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Static vacuum 3+1 black holes that cannot be put into stationary rotation

Javier Peraza, Martin Reiris

Abstract

We prove that some of the static Myers/Korotkin-Nicolai (MKN) vacuum 3+1 static black holes cannot be put into stationary rotation. Namely, they cannot be deformed into axisymmetric stationary vacuum black holes with non-zero angular momentum. We also prove that this occurs in particular for those MKN solutions for which the distance along the axis between the two poles of the horizon is sufficiently small compared to the square root of its area. The MKN solutions, sometimes called periodic Schwarzschild, are physically regular, have no struts or singularities, but are asymptotically Kasner. The static rigidity presented here appears to be the first in the literature of General Relativity.

Static vacuum 3+1 black holes that cannot be put into stationary rotation

Abstract

We prove that some of the static Myers/Korotkin-Nicolai (MKN) vacuum 3+1 static black holes cannot be put into stationary rotation. Namely, they cannot be deformed into axisymmetric stationary vacuum black holes with non-zero angular momentum. We also prove that this occurs in particular for those MKN solutions for which the distance along the axis between the two poles of the horizon is sufficiently small compared to the square root of its area. The MKN solutions, sometimes called periodic Schwarzschild, are physically regular, have no struts or singularities, but are asymptotically Kasner. The static rigidity presented here appears to be the first in the literature of General Relativity.
Paper Structure (9 sections, 4 theorems, 53 equations, 3 figures)

This paper contains 9 sections, 4 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Static Periodic Black Holes
  3. Stationary Perturbations
  4. Main results
  5. Physical interpretation of results
  6. Outlook, open problems and further work.
  7. Proof of the main results
  8. Proof of Theorem \ref{['MAIN1']}
  9. Proof of Theorem \ref{['MAIN3']}

Key Result

Theorem 1

Assume that, is the metric of a periodic stationary and axisymmetric black-hole with period $L$, horizon length $m$ and non-zero angular momentum. Assume also that the 2-metric $q=e^{2\gamma}(dz^{2}+d\rho^{2})$ is metrically complete at infinity. Then, $L/m\geq 4$.

Figures (3)

  • Figure 1: The grey region is the spatial manifold of a MKN solution. It is topologically an open solid 3-torus minus a 3-ball (the black ball). The boundary of the black 3-ball is the horizon. The border of the solid torus, marked with a dashed line, is 'infinity' and of course lies at an infinite metric distance from the horizon. The axis of the ${\rm S}^{1}$-symmetry is marked with the dashed line in the middle of the solid torus.
  • Figure 2: A representation of the the infinite periodic array of horizons of the universal MKN solutions.
  • Figure 3: The domains $D_{L,m}$.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Theorem 3
  • proof : Proof of Theorem \ref{['MAIN1']}
  • proof : Proof of Proposition \ref{['P1']}