Perturbations of higher-dimensional spacetimes
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Transcript Perturbations of higher-dimensional spacetimes
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 = e2π dt2 - e2π (dπ - πdt - q2dx2 - q3dx3) 2 - e2π (dx2)2 - e2π (dx3)2
Linearization
Regge-Wheeler & Zerilli equation
(S)
(d2 /dr2* + π2) Z = VZ
TASK:
-d2 /dr2* + V positive and self-adjoint in L2(r*,dr*)
Stability of system (S) β¦ language of spectral theory
Higher-dimensional Black Holes
ο§ Physics of event horizons is far richer:
βblack Saturnβ, S3 ,S1×S2, β¦ which solutions are
stable?
ο§ Schwarzschild-Tangherlini solution stable
against linearized gravitational PBs for all d > 4
[2003 Ishibashi, Kodama]
ο§ Stability of Myers-Perry is an open problem
Note: See the
authorβs page, he
compares this photo
with G-L instability
Photo: Vitor Cardoso
Gregory-Laflamme Instability
ο§ Prototype for situations where the size of the
horizon is much larger in some directions than
in other
ο§ Ultraspinning BH β arbitrarily large angular
momentum in dβ§6
ο§ GL instability β ultraspinning black holes are
unstable
Gauge-invariant variables
ο§ We use GHP formalism [Pravda et al. 2010]
ο§ Quantity X, X = X(0) + X(1), where X(0) is the
value in the background ST and X(1) is the PB
ο§ Let X be a ST scalar β infinitesimal coordinate
transformation with parameters ππ:
X(1) + π.πX(0)
Hence X(1) is invariant under infinitesimal
coordinate transformations, iff X(0) is
constant.
ο§ In the case of gravitational PBβs β πΊij, since
these are higher-dimensional generalization
of the 4d quantity πΉ0
ο§ Lemma: πΊ(1)ij is a gauge invariant quantity,
iff l is a multiple WAND of the background ST
ο§ Decoupling of equations ?
KUNDT: β l geodesic, such that
π=π=w=0
Near-horizon geometry
ο§ Consider an extreme black holeβ¦
where π/ππI , I=1,β¦,n are the rotational Killing
vector fields of the black hole and kI are
constants. The coordinates πI have period 2π.
ο§ The near-horizon geometry of an extreme black
hole is the Kundt spacetime β study gravitational
perturbations using our perturbed equation
β’ 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.
ο§ There exists a gauge-invariant quantity
for describing perturbations of algebraically
special spacetimes, e.g. Myers-Perry black
holes.
ο§ This quantity satisfies a decoupled equation
only in a Kundt background.
ο§ 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, isnβt it ?
Thank you for your attention