On classical and quantum scattering for field equations on the (De - PowerPoint PPT Presentation

On classical and quantum scattering for field equations on the (De Sitter) Kerr metric Dietrich H¨ afner Institut Fourier, Universit´ e Grenoble Alpes Spectral theory and mathematical physics, Cergy Pontoise, June 23 2016

1.1 Black holes ( M , g ) lorentzian manifold, sign ( g ) = (+ , − , − , − ) . Einstein equations (1915) : R µν − 1 2 g µν R + Λ g µν = κ T µν . ◮ R µν : Ricci curvature, ◮ R : scalar curvature, ◮ g : metric, ◮ Λ : cosmological constant, ◮ T µν : energy momentum tensor, ◮ κ = 8 π G : Einstein constant. c 4 ◮ T µν = 0 : Einstein vacuum equations.

The Schwarzschild solution Schwarzschild (1916). M = R t × R r > 2 M × S 2 ω g = Ndt 2 − N − 1 dr 2 − r 2 d ω 2 N = ( 1 − 2 M r ) ( M : mass of the black hole). r = 0 : curvature singularity, r = 2 M : coordinate singularity. Regge-Wheeler coordinate : dx dr = N − 1 , x ± t = const . along spherically symmetric null geodescics. g = Ndvdw − r 2 d ω 2 . v = t + x , w = t − x , ′ = exp ( v ′ = − exp ( − w ′ = v ′ + w ′ = v ′ − w ′ ′ v 4 M ) , w 4 M ) , t , x 2 2 ′ ) 2 − ( dx ′ ) 2 ) − r 2 ( t ′ , x ′ ) d σ 2 . g = 32 M 2 exp ( − r 2 M )(( dt r t' v r = 2m, t = 8 r = 2m, t = + - r = 0 8 t = constant v = constant x' w = constant w r = constant r = 0 8 r = 2m, t = - r = 2m, t = + 8

The (De Sitter) Kerr metric De Sitter Kerr metric in Boyer-Lindquist coordinates M BH = R t × R r × S 2 ω , with spacetime metric ∆ r − a 2 sin 2 θ ∆ θ dt 2 + 2 a sin 2 θ (( r 2 + a 2 ) 2 ∆ θ − a 2 sin 2 θ ∆ r ) g = dtd ϕ λ 2 ρ 2 λ 2 ρ 2 ∆ θ d θ 2 − sin 2 θσ 2 ∆ r dr 2 − ρ 2 ρ 2 d ϕ 2 , − λ 2 ρ 2 � 1 − Λ � r 2 + a 2 cos 2 θ, ( r 2 + a 2 ) − 2 Mr , ρ 2 3 r 2 = ∆ r = 1 + 1 3 Λ a 2 cos 2 θ, σ 2 = ( r 2 + a 2 ) 2 ∆ θ − a 2 ∆ r sin 2 θ, λ = 1 + 1 3 Λ a 2 . ∆ θ = Λ ≥ 0 : cosmological constant ( Λ = 0 : Kerr), M > 0 : masse, a : angular momentum per unit masse ( | a | < M ). ◮ ρ 2 = 0 is a curvature singularity, ∆ r = 0 are coordinate singularities. ∆ r > 0 on some open interval r − < r < r + . r = r − : black hole horizon, r = r + cosmological horizon. ◮ ∂ ϕ and ∂ t are Killing. There exist r 1 ( θ ) , r 2 ( θ ) s. t. ∂ t is ◮ timelike on { ( t , r , θ, ϕ ) : r 1 ( θ ) < r < r 2 ( θ ) } , ◮ spacelike on { ( t , r , θ, ϕ ) : r − < r < r 1 ( θ ) }∪{ ( t , r , θ, ϕ : r 2 ( θ ) < r < r + } =: E − ∪E + . The regions E − , E + are called ergospheres.

