Electromagnetic-gravitational perturbations of Kerr-Newman black - - PowerPoint PPT Presentation

1/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Electromagnetic-gravitational perturbations

• f 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”

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

2/31 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

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

3/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Kerr-Newman black hole

The Kerr-Newman metric gM,a,Q for

• a2 + Q2 ≤ M given by

gM,a,Q = − ∆ ρ2

• dt − a sin2 θdφ

2 + ρ2 ∆ dr2 + ρ2dθ2 + sin2 θ ρ2

• adt − (r2 + a2)dφ

2 where ∆ = r2 − 2Mr + a2 + Q2, ρ2 = r2 + a2 cos2 θ, is a solution to the Einstein-Maxwell equation: Ricµν(g) = 2FµλF λν − 1 2 gµνF αβFαβ where F is an anti-symmetric 2-tensor, called the electromagnetic tensor, satisfying the Maxwell equations: ∇[αFβγ] = 0, ∇αFαβ = 0

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

4/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Kerr-Newman metric gM,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 gM,a, for |a| ≤ M for a = 0: the Reissner–Nordstr¨

• m spacetime gM,Q for |Q| ≤ M

for both a, Q = 0: the Schwarzschild spacetime gM The Kerr-Newman metric is the most general explicit black hole solution.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

4/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Kerr-Newman metric gM,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 gM,a, for |a| ≤ M for a = 0: the Reissner–Nordstr¨

• m spacetime gM,Q for |Q| ≤ M

for both a, Q = 0: the Schwarzschild spacetime gM 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¨

• m or Kerr-Newman.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

5/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Black hole perturbation theory

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

6/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

7/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Metric perturbations of Schwarzschild

Consider metric perturbations of the form g = e2νdt2 − e2ψ dφ − ωdt − q2dx2 − q3dx32 − e2µ2(dx2)2 − e2µ3(dx3)2

• f the Schwarzschild metric, with

e2ν = e−2µ2 = 1 − 2M r , eµ3 = r, eψ = r sin θ, ω = q2 = q3 = 0. The axial perturbations (i.e. those modifying ω, q2, q3) are governed by the so-called Regge-Wheeler equation: gM ψ = 4 r2

• 1 − 2M

r

• ψ

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

7/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Metric perturbations of Schwarzschild

Consider metric perturbations of the form g = e2νdt2 − e2ψ dφ − ωdt − q2dx2 − q3dx32 − e2µ2(dx2)2 − e2µ3(dx3)2

• f the Schwarzschild metric, with

e2ν = e−2µ2 = 1 − 2M r , eµ3 = r, eψ = r sin θ, ω = q2 = q3 = 0. The axial perturbations (i.e. those modifying ω, q2, q3) are governed by the so-called Regge-Wheeler equation: gM ψ = 4 r2

• 1 − 2M

r

• ψ

In general, we call Regge-Wheeler equation an equation of the form gψ − V ψ = 0, for a positive real potential V

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

8/31 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 Ψ0 = −W (l, m, l, m), Ψ4 = −W (n, m, n, m) and for electromagnetic perturbations, the relevant electromagnetic scalars are φ0 = F(l, m), φ2 = F(m, n).

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

9/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The extreme null curvature components of spin ±2 ψ[+2] = Ψ0, ψ[−2] = (r − ia cos θ)4Ψ4 and the extreme null curvature components of spin ±1 ψ[+1] = φ0, ψ[−1] = (r − ia cos θ)2φ2 satisfy the Teukolsky equation of spin s: T [s](ψ[s]) := gM,aψ[s] + 2s ρ2 (r − M)∂rψ[s] + 2s ρ2 a(r − M) ∆ + i cos θ sin2 θ

• ∂φψ[s]

