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Boundary Value Problem for Black Rings

Yoshiyuki Morisawa, Shinya Tomizawa, Yukinori Yasui

TL;DR

This work addresses the uniqueness of asymptotically flat, stationary black rings in five-dimensional vacuum gravity under the presence of three commuting Killing vectors and S^1×S^2 horizon topology. It constructs a general four-parameter black ring solution (potentially with conical singularities) and identifies how demanding regularity removes these singularities to recover the Pomeransky-Sen'kov solution. The authors then apply the Mazur identity, within a boundary-value framework, to show that two solutions with the same mass, two angular momenta, and rod structure are isometric, thereby establishing uniqueness of the regular black ring in this class. The result narrows the landscape of possible higher-dimensional black holes by linking horizon topology, rod data, and asymptotic charges to a single regular solution, with implications for rigidity and higher-dimensional gravity.

Abstract

We study the boundary value problem for asymptotically flat stationary black ring solutions to the five-dimensional vacuum Einstein equations. Assuming the existence of two additional commuting axial Killing vector fields and the horizon topology of $S^1\times S^2$, we show that the only asymptotically flat black ring solution with a regular horizon is the Pomeransky-Sen'kov black ring solution.

Boundary Value Problem for Black Rings

TL;DR

This work addresses the uniqueness of asymptotically flat, stationary black rings in five-dimensional vacuum gravity under the presence of three commuting Killing vectors and S^1×S^2 horizon topology. It constructs a general four-parameter black ring solution (potentially with conical singularities) and identifies how demanding regularity removes these singularities to recover the Pomeransky-Sen'kov solution. The authors then apply the Mazur identity, within a boundary-value framework, to show that two solutions with the same mass, two angular momenta, and rod structure are isometric, thereby establishing uniqueness of the regular black ring in this class. The result narrows the landscape of possible higher-dimensional black holes by linking horizon topology, rod data, and asymptotic charges to a single regular solution, with implications for rigidity and higher-dimensional gravity.

Abstract

We study the boundary value problem for asymptotically flat stationary black ring solutions to the five-dimensional vacuum Einstein equations. Assuming the existence of two additional commuting axial Killing vector fields and the horizon topology of , we show that the only asymptotically flat black ring solution with a regular horizon is the Pomeransky-Sen'kov black ring solution.

Paper Structure

This paper contains 9 sections, 3 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General Black Ring
  3. Boundary conditions
  4. Rod Structure
  5. Asymptotic behavior
  6. Mazur Identity
  7. Boundary conditions and coincidence of solutions
  8. Conclusions and Discussions
  9. explicit expressions of metric functions

Key Result

Theorem 1

Consider two stationary, asymptotically flat, vacuum black objects spacetimes of the five-dimensions with commuting two axial Killing vector fields and a timelike Killing vector field. Then, if both solutions have the same topology, the same rod structure and the values of the mass $M$ and angular m

Figures (1)

  • Figure 1: The left figure show the rod structure of the five dimensional rotating black ring solution with asymptotic flatness. The vectors on the rods denote their directions. The right figure shows the black ring solution on the $(\rho,z)$-plane with the three-dimensional $(t,\phi,\psi)$ part suppressed.

Theorems & Definitions (3)

  • Theorem 1
  • Corollary 1
  • Theorem 2