What if the enterprise had complete control over the browser? What would it mean for security, for productivity, for work itself? Introducing Island, the Enterprise Browser - the ideal enterprise workplace, where work flows freely while remaining fundamentally secure. With the core needs of the enterprise naturally embedded in the browser itself, Island gives organizations complete control, visibility, and governance over the last mile, while delivering the same smooth Chromium-based browser experience users expect. Led by experienced leaders in enterprise security and browser technology and backed by leading venture funds -- Insight Partners, Sequoia Capital, Cyberstarts and Stripes Capital -- Island is redefining the future of work for some of the largest, most respected enterprises in the world. Welcome to work as it should be. www.island.io

Island

Welcome to the Startups of the Year Winner Interview Series. Congratulations for being one of the top 3000 startups worldwide! The HackerNoon team is thrilled to know more about you! What follows is an interview with Mark Zalan, CEO of GoMining, winner of the HackerNoon Startups of the Year 2023 in the British Virgin Islands Misc, Oceania. What does it mean for you to win this title? We are pleased and gratified at the recognition of our contributions. We are constantly looking for community feedback to improve our products, and this kind of appreciation helps make our efforts worthwhile. Winning the Hackernoon community recognition is a significant indicator for us that our team is moving in the right direction. Since our product targets a fairly broad audience, the trust of the audience and feedback help us better understand user needs, and Hackernoon provides the perfect environment for addressing this task. What sets you apart from the competition? We are focused on making the superior economics of Bitcoin mining accessible to everyone.  In building products that allow our users to start mining Bitcoin within a matter of minutes and at a cost as low as $25, we are unique. The explosive customer demand we’ve seen over the past 2 years is a confirmation of that. Check out our What do you love about your team, and why are you the ones to solve this problem? \We are irreverent around challenging every assumption and traditional way of doing things.  Obviously, this can be a strength and a weakness, and if not done right, can lead to reinventing the wheel.  However, we’ve managed to maintain a healthy balance between blazing new trails and inventing a new product (that hundreds of people told us couldn’t be done), making the economics of Bitcoin mining accessible to non-industry insiders and leveraging traditional approaches that we’ve confirmed are useful in what we are trying to do. This blend of questioning the way things are and inventing new and better ways while staying practical and leveraging that which works for purpose has allowed us to offer a successful product that is posting world-class growth rates. If you weren’t building your startup, what would you be doing? My other passions are history and political science, as well as understanding how our societies function and evolved to where we are today.  In another life I would likely be lecturing at a university and writing history books. At the moment, how do you measure success? What are your core metrics? We are seeing double-digit monthly growth rates while working hard to expand the business that is profitable, positioned in the middle of an industry that is projected for staggering worldwide growth.  All of the above are how we think about success and how we want to position our ecosystem to take advantage of the global and industry trends we see for the near future.  If I had to narrow it down to one metric, it is definitely the customer growth rate, which tells us we are tackling a significant pain-point that exists for a large number of people, in a way they appreciate. Tell us how your startup is changing the world. We make the excellent economics of Bitcoin mining available to regular people. Getting a financial return with Bitcoin mining traditionally required millions of dollars in capital and highly specialized technical skills (that also cost money to develop or hire).  We’ve built a product that obfuscates that complexity and allows customers to start participating in Bitcoin mining in minutes and with as little as $25, with the first profit landing in the customer’s Bitcoin wallet the very next morning.  By lowering the barriers to entry for normal people to earn rewards, while strengthening, securing the Bitcoin network infrastructure, we make our positive impact on the world. How do you or your company intend to embrace the responsibility of this title in 2024? We believe the best way to contribute and to give back is to do what we do best, continue to innovate, while lowering barriers to entry, and to be a good citizen of the Bitcoin and broader technology community. The recognition from users and the community serves as a driving force for our continued advancement and, as I mentioned earlier, provides vital feedback on the quality of our product. What goals are you looking forward to accomplishing in 2024? We expect 2024 to be transformational for us through a variety of factors. Firstly, the Bitcoin halving will impact our business and our vertical in a significant way.  The economics will change, and the market participants will change. We’ve been preparing for this event for the past three years ensuring our business is cost optimized and as efficient as possible, as we expect this will allow us and our user community to pass through the upcoming disruption, and to be best positioned to take advantage of the changing economics following the halving.  