+ 2s ρ2 M(r2 − a2) ∆ − r − ia cos θ

• ∂tψ[s] + 1

ρ2 (s − s2 cot2 θ)ψ[s] = 0

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

9/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The extreme null curvature components of spin ±2 ψ[+2] = Ψ0, ψ[−2] = (r − ia cos θ)4Ψ4 and the extreme null curvature components of spin ±1 ψ[+1] = φ0, ψ[−1] = (r − ia cos θ)2φ2 satisfy the Teukolsky equation of spin s: T [s](ψ[s]) := gM,aψ[s] + 2s ρ2 (r − M)∂rψ[s] + 2s ρ2 a(r − M) ∆ + i cos θ sin2 θ

• ∂φψ[s]

+ 2s ρ2 M(r2 − a2) ∆ − r − ia cos θ

• ∂tψ[s] + 1

ρ2 (s − s2 cot2 θ)ψ[s] = 0 In general, we call Teukolsky equation an equation of the form gψ = c1(r, θ)∂rψ + c2(r, θ)∂φψ + c3(r, θ)∂tψ + V (r, θ)ψ with c1, c2, c3 complex functions, and V a real function.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

10/31 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 gψ = 0 (1) but rather individual modes. A mode is a solution of (1) of the form ψ(t, r, θ, φ) = e−iωteimφR(r)S(θ). (2) From (1), one derives an angular ODE for S: 1 sin θ ∂θ (sin θ∂θS) − m2 sin2 θ − a2ω2 cos2 θ

• S + λS = 0

The solutions are eigenfunctions {Sωmℓ}∞

ℓ=|m| and real eigenvalues {λωmℓ}∞ ℓ=|m|.

a radial ODE for R: ∂r(∆∂r)R +

• V (ω, m, ℓ, r) − λωmℓ − a2ω2

R = 0

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

11/31 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¨

• m, 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).

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

11/31 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¨

• m, 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). Mathematician’s comment The statement of mode stability is not conclusive for general non-mode solutions because statements at the level of individual modes do not imply boundedness for the superposition of infinitely many modes. In particular, it is still consistent with general perturbations ψ with finite initial energy which grow unboundedly in time.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

12/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Stability of Kerr-Newman spacetime?

Figure: The mathematical theory of black holes by Chandrasekhar, pages 580-581

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

13/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Figure: The mathematical theory of black holes by Chandrasekhar, page 581

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

14/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Figure: The mathematical theory of black holes by Chandrasekhar, page 582

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

15/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

15/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

What goes wrong in Kerr-Newman?

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

16/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Electromagnetic-gravitational radiation: spin-1 and spin-2

The Einstein-Maxwell equation governs the interaction between the gravitational and the electromagnetic radiation emitted from the black hole: Ricµν(g)

gravitational radiation

= 2FµλF λν − 1 2 gµνF αβFαβ

electromagnetic radiation

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

16/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Electromagnetic-gravitational radiation: spin-1 and spin-2

The Einstein-Maxwell equation governs the interaction between the gravitational and the electromagnetic radiation emitted from the black hole: Ricµν(g)

gravitational radiation

= 2FµλF λν − 1 2 gµνF αβFαβ

electromagnetic radiation

The gravitational radiation is a spin-2 field. The electromagnetic radiation is a spin-1 field.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

16/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Electromagnetic-gravitational radiation: spin-1 and spin-2

The Einstein-Maxwell equation governs the interaction between the gravitational and the electromagnetic radiation emitted from the black hole: Ricµν(g)

gravitational radiation

= 2FµλF λν − 1 2 gµνF αβFαβ

electromagnetic radiation

The gravitational radiation is a spin-2 field. The electromagnetic radiation is a spin-1 field. The gravitational and electromagnetic fields, when taken independently, satisfy the Teukolsky equation of spin s, for s = ±2, ±1 respectively: T [s](ψ[s]) := gψ[s] + 2s ρ2 (r − M)∂rψ[s] + 2s ρ2 a(r − M) ∆ + i cos θ sin2 θ

• ∂φψ[s]

+ 2s ρ2 M(r2 − a2) ∆ − r − ia cos θ

• ∂tψ[s] + 1

ρ2 (s − s2 cot2 θ)ψ[s] = 0

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

17/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Interaction of spin-2 and spin-1 modes

