Black Hole Islands in Multi-Event Horizon Space-Times

The information paradox of black holes has been investigated for black holes that are asymptotically flat, however the most recent observation indicates that our universe is currently in an accelerated phase. Because of this, it is only natural to wonder how the presence of a positive value for the cosmological constant Λ influences the information paradox problem. The information paradox of a stationary black hole with a positive cosmological constant Λ or the Schwarzschild de-Sitter black hole spacetime is something that attracted our attention. The information paradox of Schwarzschild de-Sitter black holes is important due to the fact that these black holes originated during the early inflationary phase of our universe (for example, [210–213]). It additionally offers a fantastic toy model for the global framework of isolated black holes of our universe, which is quite useful while keeping in mind the current period of accelerated expansion. Additionally, similar to the situation with black holes, there are regions of de Sitter space that are causally separated from one another. Therefore, an observer is only able to access the portions of the cosmos that are constrained by their own horizon. There are two event horizons associated with the Schwarzschild de-Sitter black hole: the cosmological event horizon (CEH) and the black hole event horizon (BEH). The global causal border of the de Sitter spacetimes has been provided by the cosmological event horizon. In comparison to Λ ≤ 0 single horizon spacetime, the thermodynamics of these event horizons are distinct [214–216]. Gibbons-Hawking radiation is emitted and absorbed by cosmological event horizon similar to the Hawking radiation of black holes. Comparatively speaking, the entropy creation of the cosmic horizon is an observer-dependent characteristic, in contrast to the entropy generation of the black hole. It is caused by people’s lack of knowledge regarding the things that exist beyond the cosmological horizon.

In order to obtain the Page curve of the Schwarzschild de-Sitter black hole, we make use of the island concept. In the case of models that are not holographic, we are able to use the s-wave approximation for studying black holes in higher dimensions. Because we neglect the angular component of the metric when performing the s-wave approximation, we are left with a CFT metric that only has two dimensions. Because of this, we are able to compute the entanglement entropy of Hawking radiation using the formula for 2D CFT that is provided in [192, 193]. The purpose of this work was to investigate the information paradox and its resolution of black holes with multiple horizons. We considered the Schwarzschild de-Sitter black hole with two horizons and obtained the Page curves for the black hole and the de-Sitter patches by placing thermally opaque membranes on the two sides of the region of study. The “effect of temperature” on the Page curves and the scrambling time is something else that would be of interest to us. In this chapter, we will discuss the Page curve of black hole patch only. The entanglement entropy and complexity have been studied in [217–219] in the context of de-Sitter space.

8.2 Preliminary

In this section, we will discuss the basics of Schwarzschild de-Sitter black hole, the concept of thermal opaque membrane and the effect of gravity near the aforementioned membrane in 8.2.1, 8.2.2, and 8.2.3 respectively.

8.2.1 Schwarzschild de-Sitter Black Hole

The Schwarzschild de-Sitter (SdS) metric has the following form in spherical polar coordinates:

and,

where the Kruskal coordinates are:

Therefore (8.7) and (8.8) do not have any coordinate singularities at r = rH and r = rC. It is not possible to remove the coordinate singularities of both event horizons simultaneously. Hence, we have to study the black hole patch by freezing the de-Sitter patch using thermal opaque membrane and vice-versa.

8.2.2 Thermal Opaque Membranes in SdS Space-Time

The fact that the effective potential term disappears at both horizons and remains positive in between them is something that stands out as an interesting feature of the Schrödingerlike equation that was given before. Therefore, this bell-shaped potential acts as a barrier between the event horizons of the black hole and de-Sitter space. To be able to depict it in Penrose diagram of the extended Schwarzschild de-Sitter space-time, we will use (8.9) to describe the Kruskal timelike and spacelike coordinates as follows:

implying

Therefore, the line r =constant is a hyperbola linking i ±, and it is possible to draw it with regard to any of the Kruskal coordinates listed above. This appears in the Penrose diagram of the maximally extended SdS space-time. When we have put this hyperbola or the thermal opaque membrane, modes on each side of wall will be unable to pass through it and will continue to exist in their respective regions. A natural manifestation of this effective potential could be possible with the thermally opaque membrane.

8.2.3 Bulk Gravitational Effect Near Thermal Opaque Membranes

In the context of this discussion, we are interested in the bulk gravitational influence on thermally opaque membranes. Using the island formula, the authors of [229, 230] studied the information paradox of pure de-Sitter space, evaporating and eternal black holes in the weak gravity domain. These setups do not include an exterior bath and the authors included an anchor curve that produces both an interior and an exterior. The island formula can be applied despite the fact that gravity is extremely weak on the exterior region. We suggest that we may think of an “anchor curve” going through the locations b ± 1,2 within our setup also where we have specified the boundaries of radiation regions in Figs. 8.1 and 8.2. Our justification is based on [229,230]. The interior and exterior in our setup are marked by these “anchor curves”. The “thermal opaque membrane” is being able of blocking the radiation coming from the region, that is not relevant to our research in any way. Therefore, the applicability of the island formula to our situation can only be considered notional at this point. The information paradox that arises from black holes in multi-event horizon spacetime can be solved using this approach. Considering the membrane that was discussed before is located at a significant distance from the region affected by the black hole, it is reasonable to suppose that the gravitational effects around these membranes is insufficiently strong. As a result, 2D CFT formulas can be utilized to determine the entanglement entropy of Hawking radiation. We have discussed another approach to get the Page curve of Schwarzschild deSitter black hole in chapter 9 using wedge holography.

