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Periodic Analogs of Multiple Black Holes Solutions

Omar Ortiz, Javier Peraza

Abstract

In this article, we extend the numerical studies developed in [arXiv:2210.12898] to construct periodic stationary axisymmetric solutions containing multiple horizons in each fundamental domain. As a direct application, we consider periodic stationary axisymmetric solutions with two identical equidistant counter-rotating horizons. These solutions can be parametrized by the period $L$, the mass $M$, and the absolute value of the angular momentum $|J| >0$. We provide strong numerical evidence for the existence of such configurations, without any restriction in terms of the distance between horizons. This is in sharp contrast with the non-zero total angular momentum case, as it was recently established in \cite{Peraza:2024uto} that static single-horizon periodic solutions cannot be put into rotation if $L < 4M$. It is shown that these solutions do not have any struts on the axis, and it is also explicitly shown that, by taking non-equidistant horizons, struts develop between the black holes. Other global properties of the solutions are also presented.

Periodic Analogs of Multiple Black Holes Solutions

Abstract

In this article, we extend the numerical studies developed in [arXiv:2210.12898] to construct periodic stationary axisymmetric solutions containing multiple horizons in each fundamental domain. As a direct application, we consider periodic stationary axisymmetric solutions with two identical equidistant counter-rotating horizons. These solutions can be parametrized by the period , the mass , and the absolute value of the angular momentum . We provide strong numerical evidence for the existence of such configurations, without any restriction in terms of the distance between horizons. This is in sharp contrast with the non-zero total angular momentum case, as it was recently established in \cite{Peraza:2024uto} that static single-horizon periodic solutions cannot be put into rotation if . It is shown that these solutions do not have any struts on the axis, and it is also explicitly shown that, by taking non-equidistant horizons, struts develop between the black holes. Other global properties of the solutions are also presented.
Paper Structure (15 sections, 1 theorem, 59 equations, 6 figures, 5 tables)

This paper contains 15 sections, 1 theorem, 59 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background and definitions
  3. Periodic stationary and axisymmetric data
  4. Kasner and Lewis Solutions as asymptotic models
  5. Smarr formula and Mass formula
  6. Numerical analysis with many horizons
  7. Construction of the seed
  8. Application: two counter-rotating horizons
  9. Numerical Implementation
  10. Parabolic flow convergence to stationary state
  11. Convergence with respect to space discretization and regularity at the axis
  12. Plot of the solutions
  13. Smarr identity
  14. Mass, Kasner parameter and angular momentum
  15. Conclusions

Key Result

Proposition 1

Absence of angle defects for certain configurations. Given a periodic stationary and axisymmetric black hole data $(\mathcal{S}, \mathfrak{q} , \sigma , \omega)$ with parameters $(\{m_i/L\}_{i=1}^N, \{z_i/L\}_{i^1}^{N} , \{A_i\}_{i=1}^N, \{J_i\}_{i=1}^N)$ such that then $q = 0$ at the axis.

Figures (6)

  • Figure 1: Schematic representation of a multi-horizon set-up, for $j$ horizons. The data is either $z-$odd or $z-$even.
  • Figure 2: Convergence of the parabolic flow to stationary state as function of time steps for the solution with $N_h = 50$ of table \ref{['table_conv_series_J03']}.
  • Figure 3: Plots of solutions corresponding to $N_h=50$ of table \ref{['table_conv_series_J03']}. From left to right, from top to bottom: $\bar{\sigma}$, $\bar{\omega}$, $\sigma$, $\omega$.
  • Figure 4: Plots of $M(\rho, \tau)$ for $\tau=0$ (seed) and final $\tau$ (solution) for six selected solutions of \ref{['table_conv_series_J03']}. From left to right, from top to bottom, the solutions corresponding to $N_h = 22, 34, 46, 58, 70, 82.$
  • Figure 5: Plots of $\eta$ of the best fitting kasner solution and $z$-averaged $\eta$ of the solution (indicated as $\bar{\eta}$), for six solutions in \ref{['table_conv_series_J03']}. From left to right, from top to bottom, the solutions corresponding to $N_h = 22, 34, 46, 58, 70, 82.$
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof