FZM Johannes-Kepler-Forschungszentrum Fakult at f ur Mathematik - PowerPoint PPT Presentation

Linear stability of the non-extreme Kerr black hole Felix Finster FZM Johannes-Kepler-Forschungszentrum Fakult¨ at f¨ ur Mathematik f¨ ur Mathematik, Regensburg Universit¨ at Regensburg Invited Talk GeLoMa2016, M´ alaga, 20 September 2016 Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to General Relativity In general relativity, gravity is described by the geometry of space-time ( M , g ) : Lorentzian manifold of signature (+ − − − ) tangent space T p M is vector space with indefinite inner product  g ( u , u ) > 0 : u is timelike  g ( u , u ) = 0 : u is lightlike or null g ( u , u ) < 0 : u is spacelike  this encodes the causal structure M light cone p spacelike hypersurface H ”worldline” of particle Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to General Relativity The gravitational field is described by the curvature of M ∇ : covariant derivative, Levi-Civita connection, ik X k � ∂ � ∂ i X j + Γ j ∇ i X = ∂ x j R i jkl : Riemann curvature tensor, ∂ ∇ i ∇ j X − ∇ j ∇ i X = R l ijk X k ∂ x l R ij = R l R = R i ilj : Ricci tensor, i : scalar curvature R jk − 1 Einstein’s equations: 2 R g jk = 8 π T jk T jk : enery-momentum tensor, describes matter “matter generates curvature” Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to General Relativity vice versa: “curvature affects the dynamics of matter” equations of motion, depend on type of matter: � classical point particles: geodesic equation � dust: perfect fluid � classical waves: wave equations � quantum mechanical matter: equations of wave mechanics (Dirac or Klein Gordon equation) � . . . . . . coupling Einstein equations with equations of motion yields system of nonlinear PDEs Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to General Relativity This Einstein-matter system describes exciting effects like the gravitational collapse of a star to a black hole but nonlinear system of PDEs, extremely difficult to analyze figure taken from Kip Thorne, “Black Holes and Time Warps” Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to General Relativity Possible methods and simplifications: � numerical simulations � small initial data (Christodoulou-Klainerman, . . . ) � analytical work in spherical symmetry and a massless scalar field (Christodoulou, . . . ) In this talk: consider late-time behavior of gravitational collapse, system has nearly settled down to a stationary black hole consider linear perturbations of a stationary black hole no symmetry assumptions for perturbation! Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to Black Holes Special Solutions to Einstein’s equations describe a “star”, no matter outside vacuum solutions: R jk = 0 Schwarzschild solution (1916) spherically symmetric, static, asymptotically flat polar coordinates ( r , ϑ, ϕ ) , time t � − 1 � � � 1 − 2 M 1 − 2 M dt 2 − ds 2 dr 2 = r r − r 2 � d ϑ 2 + sin 2 ϑ d ϕ 2 � here M is the mass of the star Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to Black Holes Schwarzschild solution (figures taken from Hawking/Ellis, “The Large-Scale Structure of Space-Time”) Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to Black Holes Schwarzschild solution in Finkelstein coordinates Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to Black Holes Kerr solution (1965) again asymptotically flat, but only axisymmetric, stationary Boyer-Lindquist coordinates ( t , r , ϑ, ϕ ) � dr 2 � ∆ U ( dt − a sin 2 ϑ d ϕ ) 2 − U ds 2 ∆ + d ϑ 2 = − sin 2 ϑ ( a dt − ( r 2 + a 2 ) d ϕ ) 2 U r 2 + a 2 cos 2 ϑ U ( r , ϑ ) = r 2 − 2 Mr + a 2 , ∆( r ) = M = mass, aM = angular momentum we always consider non-extreme case M 2 > a 2 Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to Black Holes The Kerr solution two horizons: the Cauchy horizon and the event horizon ergosphere: annular region outside the event horizon (figures taken from Hawking/Ellis, “The Large-Scale Structure of Space-Time”) Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to Black Holes view from north pole Felix Finster Linear stability of the non-extreme Kerr black hole

Introduction to Black Holes Black hole uniqueness theorem (Israel, Carter, Robinson, in 1970s) Assume the following: time orientability, topology R 2 × S 2 weak asymptotic simplicity, causality condition existence of event horizon with spherical topology axi-symmetry, pseudo-stationarity Every such solution of the vacuum Einstein equations is the non-extreme Kerr solution. Thus the Kerr solution is the mathematical model of a rotating black hole in equilibrium Felix Finster Linear stability of the non-extreme Kerr black hole