The Penrose diagram ( Λ = 0) ◮ Kerr-star coordinates : dr = r 2 + a 2 , d Λ( r ) t ∗ = t + x , r , θ, ϕ ∗ = ϕ + Λ( r ) , dx = a ∆ . ∆ dr . t ∗ = ˙ Along incoming principal null geodesics : ˙ ϕ ∗ = 0 , ˙ θ = ˙ r = − 1 . ◮ Form of the metric in Kerr-star coordinates : g = g tt dt ∗ 2 + 2 g t ϕ dt ∗ d ϕ ∗ + g ϕϕ d ϕ ∗ 2 + g θθ d θ 2 − 2 dt ∗ dr + 2 a sin 2 d ϕ ∗ dr . ◮ Future event horizon : H + := R t ∗ × { r = r − } × S 2 θ,ϕ ∗ . ◮ The construction of the past event horizon H − is based on outgoing principal null geodesics (star-Kerr coordinates). Similar constructions for future and past null infinities I + and I − using the 1 conformally rescaled metric ˆ g = r 2 g .

1.2 The Dirac and Klein-Gordon equation on the (De Sitter) Kerr metric The Klein-Gordon equation We now consider the unitary transform σ 2 L 2 ( M ; L 2 ( M ; drd ω ) ∆ r ∆ θ drd ω ) → U : √ σ ψ �→ ∆ r ∆ θ ψ If ψ fulfills ( ✷ g + m 2 ) ψ = 0, then u = U ψ fulfills ( ∂ 2 (1) t − 2 ik ∂ t + h ) u = 0 . with a (∆ r − ( r 2 + a 2 )∆ θ ) k = D ϕ , σ 2 √ ∆ r ∆ θ √ ∆ r ∆ θ − (∆ r − a 2 sin 2 θ ∆ θ ) ∂ 2 h = ϕ − ∂ r ∆ r ∂ r sin 2 θσ 2 λσ λσ √ ∆ r ∆ θ √ ∆ r ∆ θ + ρ 2 ∆ r ∆ θ m 2 . − λ sin θσ ∂ θ sin θ ∆ θ ∂ θ λσ λ 2 σ 2 h is not positive inside the ergospheres. This entails that the natural conserved quantity E ( u ) = � ∂ t u � 2 + ( hu | u ) ˜ is not positive → superradiance.

Dirac equation The situation is easier for the Dirac equation ! Weyl equation : ∇ A A ′ φ A = 0 . Conserved current on general globally hyperbolic spacetimes 1 � A ′ V a = φ A φ V a T a d σ Σ t = const . , C ( t ) = √ 2 Σ t T a : normal to Σ t , M = � t Σ t foliation of the spacetime. ◮ Newman-Penrose tetrad l a , n a , m a , m a : l a l a = n a n a = m a m a = l a m a = n a m a = 0 . ◮ Normalization l a n a = 1 , m a m a = − 1 ◮ l a , n a : Scattering directions. ◮ Spin frame o A o A ′ = l a , ι A ι A ′ = n a , o A ι A ′ = m a ι A o A ′ = m a , o A ι A = 1 ◮ Components in the spin frame : φ 0 = φ A o A , φ 1 = φ A ι A ◮ Weyl equation : � n a ∂ a φ 0 − m a ∂ a φ 1 + ( µ − γ ) φ 0 + ( τ − β ) φ 1 = 0 , l a ∂ a φ 1 − m a ∂ a φ 0 + ( α − π ) φ 0 + ( ǫ − ˜ ρ ) φ 1 = 0 .

Some aspects of the study of field equations on the (De Sitter) Kerr metric ◮ Superradiance. Exists for entire spin equations (Klein-Gordon, Maxwell), no superradiance for half integer spin equations (Dirac, Rarita Schwinger). ◮ Local geometry. Trapping. Toy model for Schwarzschild ( ∂ 2 t + P ) u = 0 , P = − ∂ 2 x − V ∆ S 2 . V has a non degenerate maximum at r = 3 M (photon sphere). h − 2 = l ( l + 1 ) where l ( l + 1 ) are the eigenvalues of − ∆ S 2 is a good semiclassical parameter. Similar trapping in (De Sitter) Kerr. Normally hyperbolic trapping. ◮ Geometry at infinity. Schwarzschild. Reinterpretation of P as a perturbation of the Laplacian on a riemannian manifold with two ends : ◮ Λ = 0 : one asymptotically euclidean end (corresponding to infinity) and one asymptotically hyperbolic end (corresponding to the black hole horizon). ◮ Λ > 0 : two asymptotically hyperbolic ends. Consequence : the study of the low frequency behavior is easier in the De Sitter case (case of positive cosmological constant).

