Perturbations
of Higher-dimensional Spacetimes
Jan Novák
1.
2.
3.
4.
5.
Introduction
Stability of the Swarzschild solution
Higher-dimensional black holes
Gregory-Laflamme instability
Gauge-invariant variables and decoupling
of perturbations
6. Near-horizon geometry
7. Summary
Introduction
Schwarzschild
stable
Reissner-Nordström
unstable Cauchy horizon
Kerr
stable
Stability of Schwarzschild Solution
PBs of STs that are SS and static
ds2 = e2𝜈 dt2 - e2𝜓 (d𝜑 - 𝜔dt - q2dx2 - q3dx3) 2 - e2𝜇 (dx2)2 - e2𝜒 (dx3)2
Linearization
Regge-Wheeler & Zerilli equation
(S)
(d2 /dr2* + 𝜎2) Z = VZ
TASK:
-d2 /dr2* + V positive and self-adjoint in L2(r*,dr*)
Stability of system (S) … language of spectral theory
Higher-dimensional Black Holes
 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
Note: See the
author’s page, he
compares this photo
with G-L instability
Photo: Vitor Cardoso
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 generalization
of 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