Vol. 14, No. 8, 2021

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
This article is available for purchase or by subscription. See below.
The stable trapping phenomenon for black strings and black rings and its obstructions on the decay of linear waves

Gabriele Benomio

Vol. 14 (2021), No. 8, 2427–2496

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 gringΨ = 0, where gring is the metric of a fixed black ring solution to the five-dimensional Einstein vacuum equations. For a class 𝔤 of black ring metrics, we prove a logarithmic lower bound for the uniform energy decay rate on the black ring exterior (𝒟,gring), with gring 𝔤. 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 𝒟. 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.

PDF Access Denied

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

black rings, instability, stable trapping, wave equation
Mathematical Subject Classification 2010
Primary: 35Q76
Received: 1 February 2019
Revised: 8 June 2020
Accepted: 15 September 2020
Published: 19 December 2021
Gabriele Benomio
Imperial College London
Department of Mathematics
United Kingdom