Secondly, our user community is scaling by orders of magnitude, and we place a major focus on ensuring our support and engagement processes are peerless in creating a world-class user experience. Which trend(s) are you most excited about in 2024? Share your reasons. 2024 Is a paradigm-shifting year for the Bitcoin mining community with the ETFs and the halving taking place within three months of each other.  We are very excited to see the ways in which our industry and the broader technology space will transform to take advantage of the changes. 2023 had been another crazy year, especially in tech, with layoffs and the Generative AI takeover! Which trend are you most concerned about? Be as brief or as detailed as you like. I wouldn’t say concerned, but we are very cognizant of the pace of change. It seems things are speeding up, paradigms are shifting quickly, changes that used to take decades now take years or sometimes months. The way the world became location-independent in the past four years is very illustrative of that. We expect changes of the same scale to happen as quickly or faster going forward, so it will be important to be diligent in always learning, looking for ways to do something better, and being flexible, adjusting to take advantage of new trends, technologies, and opportunities. We don’t know what specifically those will be, five years ago, hardly anyone would have said that remote work, generative AI or Bitcoin ETFs/general institutionalization would be a thing over the next 5 years, so we don’t think it's worthwhile to predict specifics. We do think it's important to understand that changes happen quickly. They bring threats and opportunities, and figuring it out is a fun and exciting part of it. Share your biggest success so far and/or your biggest failure so far. Building a career out of always learning, developing, and pushing myself into new pursuits, taking advantage of opportunities that may have scared others off, is probably the biggest success. In terms of failure, I certainly wish I was smarter along the way, and figured out many things faster than I did! :) We would love your feedback on HackerNoon as a tech publication! How has your experience been with us? HackerNoon is an excellent platform for showcasing our technological solutions to a wide and relevant community capable of providing objective feedback. We closely monitor all materials about our product appearing on Hackernoon and pay attention to UX moments that require improvement. In a broader sense, your platform is a great place to feel the pulse of Web3, track emerging trends, and receive important insights directly from where they originate. Any words of wisdom you’d like to share with us? Not wildly original here, but the advice I most want to pass on to my children is to enjoy the journey, whatever destination one travels. This is one of the things I wish I would have figured out sooner. Welcome to the Startups of the Year Winner Interview Series. Congratulations for being one of the top 3000 startups worldwide! The HackerNoon team is thrilled to know more about you! Welcome to the Startups of the Year Winner Interview Series. Congratulations for being one of the top 3000 startups worldwide! The HackerNoon team is thrilled to know more about you! Welcome to the Startups of the Year Winner Interview Series. Congratulations for being one of the top 3000 startups worldwide! The HackerNoon team is thrilled to know more about you! What follows is an interview with Mark Zalan, CEO of GoMining, winner of the HackerNoon Startups of the Year 2023 in the British Virgin Islands Misc, Oceania. What follows is an interview with Mark Zalan, CEO of GoMining , winner of the HackerNoon Startups of the Year 2023 in the British Virgin Islands Misc, Oceania . GoMining British Virgin Islands Misc, Oceania What does it mean for you to win this title? What does it mean for you to win this title? We are pleased and gratified at the recognition of our contributions. We are constantly looking for community feedback to improve our products, and this kind of appreciation helps make our efforts worthwhile. Winning the Hackernoon community recognition is a significant indicator for us that our team is moving in the right direction. Since our product targets a fairly broad audience, the trust of the audience and feedback help us better understand user needs, and Hackernoon provides the perfect environment for addressing this task. What sets you apart from the competition? We are focused on making the superior economics of Bitcoin mining accessible to everyone.  In building products that allow our users to start mining Bitcoin within a matter of minutes and at a cost as low as $25, we are unique. The explosive customer demand we’ve seen over the past 2 years is a confirmation of that. Check out our What do you love about your team, and why are you the ones to solve this problem? \We are irreverent around challenging every assumption and traditional way of doing things.  Obviously, this can be a strength and a weakness, and if not done right, can lead to reinventing the wheel.  However, we’ve managed to maintain a healthy balance between blazing new trails and inventing a new product (that hundreds of people told us couldn’t be done), making the economics of Bitcoin mining accessible to non-industry insiders and leveraging traditional approaches that we’ve confirmed are useful in what we are trying to do. This blend of questioning the way things are and inventing new and better ways while staying practical and leveraging that which works for purpose has allowed us to offer a successful product that is posting world-class growth rates. If you weren’t building your startup, what would you be doing? My other passions are history and political science, as well as understanding how our societies function and evolved to where we are today.  