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Hidden symmetries and separability structures of Ovcharenko-Podolský and conformal-to-Carter spacetimes

Finnian Gray, David Kubiznak, Hryhorii Ovcharenko, Jiri Podolsky

Abstract

Recently, a remarkable new class of spacetimes describing black holes immersed in a non-aligned electromagnetic field has been found. While still of type D, this class goes beyond the famous Plebański--Demiański family. Here we demonstrate that the whole class admits a hidden symmetry encoded in the non-degenerate conformal Killing--Yano 2-form. Interestingly, as a direct consequence of non-alignment of the electromagnetic field and contrary to the Plebański--Demiański class (where the field is aligned), such a symmetry no longer generates the full "tower of symmetries". Despite this, it enables one to separate variables in massless Hamilton--Jacobi, conformal wave, and massless Dirac equations, as well as allows one to tackle massless vector and tensor perturbations. These results provide a useful mathematical tool for discussing numerous (astrophysical) applications to be described by these metrics. Moreover, as we shall show, the novel spacetimes provide an interesting test-ground for studying the recently defined Penrose charges whose existence is intrinsically related to hidden rather than explicit symmetries.

Hidden symmetries and separability structures of Ovcharenko-Podolský and conformal-to-Carter spacetimes

Abstract

Recently, a remarkable new class of spacetimes describing black holes immersed in a non-aligned electromagnetic field has been found. While still of type D, this class goes beyond the famous Plebański--Demiański family. Here we demonstrate that the whole class admits a hidden symmetry encoded in the non-degenerate conformal Killing--Yano 2-form. Interestingly, as a direct consequence of non-alignment of the electromagnetic field and contrary to the Plebański--Demiański class (where the field is aligned), such a symmetry no longer generates the full "tower of symmetries". Despite this, it enables one to separate variables in massless Hamilton--Jacobi, conformal wave, and massless Dirac equations, as well as allows one to tackle massless vector and tensor perturbations. These results provide a useful mathematical tool for discussing numerous (astrophysical) applications to be described by these metrics. Moreover, as we shall show, the novel spacetimes provide an interesting test-ground for studying the recently defined Penrose charges whose existence is intrinsically related to hidden rather than explicit symmetries.

Paper Structure

This paper contains 40 sections, 3 theorems, 218 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Hidden symmetries
  3. Two classes of type D spacetimes with aligned electromagnetic fields
  4. Carter spacetime
  5. Plebański--Demiański spacetime
  6. Novel family of black hole spacetimes and its hidden symetries
  7. Ovcharenko--Podolský spacetime
  8. Hidden symmetries
  9. Conformal-to-Carter spacetimes and the Killing tower
  10. Hidden symmetries of the conformal spacetime
  11. Candidates on Killing vectors
  12. Novel spacetime
  13. Separability structures
  14. Massless Hamilton--Jacobi equation
  15. Conformal scalar field equation
  16. ...and 25 more sections

Key Result

Lemma 1

Starting from the conformal-to-Carter spacetime off-shell-PD with $\Omega$ given by there exists a coordinate transformation that brings this metric to the off-shell Plebański--Demiański form PD with ${\Omega=1-p\,q}$ .

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof