Perturbations of Higher-dimensional Spacetimes

Jan Novák

1. Introduction2. Stability of the Swarzschild solution3. Higher-dimensional black holes4. Gregory-Laflamme instability5. Gauge-invariant variables and decoupling

of perturbations6. Near-horizon geometry7. Summary

Introduction

Schwarzschild

Reissner-Nordström

Kerr

unstable Cauchy horizon

stable

stable

Stability of Schwarzschild Solution

PBs of STs that are SS and static ds2 = e2𝜈 dt2 - e2𝜓 (d - dt – q𝜑 𝜔 2dx2 – q3dx3) 2 – e2𝜇 (dx2)2 - e2 𝜒 (dx3)2

Linearization

Regge-Wheeler & Zerilli equation

(S) (d2 /dr2* + 𝜎2) Z = VZTASK: -d2 /dr2* + V positive and self-adjoint in L2(r*,dr*) Stability of system (S) … language of spectral theory

Physics of event horizons is far richer: ‘black Saturn’, S3 ,S1×S2, … which solutions are stable?

Schwarzschild-Tangherlini solution stable against linearized gravitational PBs for all d > 4 [2003 Ishibashi, Kodama]

Stability of Myers-Perry is an open problem

Higher-dimensional Black Holes

Photo: Vitor Cardoso

Note: See the author’s page, he compares this photo with G-L instability

Gregory-Laflamme Instability

Prototype for situations where the size of the horizon is much larger in some directions than in other

Ultraspinning BH → arbitrarily large angular momentum in d 6 ≧

GL instability ultraspinning black holes are ⇒unstable

Gauge-invariant variables

We use GHP formalism [Pravda et al. 2010]

Quantity X, X = X(0) + X(1), where X(0) is the value in the background ST and X(1) is the PB

Let X be a ST scalar → infinitesimal coordinate transformation with parameters 𝜉𝜇:X(1) + 𝜉.𝜕X(0) Hence X(1) is invariant under infinitesimal coordinate transformations, iff X(0) is constant.

In the case of gravitational PB’s → 𝛺ij, since these are higher-dimensional generalizationof the 4d quantity 𝛹0

Lemma: 𝛺(1)ij is a gauge invariant quantity,

iff l is a multiple WAND of the background ST

Decoupling of equations ?KUNDT: ∃ l geodesic, such that 𝜃=𝜎=w=0

Near-horizon geometry

Consider an extreme black hole…

where 𝜕/𝜕𝜙I , I=1,…,n are the rotational Killing vector fields of the black hole and kI are constants. The coordinates 𝜙I have period 2𝜋. The near-horizon geometry of an extreme black hole is

the Kundt spacetime → study gravitational perturbations using our perturbed equation

• Under certain circumstances, instability of near-horizon geometry implies instability of the full extreme black hole !! [Reall et al.2002-2010]

Summary Heuristic arguments suggest that Myers-Perry black holes might be unstable for sufficiently large angular momentum.

There exists a gauge-invariant quantity

for describing perturbations of algebraically special spacetimes, e.g. Myers-Perry black holes.

This quantity satisfies a decoupled equation only in a Kundt background.

This decoupled equation can be used to study gravitational perturbations of the so called near - horizon geometries of extreme black holes: much easier than studying full black hole, isn’t it ?

Thank you for your attention