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Non-axisymmetric (2+1) black holes with Dym boundary conditions

Miguel Pino, Francisco Reyes

TL;DR

This work analyzes stationary, non-axisymmetric black holes in AdS$_3$ gravity under the first non-trivial Dym boundary conditions. It combines holonomy conditions from the Chern-Simons formulation with periodic solutions of the Dym integrable hierarchy (N=1) to construct solid-torus Euclidean geometries describing BH spacetimes. The solution space is parameterized by two positive real numbers, two positive integers, and a relative phase, yielding explicit non-axisymmetric horizons and computable charges M, J, and temperatures from the two CS copies. This provides a concrete bridge between integrable boundary conditions and black-hole spectra in three-dimensional gravity, and outlines a path to higher-order Dym solutions.

Abstract

We present the spectrum of stationary black hole solutions associated with the first non-trivial member of the Dym family of boundary conditions. The analysis combines the holonomy conditions provided by the Chern-Simons formulation of AdS$_3$ gravity with the periodic solutions of the stationary Dym equation. The resulting spacetimes are characterized by two positive real parameters, two positive integers, and a relative phase, generically describing stationary, non-axisymmetric black holes.

Non-axisymmetric (2+1) black holes with Dym boundary conditions

TL;DR

This work analyzes stationary, non-axisymmetric black holes in AdS gravity under the first non-trivial Dym boundary conditions. It combines holonomy conditions from the Chern-Simons formulation with periodic solutions of the Dym integrable hierarchy (N=1) to construct solid-torus Euclidean geometries describing BH spacetimes. The solution space is parameterized by two positive real numbers, two positive integers, and a relative phase, yielding explicit non-axisymmetric horizons and computable charges M, J, and temperatures from the two CS copies. This provides a concrete bridge between integrable boundary conditions and black-hole spectra in three-dimensional gravity, and outlines a path to higher-order Dym solutions.

Abstract

We present the spectrum of stationary black hole solutions associated with the first non-trivial member of the Dym family of boundary conditions. The analysis combines the holonomy conditions provided by the Chern-Simons formulation of AdS gravity with the periodic solutions of the stationary Dym equation. The resulting spacetimes are characterized by two positive real parameters, two positive integers, and a relative phase, generically describing stationary, non-axisymmetric black holes.

Paper Structure

This paper contains 8 sections, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Euclidean Black Holes
  3. Periodic Solutions
  4. Metric Formalism
  5. Final Comments
  6. Acknowledgments
  7. Dym Conserved Charges $H^\pm_n$
  8. Integration of the $N$-th Stationary Dym Equation

Figures (4)

  • Figure 1: Plot of the function $f(u)$, showing two turning points $u_\pm$ that surround the region where $u'^2 > 0$, which is necessary for periodic solutions to exist. In this example $A>0$ and $B>\sqrt{A}$.
  • Figure 2: Numerical integration of equation \ref{['up2']}. In this example, $A=2$ and $B=1.8945$, thus rendering a periodic solution with $T=\frac{2\pi}{3}$.
  • Figure 3: Exploring the parameter space of stationary periodic solutions of the Dym equation. As can be seen in the plot on the right, for $A = 5$, the maximum attainable mode is $m = 6$, which is the highest displayed on the left plot. The minimum values $A_{min}^{(m)}$ for the parameter $A$ supporting the $m$-th mode (represented by small white circles in the plot of the right), were obtained numerically and correspond to $\{0.40,1.00,1.72,2.52,3.39,4.33,5.31,6.35,7.43,8.55,9.71,10.90,12.13,13.39,14.68, ...\}$ in order of increasing $m$. The corresponding minimum values for $B$ are given by $B_{min}^{(m)}=\sqrt{A_{min}^{(m)}}$.
  • Figure 4: Examples of surfaces at fixed $\rho = \rho_0 \ll 1$, illustrating the non-axisymmetric features of the spacetime. The plots are not to scale with respect to each other.