Linear Hyperbolic Equations in Kerr Questions of Physical Interest � gravitational wave detectors (LIGO, LISA) What signals can one expect? general interest in propagation of gravitational waves � Hawking radiation Do black holes emit Dirac particles? � superradiance Can one extract energy from rotating black holes using gravitational or electromagnetic waves? � problem of stability of black holes under electromagnetic or gravitational perturbations general problem: understand long-time dynamics Felix Finster Linear stability of the non-extreme Kerr black hole

Methods � Vector field method: Rodnianski, Dafermos, . . . � Strichartz estimates, local decay estimates: Tataru, Sterbenz, . . . � microlocal analysis, quasi-normal modes: Zworski, Vasy, Dyatlov, . . . � Analysis of the Maxwell equations: Tataru, Metcalfe, Tohaneanu, . . . Anderson and Blue Here focus on spectral methods in the Teukolsky formulation Felix Finster Linear stability of the non-extreme Kerr black hole

Linear Hyperbolic Equations in Kerr Newman-Penrose formalism: characterize by spin spin s massless massive 0 scalar waves Klein-Gordon field 1 neutrino field Dirac field 2 1 electromagnetic waves vector bosons 3 Rarita-Schwinger field 2 2 gravitational waves Felix Finster Linear stability of the non-extreme Kerr black hole

Structural Results massless equations of spin s : system of ( 2 s + 1 ) first order PDEs, write symbolically as   Ψ s . . D  = 0   .  Ψ − s Teukolsky (1972): After multiplying by first-order operator, the first and last components decouple,   T s 0 · · · 0 0 ∗ ∗ · · · ∗ ∗    . . ... . .  . . . . D D =   . . . .     ∗ ∗ · · · ∗ ∗   · · · 0 0 0 T − s Felix Finster Linear stability of the non-extreme Kerr black hole

Structural Results � gives one second order complex equation � other components obtained by differentiation (Teukolsky-Starobinsky identity) � combine equations for different s into one so-called Teukolsky master equation, s enters as a parameter this method does not work for massive equations All the equations (massive and massless) can be separated into ODEs: Carter (1968) scalar waves, Klein-Gordon field G¨ uven neutrino field Teukolsky (1972) massless eqns, general spin Chandrasekhar (1976) Dirac field Felix Finster Linear stability of the non-extreme Kerr black hole

Separation of Variables explain for the Teukolsky Equation � � 2 ∂ r ∆ ∂ ∂ ∂ r − 1 � ( r 2 + a 2 ) ∂ ∂ t + a ∂ ∂ϕ − ( r − M ) s ∆ − 4 s ( r + ia cos ϑ ) ∂ ∂ ∂ ∂ cos ϑ sin 2 ϑ ∂ t + ∂ cos ϑ � 2 � � 1 a sin 2 ϑ ∂ ∂ t + ∂ + ∂ϕ + i cos ϑ s φ = 0 sin 2 ϑ metric function ∆( r ) = r 2 − 2 Mr − a 2 spin parameter s = 0 , 1 2 , 1 , 3 2 , 2 , . . . Felix Finster Linear stability of the non-extreme Kerr black hole

Separation of Variables standard separation ansatz: φ ( t , r , ϑ, ϕ ) = e − i ω t e − ik ϕ Φ( ϑ, r ) yields � ∂ ∂ r ∆ ∂ ∂ r + 1 � 2 � ( r 2 + a 2 ) ω + ak + i ( r − M ) s ∆ ∂ ∂ ∂ cos ϑ sin 2 ϑ + 4 is ( r + ia cos ϑ ) ω + ∂ cos ϑ 1 � 2 � � a sin 2 ϑ ω + k − cos ϑ s − Φ = 0 sin 2 ϑ Separation of r and ϑ possible, Φ( r , ϑ ) = X ( r ) Y ( ϑ ) last separation does not correspond to space-time symmetry! Felix Finster Linear stability of the non-extreme Kerr black hole

A lot of work has been done on the separated equations (= ODEs for fixed separation constants ω , k , λ ) Mode Analysis � Regge & Wheeler (1957): metric perturbations of Schwarzschild mode stability: rule out complex ω � Starobinsky (1973) superradiance for scalar waves (see later) � Teukolsky & Press (1973): perturbations of Weyl tensor show mode stability in Kerr numerically superradiance for higher spin � Whiting (1989): mode stability in Kerr show mode stability analytically for general s many calculations and numerics in Chandrasekhar’s book Felix Finster Linear stability of the non-extreme Kerr black hole