe linear time growth of the entanglement entropy of the Hawking radiation at the beginning. This is followed by the emergence of an island at later times, that dominates the linear time growth of the Hawking radiation. As a result, the entanglement entropy of the Hawking radiation eventually reaches exactly two times the Bekenstein-Hawking entropy of the black hole, and we obtained the Page curve exactly similar to the first case. It is interesting that we found that the Page curves with Gauss-Bonnet coupling behave in a manner that is comparable to that which was explained earlier in the first case.

3. In the first scenario, the higher derivative terms have an effect on how long it takes the charged black hole to scramble its information. If the correction term is positive, it will make it so that it is larger, but if it is negative, it will make it so that it is smaller. In addition, if we take the coupling that appears in scrambling time and set it to zero, then we are able to get back the scrambling time of the ReissnerNordström black hole. Scrambling time is unaffected by the higher derivative term in the second scenario, which is when we look at solely the Gauss-Bonnet term as the higher derivative term. In this scenario, scrambling time remains identical, just like the scrambling time associated with the Reissner-Nordström black hole.

4. When the Gauss-Bonnet coupling goes to zero, we are able to recover the Page curve of the Reissner-Nordsrtöm black hole, which was obtained in [172]. Further, all of our results reduces to that of [172] in the same limit.

This paper is available on arxiv under CC 4.0 license.

L O A D I N G

. . . comments & more!

. . . comments & more!