In another life I would likely be lecturing at a university and writing history books. At the moment, how do you measure success? What are your core metrics? We are seeing double-digit monthly growth rates while working hard to expand the business that is profitable, positioned in the middle of an industry that is projected for staggering worldwide growth.  All of the above are how we think about success and how we want to position our ecosystem to take advantage of the global and industry trends we see for the near future.  If I had to narrow it down to one metric, it is definitely the customer growth rate, which tells us we are tackling a significant pain-point that exists for a large number of people, in a way they appreciate. Tell us how your startup is changing the world. We make the excellent economics of Bitcoin mining available to regular people. Getting a financial return with Bitcoin mining traditionally required millions of dollars in capital and highly specialized technical skills (that also cost money to develop or hire).  We’ve built a product that obfuscates that complexity and allows customers to start participating in Bitcoin mining in minutes and with as little as $25, with the first profit landing in the customer’s Bitcoin wallet the very next morning.  By lowering the barriers to entry for normal people to earn rewards, while strengthening, securing the Bitcoin network infrastructure, we make our positive impact on the world. How do you or your company intend to embrace the responsibility of this title in 2024? We believe the best way to contribute and to give back is to do what we do best, continue to innovate, while lowering barriers to entry, and to be a good citizen of the Bitcoin and broader technology community. The recognition from users and the community serves as a driving force for our continued advancement and, as I mentioned earlier, provides vital feedback on the quality of our product. What goals are you looking forward to accomplishing in 2024? We expect 2024 to be transformational for us through a variety of factors. Firstly, the Bitcoin halving will impact our business and our vertical in a significant way.  The economics will change, and the market participants will change. We’ve been preparing for this event for the past three years ensuring our business is cost optimized and as efficient as possible, as we expect this will allow us and our user community to pass through the upcoming disruption, and to be best positioned to take advantage of the changing economics following the halving.  Secondly, our user community is scaling by orders of magnitude, and we place a major focus on ensuring our support and engagement processes are peerless in creating a world-class user experience. Which trend(s) are you most excited about in 2024? Share your reasons. 2024 Is a paradigm-shifting year for the Bitcoin mining community with the ETFs and the halving taking place within three months of each other.  We are very excited to see the ways in which our industry and the broader technology space will transform to take advantage of the changes. 2023 had been another crazy year, especially in tech, with layoffs and the Generative AI takeover! Which trend are you most concerned about? Be as brief or as detailed as you like. I wouldn’t say concerned, but we are very cognizant of the pace of change. It seems things are speeding up, paradigms are shifting quickly, changes that used to take decades now take years or sometimes months. The way the world became location-independent in the past four years is very illustrative of that. We expect changes of the same scale to happen as quickly or faster going forward, so it will be important to be diligent in always learning, looking for ways to do something better, and being flexible, adjusting to take advantage of new trends, technologies, and opportunities. We don’t know what specifically those will be, five years ago, hardly anyone would have said that remote work, generative AI or Bitcoin ETFs/general institutionalization would be a thing over the next 5 years, so we don’t think it's worthwhile to predict specifics. We do think it's important to understand that changes happen quickly. They bring threats and opportunities, and figuring it out is a fun and exciting part of it. Share your biggest success so far and/or your biggest failure so far. Building a career out of always learning, developing, and pushing myself into new pursuits, taking advantage of opportunities that may have scared others off, is probably the biggest success. In terms of failure, I certainly wish I was smarter along the way, and figured out many things faster than I did! :) We would love your feedback on HackerNoon as a tech publication! How has your experience been with us? HackerNoon is an excellent platform for showcasing our technological solutions to a wide and relevant community capable of providing objective feedback. We closely monitor all materials about our product appearing on Hackernoon and pay attention to UX moments that require improvement. In a broader sense, your platform is a great place to feel the pulse of Web3, track emerging trends, and receive important insights directly from where they originate. Any words of wisdom you’d like to share with us? Not wildly original here, but the advice I most want to pass on to my children is to enjoy the journey, whatever destination one travels. This is one of the things I wish I would have figured out sooner.

Meet GoMining: Winner of the Startups of the Year, British Virgin Islands 

Authors: (1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute. 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 8 - BLACK HOLE ISLANDS IN MULTI-EVENT HORIZON SPACE-TIMES 8.1 Introduction and Motivation 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 entanglement entropy and complexity have been studied in [217–219] in the context of de-Sitter space. 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. 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, 2 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. D 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 G ). N 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: Page 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: Scrambling time 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 under CC 4.0 license. available on arxiv

Black Hole Islands in Multi-Event Horizon Space-Times

The Noonification: Gravity Islands and the Multiverse (2/22/2024)

Authors: (1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute. 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 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 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 in Quantum Mechanics (QM): 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: 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: Entanglement Entropy in Quantum Field Theory (QFT): • 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. The Planck brane is coupled to the external CFT bath, which acts as the sink for the Hawking radiation. • 2D-Gravity: Two-dimensional conformal field theory has the three-dimensional gravity dual with metric AdS3 via AdS/CFT correspondence. • 3D-Gravity: The boundary of the external CFT bath is one dimensional where quantum mechanics (QM) is present. • QM: 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 boundary as well. The generalised entropy in the presence of island surface is written as follows: n 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 S (r) = S r(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]. gen matte 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 S (R ∪ I) of (5.11) and S will be calculated using the Dong’s formula [127]. For the AdSd+1/CF Td correspondence the Dong’s formulas is given below[1]. matter gravity where 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 AdS correspondence using a bottom-up approach, and the setup is shown in figure 5.1. d+1/BCFTd The doubly holographic setup has the three descriptions summarised below. -dimensional BCFT at the conformal boundary of the bulk AdS . The boundary of is the ( ) dimensional defect. • Boundary description: d d+1 BCFTd d − 1 Gravity on the end-of-the-world brane is coupled to BCFT via transparent boundary condition at the defect. • Intermediate description: The holographic dual of is spacetime. • Bulk description: BCFTd AdSd+1 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 (S (R ∪ I)) is [126]: gen A doubly holographic setup is advantageous in the sense that we can obtain very easily using the classical Ryu-Takayanagi formula [107]. When bulk is (d + 1) dimensional then [107]: S(R) 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. 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: BCF Td on the conformal boundary of the bulk with the ( ) dimensional defect. • Boundary description: AdSd+1 d − 1 two gravitating systems are connected with each other via the transparent boundary condition at the defect. • Intermediate description: the holographic dual of is classical gravity spacetime. • Bulk description: BCFTd AdSd+1 The wedge holographic dictionary for the ( + 1)- dimensional bulk is stated as: . Therefore it is a co-dimension two holography. Now let us understand the how this duality exists. d holographic dual of the (d−1)-dimensional defect conformal field theory is the classical gravity in (d+1)- dimensions 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, ( + 1) 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. classical gravity in d bulk is dual to CFTd−1 at the defect. [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. This paper is under CC 4.0 license. available on arxiv

Gravity Islands and the Multiverse

গ্র্যাভিটি দ্বীপপুঞ্জ এবং মাল্টিভার্স

Schwerkraftinseln und das Multiversum

Islas de gravedad y el multiverso

Îles de gravité et multivers

ग्रेविटी द्वीप और मल्टीवर्स

重力諸島と多元宇宙

중력섬과 다중우주

Ilhas Gravitacionais e o Multiverso

Гравитационные острова и Мультивселенная

Yerçekimi Adaları ve Çoklu Evren

Quần đảo trọng lực và đa vũ trụ

重力岛和多元宇宙

The United States v Meta Platforms Court Filing October 24, 2023 is part of . You can jump to any part in this filing . This is part 94 of 100. HackerNoon’s Legal PDF Series here COUNT XLIX: VIOLATIONS OF RHODE ISLAND DECEPTIVE TRADE PRACTICES ACT, R.I. GEN. L. §§ 6-13.1-1 TO 6-13.1-10 1133. The State of Rhode Island incorporates and realleges each of the paragraphs 1 through 850 as if fully set forth herein. 1134. The Rhode Island Deceptive Trade Practices Act (RI DTPA) makes unfair methods of competition and unfair or deceptive acts or practices in the conduct of any trade or commerce unlawful. R.I. Gen. Laws § 6-13.1-2. 1135. The RI DTPA defines “unfair methods of competition and unfair and deceptive acts or practices” as, among other things, “conduct that . . . creates a likelihood of confusion or of misunderstanding,” “any other methods, acts, or practices that mislead or deceive members of the public in a material respect,” and “any act or practice that is unfair or deceptive to the consumer.” R.I. Gen. Laws § 6-13.1-1(6)(xii), (xiv), (xiii). 1136. Any person, firm, or corporation who violates the RI DTPA is liable for a civil penalty up to $10,000 for each violation. R.I. Gen. Laws § 6-13.1-8. 1137. Meta’s acts or practices enumerated in the foregoing paragraphs have been in the conduct of trade or commerce, directly or indirectly, in Rhode Island. 1138. As alleged herein, Meta made representations including that Meta’s Social Media Platforms are not designed to harm young users or to induce compulsive use, that Meta’s Social Media Platforms are less addictive than they actually are, that the incidence of negative user experiences was lower than it actually was, that Meta was not prioritizing profit maximization over young users’ well-being when it was, that Meta effectively excluded under-13 users when its safeguards were insufficient, that Meta complied with federal laws and regulations related to the exclusion of under-13 users when it did not, and that Meta did not collect user data for the purpose of causing addiction to its Social Media Platforms when it had such a purpose. These representations constitute conduct that creates a likelihood of confusion or misunderstanding and that deceive and mislead members of the public regarding Meta’s Social Media Platforms. R.I. Gen. Laws § 6-13.1-1(6)(xii), (xiii), (xiv). 1139. Similarly, as alleged herein, Meta is engaging in unfair acts to consumers, including implementing psychologically manipulative, engagement-inducing features that harm consumers and targeting young users despite knowing their specific vulnerability to compulsive and unhealthy platform use. These unfair acts constitute conduct that is especially unfair to younger users. R.I. Gen. Laws § 6-13.1-1(6)(xiii). 1140. Meta’s acts or practices, both past and continuing, are immoral, unethical, oppressive, unscrupulous, and substantially injurious to Rhode Island consumers. Pursuant to R.I. Gen. Laws § 6-13.1-2 and § 6-13.1-5, the acts, practices, representations, and omissions of Meta described herein are unlawful, violate the prohibition against unfair or deceptive acts or practices found in RI DTPA, and restraint of these practices is in the public interest. Continue Reading . Here About HackerNoon Legal PDF Series: We bring you the most important technical and insightful public domain court case filings. This court case 4:23-cv-05448 retrieved on October 25, 2023, from is part of the public domain. The court-created documents are works of the federal government, and under copyright law, are automatically placed in the public domain and may be shared without legal restriction. Washingtonpost.com Lead image by Dima Solomin on Unsplash

