Perturbations of higher-dimensional spacetimes

TL;DR

This paper investigates gravitational perturbations in higher-dimensional spacetimes, identifying conditions under which decoupled equations similar to 4D Newman-Penrose scalars exist, especially in near-horizon geometries of extreme black holes.

Contribution

It generalizes the concept of gauge-invariant perturbation quantities to higher dimensions and establishes criteria for decoupling equations in these spacetimes.

Findings

Decoupling occurs if the spacetime admits a null geodesic congruence with zero expansion, rotation, and shear.

Such decoupling is not present in black hole spacetimes but occurs in near-horizon geometries of extreme black holes.

Higher-dimensional generalizations of Newman-Penrose scalars satisfy decoupled equations under specific geometric conditions.

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.