The geometry of solutions to the higher-dimensional Einstein vacuum
equations presents aspects that are absent in four dimensions, one
of the most remarkable being the existence of stably trapped null
geodesics in the exterior of asymptotically flat black holes. This
paper investigates the stable trapping phenomenon for two families
of higher-dimensional black holes, namely black strings and black
rings, and how this trapping structure is responsible for the slow
decay of linear waves on their exterior. More precisely, we study decay
properties for the energy of solutions to the scalar, linear wave equation
, where
is the metric
of a fixed black ring solution to the five-dimensional Einstein vacuum equations. For a class
of black ring
metrics, we prove a logarithmic lower bound for the uniform energy decay rate on the black
ring exterior
,
with
.
The proof generalizes the perturbation argument and quasimode construction of
Holzegel and Smulevici (Anal. PDE 7:5 (2014), 1057–1090) to the case of a nonseparable
wave equation and crucially relies on the presence of stably trapped null geodesics
on .
As a by-product, the same logarithmic lower bound can be established for
any five-dimensional black string.
Our result is the first mathematically rigorous statement supporting the
expectation that black rings are dynamically unstable to generic perturbations. In
particular, we conjecture a new
nonlinear instability for five-dimensional black
strings and thin black rings which is already present at the level of scalar
perturbations and clearly differs from the mechanism driven by the well-known
Gregory–Laflamme instability.
Keywords
black rings, instability, stable trapping, wave equation