8.3 Black Hole Islands in Schwarzschild de-Sitter Black Hole

The conformal factor and Kruskal coordinates, both of which exist in (8.13), are defined as follows:

It has been calculated that thermal entropy of the black hole is:

8.3.1 No Island Phase

In this case, the region of interest for the black hole patch is the area that is completely encompassed by the thermally opaque membrane on both sides in the C and L domains. Without island surface, we were able to calculate the entanglement entropy of Hawking

radiation by utilizing the two-dimensional CFT formula that is given below [192, 193]:

By utilizing (8.14), (8.5) and (8.17), we are able to simplify the equation (8.16). The form in its refined version is written as follows:

The above equation has a significant physical meaning that it predicts that there is going to be a limitless quantity of Hawking radiation due to its linear time dependence upon entanglement entropy. This is in contrast to the fact that the Page curve of an eternal black hole predicts that there will only be a finite amount of Hawking radiation [132].

8.3.2 Island Phase

Taking into account the island surface as part of the setup, we will now calculate the entanglement entropy of Hawking radiation. With inclusion of the island surface, the entanglement

entropy of the Hawking radiation can be determined using the formula [192, 193], which is the matter component of the generalized entropy (5.11).

Simplifying the expression for the matter contribution to the generalized entropy (5.11) from the radiation and island regions by utilizing (8.13), and (8.17). We obtained:

As a result, the area of the boundary of the island surface and the matter component (8.22) are added together to generate the generalized entropy (5.11):

The previous equation has the following solution:

Using the above equation, we can determine that the island exists at the following location:

Since the second component in the above equation has a negative sign in front of it, the island must be found inside the black hole horizon. We have showed in the Penrose diagram where islands are beyond the black hole event horizon, but we have found that it lies within the horizon itself. Substituting a1 from (8.28) into the definition of generalized entropy yields:

Because of this, we may deduce that the entanglement entropy of Hawking radiation remains constant when the island is incorporated into the black hole’s interior and is thus equal to twice the black hole’s thermal entropy (to leading order in GN).

Page time: Hawking radiation’s entanglement entropy without the island surface is equivalent to its entanglement entropy with the island surface present at a point in time called “Page time” as discussed in chapter 6. By equating (8.19) and (8.29), we obtained the Page time as given below:

Scrambling time: The scrambling time is discussed in detail in chapter 6. For the black hole patch in Schwarzschild de-Sitter black hole, scrambling time can be obtained using the following equation:

From (8.5) and (8.28), the black hole patch’s scrambling time is therefore obtained as:

8.3.3 Page Curves

The black hole patch Page curves are now obtained using the results of 8.3.1 and 8.3.2. (8.19) and (8.29) are the key equations. When we incorporate the island surface, the entanglement entropy of Hawking radiation changes from a time-dependent function of entanglement entropy to a constant value of (2S BEH th ) (8.29). As a result, if we plot all of these contributions together, we get the Page curves of eternal black holes that are in accord with their unitary evolution. From the use of (8.2), (8.3), (8.19) and (8.29), we obtained the Page curves for black hole patch when Q = GN = 1, 3M √ Λ = 0.4 (green) and 3M √ Λ = 0.5 (magenta), shown in Fig. 8.3. As the mass of the black hole grows (as shown in 8.3), the Page curves move later in time. As a result, information recovered from large black holes takes much more time than that from smaller black holes. This finding can also be interpreted in terms of the temperature of black holes. Information can be recovered faster from black holes with higher temperatures than from those with lower temperatures because the black hole temperature is inversely related to black hole mass. To rephrase, island emerges earlier for the black holes with higher temperatures compared to the black holes with lower temperature.

8.4 Information Problem in SdS Black Hole as a Whole

8.5 Conclusion

Using the Island concept, we have investigated the information paradox that is associated with the black hole patch in the Schwarzschild-deSitter black hole spacetime. On both sides of the black hole patch, we used thermally opaque membranes to create an isolation barrier between the black hole and the de-Sitter patches, and vice versa. We calculated the entanglement entropy of Hawking radiation without and with the inclusion of an island surface and found that, as is typical, the entanglement entropy has a linear time dependency as long as there isn’t an island surface, but it turns into constant value when there is an island surface present. As a result, we are able to produce the Page curve in a manner that accords with the unitary evolution of the black holes. In this particular instance, the island can be found inside the horizon of the black hole, in contrast to the universal finding, which states that the island must be found outside the horizon of the black hole in the case of eternal black holes [197]. Our results are likewise comparable to those presented in [235], in which the researchers calculated the Page curve of a one-sided asymptotically flat black hole and found that the island lies within the horizon of the black hole.

In addition, we have investigated the “effect of temperature” on the Page curves depicted by the black hole patch. We noticed that as the temperature of the black hole patch rose, a shift toward later times occurred in the Page curves. When compared to the black hole patch with the greater temperature, the one with the lower temperature will take a significantly longer amount of time to send the information to the observer. This suggests that the island of the black hole patch doesn’t make its appearance until much later for the black hole patch with the lower temperature, but the island makes its debut much sooner for the black hole patch with the higher temperature. We are able to begin the process of recovering the information that was flung into the black holes horizon as soon as an island surface enters into the picture [194, 195]. As a result, we get to the conclusion that the temperature of both event horizons has a role in determining the “dominance of islands” and the “scrambling time”.

This paper is available on arxiv under CC 4.0 license.