Some aspects of the study of field equations on the (De Sitter) Kerr metric ◮ Superradiance. Exists for entire spin equations (Klein-Gordon, Maxwell), no superradiance for half integer spin equations (Dirac, Rarita Schwinger). ◮ Local geometry. Trapping. Toy model for Schwarzschild ( ∂ 2 t + P ) u = 0 , P = − ∂ 2 x − V ∆ S 2 . V has a non degenerate maximum at r = 3 M (photon sphere). h − 2 = l ( l + 1 ) where l ( l + 1 ) are the eigenvalues of − ∆ S 2 is a good semiclassical parameter. Similar trapping in (De Sitter) Kerr. Normally hyperbolic trapping. ◮ Geometry at infinity. Schwarzschild. Reinterpretation of P as a perturbation of the Laplacian on a riemannian manifold with two ends : ◮ Λ = 0 : one asymptotically euclidean end (corresponding to infinity) and one asymptotically hyperbolic end (corresponding to the black hole horizon). ◮ Λ > 0 : two asymptotically hyperbolic ends. Consequence : the study of the low frequency behavior is easier in the De Sitter case (case of positive cosmological constant).

Some aspects of the study of field equations on the (De Sitter) Kerr metric ◮ Superradiance. Exists for entire spin equations (Klein-Gordon, Maxwell), no superradiance for half integer spin equations (Dirac, Rarita Schwinger). ◮ Local geometry. Trapping. Toy model for Schwarzschild ( ∂ 2 t + P ) u = 0 , P = − ∂ 2 x − V ∆ S 2 . V has a non degenerate maximum at r = 3 M (photon sphere). h − 2 = l ( l + 1 ) where l ( l + 1 ) are the eigenvalues of − ∆ S 2 is a good semiclassical parameter. Similar trapping in (De Sitter) Kerr. Normally hyperbolic trapping. ◮ Geometry at infinity. Schwarzschild. Reinterpretation of P as a perturbation of the Laplacian on a riemannian manifold with two ends : ◮ Λ = 0 : one asymptotically euclidean end (corresponding to infinity) and one asymptotically hyperbolic end (corresponding to the black hole horizon). ◮ Λ > 0 : two asymptotically hyperbolic ends. Consequence : the study of the low frequency behavior is easier in the De Sitter case (case of positive cosmological constant).

2 Asymptotic completeness for the Klein-Gordon equation on the De Sitter Kerr metric (with C. G´ erard and V. Georgescu) 2.1 : 3+1 decomposition, energies, Killing fields Let v = e − ikt u . Then u is solution of (1) if and only if v is solution of h 0 = h + k 2 ≥ 0 . ( ∂ 2 h ( t ) = e − ikt h 0 e ikt , t + h ( t )) v = 0 , Natural energy : � ∂ t v � 2 + ( h ( t ) v | v ) . Rewriting for u : E ( u ) = � ( ∂ t − ik ) u � 2 + ( h 0 u | u ) . ˙ This energy is positive, but may grow in time → superradiance. Remark k = Ω D ϕ and Ω has finite limits Ω − / + when r → r ∓ . These limits are called angular velocities of the horizons. The Killing fields ∂ t − Ω − / + ∂ ϕ on the De Sitter Kerr metric are timelike close to the black hole (-) resp. cosmological (+) horizon. Working with these Killing fields rather than with ∂ t leads to the conserved energies : E − / + ( u ) = � ( ∂ t − Ω − / + ∂ ϕ ) u � 2 + ( h 0 − ( k − Ω − / + D ϕ ) 2 u | u ) . ˜ Note that in the limit k → Ω − / + D ϕ the expressions of ˙ E ( u ) and ˜ E − / + ( u ) coincide.