Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Couch--Torrence inversions

Panagiotis Charalambous, Laura Donnay, Alexandru Lupsasca

TL;DR

This paper broadens the Couch–Torrence (CT) inversion symmetry beyond extremal black holes by recasting CT as a photon-sphere mapping and showing that the symmetry can persist without horizon–infinity interchange or preservation of the radial equation’s singularity structure. It demonstrates CT symmetry in the static lukewarm de Sitter background, reveals a phase-space–dependent CT for extremal Kerr–Newman, and uncovers a hidden CT symmetry in the static sector of Kerr perturbations using adapted coordinates where the problem becomes effectively flat. A key finding is that CT fixed points align with photon orbits, linking geometric optics with scattering data, including superradiance via a factorized CT map. Together, these results imply a broader, potentially universal CT framework with implications for black hole scattering, perturbation theory, and holography.

Abstract

The Laplace equation on Euclidean flat space admits a discrete radial inversion symmetry. In 1983, Couch and Torrence (CT) found -- surprisingly -- that the massless wave equation continues to display this symmetry on the background of an extremal (and asymptotically flat) black hole, where the inversion interchanges horizon and infinity while preserving the singularity structure of the separated radial mode equation. We revisit this CT inversion symmetry and investigate its possible extensions beyond the extremal (Reissner-Nordström or Kerr) setting in which it was originally identified. Using the example of the static lukewarm de Sitter black hole, we show that neither the exchange of horizon with infinity, nor the preservation of radial singularities, are essential features needed for a CT inversion to exist. Instead, we interpret CT transformations through their action on photon spheres, providing a unified viewpoint that extends to the (phase-space-dependent) CT inversions of the extremal Kerr-Newman geometry. For scalar fields on that spacetime, we find a simple relation between the fixed point of a CT inversion and the coefficient of superradiant scattering. Finally, we exhibit a hidden CT inversion symmetry that arises in the static limit of the Kerr Laplacian for all spins. Together, these results suggest that CT symmetry may admit a broader generalization than previously understood.

Generalized Couch--Torrence inversions

TL;DR

This paper broadens the Couch–Torrence (CT) inversion symmetry beyond extremal black holes by recasting CT as a photon-sphere mapping and showing that the symmetry can persist without horizon–infinity interchange or preservation of the radial equation’s singularity structure. It demonstrates CT symmetry in the static lukewarm de Sitter background, reveals a phase-space–dependent CT for extremal Kerr–Newman, and uncovers a hidden CT symmetry in the static sector of Kerr perturbations using adapted coordinates where the problem becomes effectively flat. A key finding is that CT fixed points align with photon orbits, linking geometric optics with scattering data, including superradiance via a factorized CT map. Together, these results imply a broader, potentially universal CT framework with implications for black hole scattering, perturbation theory, and holography.

Abstract

The Laplace equation on Euclidean flat space admits a discrete radial inversion symmetry. In 1983, Couch and Torrence (CT) found -- surprisingly -- that the massless wave equation continues to display this symmetry on the background of an extremal (and asymptotically flat) black hole, where the inversion interchanges horizon and infinity while preserving the singularity structure of the separated radial mode equation. We revisit this CT inversion symmetry and investigate its possible extensions beyond the extremal (Reissner-Nordström or Kerr) setting in which it was originally identified. Using the example of the static lukewarm de Sitter black hole, we show that neither the exchange of horizon with infinity, nor the preservation of radial singularities, are essential features needed for a CT inversion to exist. Instead, we interpret CT transformations through their action on photon spheres, providing a unified viewpoint that extends to the (phase-space-dependent) CT inversions of the extremal Kerr-Newman geometry. For scalar fields on that spacetime, we find a simple relation between the fixed point of a CT inversion and the coefficient of superradiant scattering. Finally, we exhibit a hidden CT inversion symmetry that arises in the static limit of the Kerr Laplacian for all spins. Together, these results suggest that CT symmetry may admit a broader generalization than previously understood.

Paper Structure

This paper contains 17 sections, 94 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. CT symmetry of the static lukewarm de Sitter black hole
  3. Conformal inversion for a nonextremal horizon
  4. CT inversions need not preserve the ODE singularity structure
  5. CT inversions preserve photon spheres
  6. ERN black hole
  7. Static lukewarm geometry
  8. EKN black hole
  9. Fixed points of CT inversions and superradiance
  10. CT symmetry for static perturbations of subextremal Kerr
  11. Conclusion
  12. Acknowledgments
  13. Geodesics in Kerr--Newman-(A)dS
  14. Constants of motion
  15. System of geodesics equations
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Penrose diagram for the static lukewarm black hole showing surfaces of constant $r$ that are of particular interest (see Refs. Mellor:1989giMellor:1989wcRunarsson:2012yh), color-coded in CT-exchanged pairs.
  • Figure 2: Singularity structure of the wave equation for conformally coupled scalar perturbations of the static lukewarm black hole. Top: Fig. \ref{['fig:LukewarmODESingularityPhi']} shows this structure for the wave operator \ref{['eq:WaveEqSingularities1']}--\ref{['eq:WaveEqSingularities2']} describing the propagation of the canonical scalar field $\Phi$. Bottom: Fig. \ref{['fig:LukewarmODESingularityVarphi']} shows this structure for the wave operator \ref{['eq:WaveEqSingularitiesRed1']}--\ref{['eq:WaveEqSingularitiesRed2']} governing the field $\varphi=\sqrt{\frac{M}{r-M}}\,\Phi$ that is conformally weightless under CT inversion. A "$\times$" or "$\bullet$" represents a regular singular point or an ordinary point of the ODE, respectively. The arrows connect radial points that are related by a CT inversion.