Meta Accused of Violating Rhode Island Deceptive Trade Practices Act

FTC v. Amazon Court Filing, retrieved on Sep 26, 2023, is part of . You can jump to any part in this filing . This is part 78 of 80. HackerNoon’s Legal PDF Series here COUNT XIX: VIOLATIONS OF RHODE ISLAND LAW 557. Plaintiff State of Rhode Island repeats and re-alleges and incorporates by reference each and every preceding paragraph and allegation in the Complaint as if fully set forth herein. 558. Amazon’s actions as alleged herein violate the Rhode Island Antitrust Act, R.I. Gen. Laws § 6-36-1, et seq. 559. Plaintiff State of Rhode Island seeks all remedies available under the Rhode Island Antitrust Act, R.I. Gen. Laws §§ 6-36-10, 6-36-11 and 6-36-12 and seeks relief, including but not limited to injunctive relief, equitable monetary relief, fees, costs, and such other relief as this Court deems just and equitable. 560. Amazon’s actions as alleged herein constitute unfair methods of competition and unfair or deceptive acts or practices as defined in the Rhode Island Deceptive Trade Practices Act, R.I. Gen. Laws § 6-13.3-1, et seq. 561. The State of Rhode Island brings this action pursuant to R.I. Gen. Laws § 6-13.1- 5, and seeks relief, including but not limited to injunctive relief, equitable monetary relief, fees, costs, and such other relief as this Court deems just and equitable. Continue Reading . Here About HackerNoon Legal PDF Series: We bring you the most important technical and insightful public domain court case filings. This court case 2:23-cv-01495 retrieved on October 2, 2023, from is part of the public domain. The court-created documents are works of the federal government, and under copyright law, are automatically placed in the public domain and may be shared without legal restriction. ftc.gov

Rhode Island Takes Legal Action Against Amazon for Unfair Trade Practices 

Return to Pleasure Island by Cory Doctorow is part of the HackerNoon Books Series. Read this book online for free on HackerNoon! : Return to Pleasure Island Title : Cory Doctorow Author : November 20, 2005 [eBook #17027] Most recently updated: December 12, 2020 Release date : English Language TABLE OF LINKS Return to Pleasure Island About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Cory Doctorow (2005). Return to Pleasure Island. Urbana, Illinois: Project Gutenberg. Retrieved October 2022* https://www.gutenberg.org/cache/epub/620/pg620-images.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at , located at . www.gutenberg.org https://www.gutenberg.org/policy/license.html

ISLAND

island.io

51-200 employees

Since 2020

COMPANY RANKING

ISLAND

EVERGREEN INDEX #5441

Representing this company? Click here to claim and customize this page!

Island WIKI

Apply to get your Tech Company News Page live on HackerNoon.com today!

ISLAND

island.io

51-200 employees

Since 2020

COMPANY RANKING

ISLAND

EVERGREEN INDEX #5441

RELATED COMPANIESRANK

STORIES ABOUT ISLAND

Representing this company? Click here to claim and customize this page!

Island WIKI

Apply to get your Tech Company News Page live on HackerNoon.com today!