[Paper Review] The black hole challenge in Randall-Sundrum II model

Νικόλαος Παππάς|arXiv (Cornell University)|Sep 2, 2014

Black Holes and Theoretical Physics3 citations

TL;DR

This paper investigates the longstanding challenge of finding a stable, regular black hole solution in the Randall-Sundrum II model, a higher-dimensional gravity framework with an infinite warped extra dimension. Despite extensive analytic and numerical efforts, no such solution has been found using conventional matter fields; the study reviews past approaches, presents recent numerical progress, and explores the failure of attempts using unconventional scalar fields, highlighting the need for new theoretical frameworks.

ABSTRACT

Models postulating the existence of additional spacelike dimensions of macroscopic or even infinite size, while viewing our observable universe as merely a 3-brane living in a higher-dimensional bulk were a major breakthrough when proposed some 15 years ago. The most interesting among them both in terms of elegance of the setup and of the richness of the emerging phenomenology is the Randall-Sundrum II model where one infinite extra spacelike dimension is considered with an AdS topology, characterized by the warping effect caused by the presence of a negative cosmological constant in the bulk. A major drawback of this model is that despite numerous efforts no line element has ever been found that could describe a stable, regular, realistic black hole. Finding a smoothly behaved such solution supported by the presence of some more or less conventional fields either in the bulk and/or on the brane is the core of the black hole challenge. After a comprehensive presentation of the details of the model and the analysis of the significance and the utility of getting a specific analytic black hole solution, several (unsuccessful) analytic and numerical approaches to the problem developed over the years are presented with some discussion about their results. The chapter closes with the latest numerical results that actually consists a major advancement in the effort to address the challenge, the presentation of the most recent analytic work trying (and unfortunately failing) to build a solution assuming the existence of unconventional scalar fields and some ideas about the routes the forthcoming analytic approaches should explore.

Motivation & Objective

• To address the persistent difficulty in constructing a stable, regular black hole solution within the Randall-Sundrum II model, a prominent higher-dimensional gravity theory with an infinite, warped extra dimension.
• To review and analyze the history of analytic and numerical approaches aimed at solving the black hole challenge, assessing their successes and limitations.
• To present the latest numerical results that represent a significant advancement in simulating black hole-like configurations in the model.
• To examine recent analytic attempts that fail to construct a solution under the assumption of unconventional scalar fields, identifying gaps in current theoretical approaches.

Proposed method

• Systematic review of the Randall-Sundrum II model's geometric and field-theoretic foundations, emphasizing the role of the negative cosmological constant in generating the warping of the extra dimension.
• Analysis of the Einstein field equations in the bulk and on the brane, including junction conditions that couple the brane to the bulk geometry.
• Application of numerical relativity techniques to solve the coupled system of bulk and brane equations, focusing on constructing horizon-like structures and regularity conditions.
• Use of perturbative and non-perturbative methods to explore solutions with matter fields, including scalar fields with non-standard kinetic terms.
• Evaluation of the stability and regularity of candidate solutions by examining curvature invariants and asymptotic behavior.
• Comparison of numerical results with analytical expectations, particularly in the limit of small brane tension and weak coupling.

Experimental results

Research questions

• RQ1 Can a stable, regular black hole solution be consistently constructed in the Randall-Sundrum II model using conventional matter fields in the bulk and on the brane?
• RQ2 What are the key obstacles that prevent the existence of such solutions, and how do they relate to the geometry of the warped extra dimension?
• RQ3 To what extent do recent numerical simulations provide evidence for the existence of black hole-like configurations in the model, even if not fully regular or stable?
• RQ4 Why do analytic attempts relying on unconventional scalar fields fail to yield viable solutions, and what do these failures reveal about the underlying physics?
• RQ5 What new theoretical or computational approaches are required to overcome the current impasse in finding a fully consistent black hole solution?

Key findings

• No fully regular, stable black hole solution has been found in the Randall-Sundrum II model using conventional matter fields, despite extensive analytical and numerical efforts.
• Recent numerical simulations have produced configurations that resemble black holes, showing apparent horizons and localized energy density, representing a major advancement in the field.
• Analytic attempts assuming the presence of unconventional scalar fields with non-standard kinetic terms have failed to produce a regular, finite-energy solution.
• The failure of these analytic approaches suggests that the black hole challenge may require new field content or a radical rethinking of the boundary conditions in the bulk.
• The results underscore the tension between the geometric constraints of the warped bulk and the physical requirements for a regular, localized black hole on the brane.