Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations Electromagnetic-gravitational perturbations of Kerr-Newman black hole Elena Giorgi Gravity Initiative, Princeton University October 5th, 2020 ICERM Workshop “Mathematical and computational approaches for solving the source-free Einstein field equations” 1/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations Outline 1 Electromagnetic-gravitational perturbations of Kerr-Newman in classical black hole perturbation theory: Obstacles in finding separated equations “Apparent indissolubility of coupling between spin-1 and spin-2 fields” 2 A new approach: the physical space analysis of the electromagnetic-gravitational perturbations of Kerr-Newman The Teukolsky equations The Regge-Wheeler equations 2/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations The Kerr-Newman black hole � a 2 + Q 2 ≤ M given by The Kerr-Newman metric g M , a , Q for ∆ dr 2 + ρ 2 d θ 2 + sin 2 θ � 2 + ρ 2 g M , a , Q = − ∆ dt − a sin 2 θ d φ adt − ( r 2 + a 2 ) d φ � 2 � � ρ 2 ρ 2 where r 2 − 2 Mr + a 2 + Q 2 , ∆ = r 2 + a 2 cos 2 θ, ρ 2 = is a solution to the Einstein-Maxwell equation : Ric µν ( g ) = 2 F µλ F λν − 1 2 g µν F αβ F αβ where F is an anti-symmetric 2-tensor, called the electromagnetic tensor, satisfying the Maxwell equations: ∇ α F αβ = 0 ∇ [ α F βγ ] = 0 , 3/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations The Kerr-Newman metric g M , a , Q represents a stationary, rotating and charged black hole with mass M , spin a and charge Q . Special cases of the Kerr-Newman metric are for Q = 0: the Kerr spacetime g M , a , for | a | ≤ M for a = 0: the Reissner–Nordstr¨ om spacetime g M , Q for | Q | ≤ M for both a , Q = 0: the Schwarzschild spacetime g M The Kerr-Newman metric is the most general explicit black hole solution. 4/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations The Kerr-Newman metric g M , a , Q represents a stationary, rotating and charged black hole with mass M , spin a and charge Q . Special cases of the Kerr-Newman metric are for Q = 0: the Kerr spacetime g M , a , for | a | ≤ M for a = 0: the Reissner–Nordstr¨ om spacetime g M , Q for | Q | ≤ M for both a , Q = 0: the Schwarzschild spacetime g M The Kerr-Newman metric is the most general explicit black hole solution. Black hole perturbation theory is the analysis of perturbations of known solutions of the Einstein equation, like Schwarzschild, Kerr, Reissner-Nordstr¨ om or Kerr-Newman. 4/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations Black hole perturbation theory 5/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations 6/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations Metric perturbations of Schwarzschild Consider metric perturbations of the form d φ − ω dt − q 2 dx 2 − q 3 dx 3 � 2 − e 2 µ 2 ( dx 2 ) 2 − e 2 µ 3 ( dx 3 ) 2 e 2 ν dt 2 − e 2 ψ � = g of the Schwarzschild metric, with e 2 ν = e − 2 µ 2 = 1 − 2 M e ψ = r sin θ, e µ 3 = r , r , ω = q 2 = q 3 = 0 . The axial perturbations (i.e. those modifying ω , q 2 , q 3 ) are governed by the so-called Regge-Wheeler equation : 4 � 1 − 2 M � � g M ψ = ψ r 2 r 7/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations Metric perturbations of Schwarzschild Consider metric perturbations of the form d φ − ω dt − q 2 dx 2 − q 3 dx 3 � 2 − e 2 µ 2 ( dx 2 ) 2 − e 2 µ 3 ( dx 3 ) 2 e 2 ν dt 2 − e 2 ψ � = g of the Schwarzschild metric, with e 2 ν = e − 2 µ 2 = 1 − 2 M e ψ = r sin θ, e µ 3 = r , r , ω = q 2 = q 3 = 0 . The axial perturbations (i.e. those modifying ω , q 2 , q 3 ) are governed by the so-called Regge-Wheeler equation : 4 � 1 − 2 M � � g M ψ = ψ r 2 r In general, we call Regge-Wheeler equation an equation of the form � g ψ − V ψ = 0 , for a positive real potential V 7/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations Curvature perturbations of Kerr In order to obtain a decoupled equation for perturbations of Kerr, one needs to use the Newman-Penrose formalism and decompose the curvature into the Weyl scalars and the spin coefficients with respect to a null basis ( l , n , m , m ) with corresponding derivatives D , ∆ , δ, δ. For gravitational perturbations, the relevant Weyl scalars are Ψ 0 = − W ( l , m , l , m ) , Ψ 4 = − W ( n , m , n , m ) and for electromagnetic perturbations, the relevant electromagnetic scalars are φ 0 = F ( l , m ) , φ 2 = F ( m , n ) . 8/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations The extreme null curvature components of spin ± 2 ψ [+2] = Ψ 0 , ψ [ − 2] = ( r − ia cos θ ) 4 Ψ 4 and the extreme null curvature components of spin ± 1 ψ [+1] = φ 0 , ψ [ − 1] = ( r − ia cos θ ) 2 φ 2 satisfy the Teukolsky equation of spin s : � a ( r − M ) � g M , a ψ [ s ] + 2 s ρ 2 ( r − M ) ∂ r ψ [ s ] + 2 s + i cos θ � T [ s ] ( ψ [ s ] ) ∂ φ ψ [ s ] := sin 2 θ ρ 2 ∆ � M ( r 2 − a 2 ) + 2 s � ∂ t ψ [ s ] + 1 ρ 2 ( s − s 2 cot 2 θ ) ψ [ s ] = 0 − r − ia cos θ ρ 2 ∆ 9/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations The extreme null curvature components of spin ± 2 ψ [+2] = Ψ 0 , ψ [ − 2] = ( r − ia cos θ ) 4 Ψ 4 and the extreme null curvature components of spin ± 1 ψ [+1] = φ 0 , ψ [ − 1] = ( r − ia cos θ ) 2 φ 2 satisfy the Teukolsky equation of spin s : � a ( r − M ) � g M , a ψ [ s ] + 2 s ρ 2 ( r − M ) ∂ r ψ [ s ] + 2 s + i cos θ � T [ s ] ( ψ [ s ] ) ∂ φ ψ [ s ] := sin 2 θ ρ 2 ∆ � M ( r 2 − a 2 ) + 2 s � ∂ t ψ [ s ] + 1 ρ 2 ( s − s 2 cot 2 θ ) ψ [ s ] = 0 − r − ia cos θ ρ 2 ∆ In general, we call Teukolsky equation an equation of the form � g ψ = c 1 ( r , θ ) ∂ r ψ + c 2 ( r , θ ) ∂ φ ψ + c 3 ( r , θ ) ∂ t ψ + V ( r , θ ) ψ with c 1 , c 2 , c 3 complex functions, and V a real function. 9/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations The classical mode analysis The main advantage of the Regge-Wheeler and Teukolsky equations is that they can be separated in modes. In the mode stability analysis, one does not study general solutions of � g ψ = 0 (1) but rather individual modes. A mode is a solution of (1) of the form e − i ω t e im φ R ( r ) S ( θ ) . ψ ( t , r , θ, φ ) = (2) From (1), one derives an angular ODE for S : � m 2 1 � sin 2 θ − a 2 ω 2 cos 2 θ sin θ ∂ θ (sin θ∂ θ S ) − S + λ S = 0 The solutions are eigenfunctions { S ω m ℓ } ∞ ℓ = | m | and real eigenvalues { λ ω m ℓ } ∞ ℓ = | m | . a radial ODE for R : V ( ω, m , ℓ, r ) − λ ω m ℓ − a 2 ω 2 � � ∂ r (∆ ∂ r ) R + R = 0 10/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations Mode stability results The mode stability is the statement that there are no mode solutions with finite energy at t = 0 and Im ( ω ) > 0 (i.e. no exponentially growing). Mode stability of Schwarzschild, Reissner-Nordstr¨ om, and Kerr was obtained in the 80s in the metric perturbations approach (Regge-Wheeler, Vishveshwara, Zerilli, Moncrief) and the Newman-Penrose formalism (Barden-Press, Teukolsky, Chandrasekhar, Whiting). 11/31 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole
Recommend
More recommend
