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Uniqueness theorem for 5-dimensional black holes with two axial Killing fields (0707.2775v2)

Published 18 Jul 2007 in gr-qc

Abstract: We show that two stationary, asymptotically flat vacuum black holes in 5 dimensions with two commuting axial symmetries are identical if and only if their masses, angular momenta, and their ``rod structures'' coincide. We also show that the horizon must be topologically either a 3-sphere, a ring, or a Lens-space. Our argument is a generalization of constructions of Morisawa and Ida (based in turn on key work of Maison) who considered the spherical case, combined with basic arguments concerning the nature of the factor manifold of symmetry orbits.

Citations (173)

Summary

  • The paper establishes that 5D black holes are uniquely characterized by mass, angular momenta, and rod structures when additional geometric parameters are considered.
  • It employs a σ-model formulation of the reduced Einstein equations to extend uniqueness methods from four to five dimensions.
  • The research underscores the significance of rod structures in distinguishing complex horizon topologies such as 3-spheres, rings, and Lens-spaces.

Uniqueness Theorem for 5-Dimensional Black Holes with Two Axial Symmetries

The research paper, authored by Stefan Hollands and Stoytcho Yazadjiev, explores a significant topic in the field of higher-dimensional black hole physics: the uniqueness theorem for 5-dimensional stationary, asymptotically flat vacuum black hole spacetimes with two commuting axial Killing fields. The central result established in this paper is that these black holes are characterized uniquely by their mass, angular momenta, and rod structures, provided the spacetime contains no discrete isotropy group points. In doing so, the authors extend earlier efforts and methodologies initiated by Maison, Morisawa, and Ida, who concentrated on a spherical horizon case.

Core Findings

The paper posits that the uniqueness of 4-dimensional black holes, in terms of their conserved asymptotic charges, does not hold in the more complex 5-dimensional setting. To address this, Hollands and Yazadjiev provide a modified uniqueness theorem by incorporating additional structural parameters—referred to as rod structures—besides mass and angular momentum. Rod structures consist of global parameters that characterize the geometry around the axis of rotation and horizons, as introduced by Harmark in a localized manner. These parameters play a crucial role in identifying the topologies of the black hole horizons, which can be a 3-sphere, S × S, or a Lens-space L(p,q), each with specific implications for the geometry and symmetry of the spacetime.

The Mathematical and Physical Framework

The paper showcases a sophisticated approach utilizing the σ-model formulation of the reduced Einstein equations in five dimensions. This mathematical framework aligns with methods previously applied in four-dimensional scenarios, encompassing twist potentials that are used to derive matrix fields (Φ) capturing essential properties of the geometry. By integrating the divergence identity of these fields across the orbit space—a simply connected 2-dimensional manifold—the authors establish that the solutions are isometric when rod structures and asymptotic charges coincide.

Implications and Speculation on Broader Theories

The implications of this research are profound, suggesting that additional parameters must be considered for a comprehensive classification of 5-dimensional black holes. The theorem helps delineate the landscape of possible solutions, shedding light on non-trivial and potentially complex configurations that would otherwise remain indistinguishable from simpler counterparts based solely on conserved charges. The insights derived have potential applications in extending this framework to black holes surrounded by matter fields and refining the understanding of spacetime rigidity in higher dimensions.

Future Directions

While the paper lays a solid foundation by confirming the uniqueness based on mass, angular momenta, and rod structures, it leaves several open questions. Notably, it raises the prospect of black hole solutions with singular orbit points, suggesting that a similar uniqueness result might still hold with further refinements in the treatment of orbifold points. Additionally, exploring the realization of exotic horizon topologies like non-trivial Lens-spaces within vacuum solutions or even in Einstein-Maxwell settings remains an intriguing avenue for future research.

In summary, this paper contributes significantly to higher-dimensional relativity by establishing the conditions under which the uniqueness of black hole solutions can be assured. Its approach marries geometrical intuition with rigorous mathematical descriptions, paving the way for more sophisticated treatments of black hole topologies in multi-dimensional frameworks.