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Perturbations of higher-dimensional spacetimes

Mark Durkee, Harvey S. Reall

TL;DR

The paper extends the Teukolsky/NP framework to higher-dimensional Einstein spacetimes using a GHP-based formalism, identifying gauge-invariant perturbation quantities Ω_{ij}^{(1)} that capture the gravitational field's physical degrees of freedom. It shows a decoupled, second-order equation for these perturbations exists if and only if the background is Kundt (null geodesic, non-expanding, non-twisting, non-shearing), with a parallel result for Maxwell fields. This Kundt-only decoupling contrasts with the four-dimensional case and implies that decoupled dynamics are not generally available for Myers-Perry black holes, though near-horizon geometries of extreme black holes are doubly Kundt and amenable to such analysis. The work provides a route to instability predictions in near-horizon regimes and lays groundwork for reconstructing full perturbations from the gauge-invariant variables.

Abstract

We discuss linearized gravitational perturbations of higher dimensional spacetimes. For algebraically special spacetimes (e.g. Myers-Perry black holes), we show that there exist local gauge invariant quantities linear in the metric perturbation. These are the higher dimensional generalizations of the 4d Newman-Penrose scalars that (in an algebraically special vacuum spacetime) satisfy decoupled equations of motion. We show that decoupling occurs in more than four dimensions if, and only if, the spacetime admits a null geodesic congruence with vanishing expansion, rotation and shear. Decoupling of electromagnetic perturbations occurs under the same conditions. Although these conditions are not satisfied in black hole spacetimes, they are satisfied in the near-horizon geometry of an extreme black hole.

Perturbations of higher-dimensional spacetimes

TL;DR

The paper extends the Teukolsky/NP framework to higher-dimensional Einstein spacetimes using a GHP-based formalism, identifying gauge-invariant perturbation quantities Ω_{ij}^{(1)} that capture the gravitational field's physical degrees of freedom. It shows a decoupled, second-order equation for these perturbations exists if and only if the background is Kundt (null geodesic, non-expanding, non-twisting, non-shearing), with a parallel result for Maxwell fields. This Kundt-only decoupling contrasts with the four-dimensional case and implies that decoupled dynamics are not generally available for Myers-Perry black holes, though near-horizon geometries of extreme black holes are doubly Kundt and amenable to such analysis. The work provides a route to instability predictions in near-horizon regimes and lays groundwork for reconstructing full perturbations from the gauge-invariant variables.

Abstract

We discuss linearized gravitational perturbations of higher dimensional spacetimes. For algebraically special spacetimes (e.g. Myers-Perry black holes), we show that there exist local gauge invariant quantities linear in the metric perturbation. These are the higher dimensional generalizations of the 4d Newman-Penrose scalars that (in an algebraically special vacuum spacetime) satisfy decoupled equations of motion. We show that decoupling occurs in more than four dimensions if, and only if, the spacetime admits a null geodesic congruence with vanishing expansion, rotation and shear. Decoupling of electromagnetic perturbations occurs under the same conditions. Although these conditions are not satisfied in black hole spacetimes, they are satisfied in the near-horizon geometry of an extreme black hole.

Paper Structure

This paper contains 16 sections, 1 theorem, 75 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Gauge-invariant variables
  4. Decoupling of electromagnetic perturbations
  5. Main Result
  6. Derivation of main result
  7. The Schwarzschild solution
  8. Decoupling of gravitational perturbations
  9. Introduction and main result
  10. Derivation of main result
  11. Comment on the expanding case
  12. Discussion
  13. Useful GHP equations
  14. Newman-Penrose equations
  15. Bianchi equations
  16. ...and 1 more sections

Key Result

Lemma 1

$\Omega^{(1)}_{ij}$ is a gauge invariant quantity if and only if ${\ell}$ is a multiple WAND of the background spacetime (or equivalently, if and only if $\Psi^{(0)}_{ijk} = 0 = \Omega^{(0)}_{ij}$).

Theorems & Definitions (1)

  • Lemma 1