Perturbations of Higher-dimensional Spacetimes

1. Perturbations • of Higher-dimensional Spacetimes Jan Novák

2. 1. Introduction 2. Stability of the Swarzschild solution 3. Higher-dimensional black holes 4. Gregory-Laflamme instability 5. Gauge-invariant variables and decoupling of perturbations 6. Near-horizon geometry 7. Summary

3. stable • stable • unstable Cauchy horizon Introduction

4. Stability of Schwarzschild Solution

7. 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

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

10. 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

11. 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, iffX(0) is constant.

12. 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

14. Near-horizon geometry • Consideranextremeblack hole… where𝜕/𝜕𝜙I , I=1,…,n are the rotational Killing vector fields of the black hole and kIare constants. The coordinates 𝜙I have period 2𝜋. • Thenear-horizon geometry ofanextremeblack hole istheKundtspacetime→ study gravitationalperturbationsusingourperturbedequation

15. Undercertaincircumstances, instabilityofnear-horizon geometry impliesinstabilityofthefullextremeblack hole !![Reall et al.2002-2010]

16. 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 Kundtbackground. • This decoupled equation can be used to study gravitational perturbations ofthesocalled near - horizon geometries of extreme blackholes: much easier than studying full black hole, isn’t it ?

17. Thank you for your attention