Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane (2010.00018v1)

Abstract: We discuss holographic models of extremal and non-extremal black holes in contact with a bath in d dimensions, based on a brane world model introduced in arXiv:2006.04851. The main benefit of our setup is that it allows for a high degree of analytic control as compared to previous work in higher dimensions. We show that the appearance of quantum extremal islands in those models is a consequence of the well-understood phase transition of RT surfaces, and does not make any direct reference to ensemble averaging. For non-extremal black holes the appearance of quantum extremal islands has the right behaviour to avoid the information paradox in any dimension. We further show that for these models the calculation of the full Page curve is possible in any dimension. The calculation reduces to numerically solving two ODEs. In the case of extremal black holes in higher dimensions, we find no quantum extremal islands for a wide range of parameters. In two dimensions, our results agree with arXiv:1910.11077 at leading order; however a finite UV cutoff introduced by the brane results in subleading corrections. For example, these corrections result in the quantum extremal surfaces moving further outward from the horizon, and shifting the Page transition to a slightly earlier time.

An Analysis of Quantum Extremal Islands in Higher Dimensional Black Holes

This paper presents a systematic paper of quantum extremal islands in higher-dimensional black holes within a holographic framework. Leveraging a Randall-Sundrum brane-world setup, the authors extend the exploration of quantum extremal islands to dimensions greater than two, building upon previous foundational work which was limited to lower-dimensional gravitational systems. The paper elucidates on how this novel configuration enables significant analytical control and facilitates deeper exploration into the entropic dynamics associated with black hole evaporation and the resolution of the information paradox.

The conceptual backdrop is grounded in the AdS/CFT correspondence where the gravitational behavior in the bulk AdS space is dual to the dynamics of a conformal field theory living on the boundary. Crucially, the presence of a codimension-one brane introduces a new brane-world scenario where gravity becomes localized. This construction is thoroughly generalized to higher-dimensional bulk spacetimes and scrutinizes its implications through several geometric configurations, particularly focusing on black holes described in AdS-Rindler coordinates.

Key Findings and Analytical Insights

1. Holographic Model and Setup:
   • The paper constructs a model where a gravitational brane spans a slice within an AdS space, creating a two-sided configuration akin to a wormhole. This setup is described across various perspectives: boundary CFT with a conformal defect, bulk AdS gravity intersected by the brane, and gravitational theories induced on the brane.
2. Quantum Extremal Islands and Entropy Calculations:
   • Quantum extremal surfaces play a pivotal role, acting as boundary markers for islands that contribute to the entropic landscape. The authors solve the extremization problems analytically, yielding simplified expressions for generalized entropy due to quantum extremal islands.
   • Calculations are reduced to solving ordinary differential equations and analyzing phase transitions in the geometry of extremal surfaces. This simplification is instrumental in understanding the information paradox, particularly in how islands evolve to dominate over direct connections across horizons.
3. Page Curve and Information Paradox:
   • Analyzing the non-extremal scenario, the entropy initially grows linearly, indicative of Hawking radiation processes. Nonetheless, the island phase becomes dominant after a certain Page time, ensuring the entropy saturates consistent with unitarity.
   • The paper outlines how entanglement wedge considerations and entropy inequalities can construct bounds explaining the dynamic entropic behavior, resolving potential paradoxes related to eternal black holes.
4. Extremal Black Hole Considerations:
   • Exploring extremal black holes wherein radial Poincaré coordinates describe the spacetime, the lack of radiating dynamics due to zero temperature is discussed. Here, the analysis reveals that islands do not form in high dimensions, emphasizing variance with two-dimensional models where such formations remain pertinent.

Implications and Future Directions

The paper's implications reach beyond theoretical resolution of the information paradox, suggesting practical insights into holography and quantum gravity dynamics. Notably, the paper suggests extending analysis to include massive black holes, potentially highlighting further nuances of gravitational couplings on branes. The intersections of these novel insights with tensor networks, data reconstruction through entanglement wedges, or probing holographic complexity under dynamics offer fertile ground for future investigation.

Overall, this paper provides a robust holographic framework for understanding quantum extremal islands in diverse gravitational scenarios. By extending analytical control to higher dimensions, it lays groundwork for nuanced explorations of black hole entropy, quantum information, and the intricate dance of gravity and quantum fields across our universe's holographic curtain.