In electromagnetic-gravitational perturbations of a black hole, there is going to be some interaction between the spin-2 gravitational field ψ[2] and the spin-1 electromagnetic field ψ[1]. Instead of having two independent Teukolsky equations: T [1](ψ[1]) = for electromagnetic perturbations T [2](ψ[2]) = for gravitational perturbations there will be a system of coupled Teukolsky equations: T [1](ψ[1]) = δ(ψ[2]) for electromagnetic-gravitational T [2](ψ[2]) = δ(ψ[1]) perturbations where δ is the raising-spin operator and δ is the lowering-spin operator.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

18/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The mode decomposition of the Teukolsky variables ψ[s](t, r, θ, φ) = e−iωteimφR[s](r)S[s]

mℓ(aω, cos θ)

involves the spin s-weighted spheroidal harmonics S[s]

mℓ(aω, cos θ) which are

eigenfunctions of / ∆[s] = 1 sin θ ∂θ(sin θ∂θ) − m2 + 2ms cos θ + s2 sin2 θ + a2ω2 cos2 θ − 2aωs cos θ For a = 0, they reduce to the spherical harmonics S[s]

mℓ(0, cos θ) = Y [s] mℓ(cos θ).

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

18/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The mode decomposition of the Teukolsky variables ψ[s](t, r, θ, φ) = e−iωteimφR[s](r)S[s]

mℓ(aω, cos θ)

involves the spin s-weighted spheroidal harmonics S[s]

mℓ(aω, cos θ) which are

eigenfunctions of / ∆[s] = 1 sin θ ∂θ(sin θ∂θ) − m2 + 2ms cos θ + s2 sin2 θ + a2ω2 cos2 θ − 2aωs cos θ For a = 0, they reduce to the spherical harmonics S[s]

mℓ(0, cos θ) = Y [s] mℓ(cos θ).

Spin-weighted spherical harmonics of different spins are simply related and have the same eigenvalues. Schematically δ

• Y [+1]

mℓ

• = −λY [+2]

mℓ ,

δ

• Y [+2]

mℓ

• = λY [+1]

mℓ

On the other hand, in the general axisymmetric case, the spin-weighted spheroidal harmonics of different spins are not simply related.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

19/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Reissner-Nordstr¨

• m, Kerr, Kerr-Newman

In a spherically symmetric background, as in Reissner-Nordstr¨

• m, the fact that

the spherical harmonics of different spins are simply related implies that the decomposition in modes passes through: T [1](Y [+1]

mℓ ) = δ(Y [+2] mℓ ) = λY [+1] mℓ

T [2](Y [+2]

mℓ ) = δ(Y [+1] mℓ ) = −λY [+2] mℓ

giving decoupled equations for the spin-1 and spin-2 fields.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

19/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Reissner-Nordstr¨

• m, Kerr, Kerr-Newman

In a spherically symmetric background, as in Reissner-Nordstr¨

• m, the fact that

the spherical harmonics of different spins are simply related implies that the decomposition in modes passes through: T [1](Y [+1]

mℓ ) = δ(Y [+2] mℓ ) = λY [+1] mℓ

T [2](Y [+2]

mℓ ) = δ(Y [+1] mℓ ) = −λY [+2] mℓ

giving decoupled equations for the spin-1 and spin-2 fields. For gravitational perturbations of Kerr one only uses the spin-2 decomposition for T [2](S[+2]

mℓ ) = 0, so this problem does not arise.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

19/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Reissner-Nordstr¨

• m, Kerr, Kerr-Newman

In a spherically symmetric background, as in Reissner-Nordstr¨

• m, the fact that

the spherical harmonics of different spins are simply related implies that the decomposition in modes passes through: T [1](Y [+1]

mℓ ) = δ(Y [+2] mℓ ) = λY [+1] mℓ

T [2](Y [+2]

mℓ ) = δ(Y [+1] mℓ ) = −λY [+2] mℓ

giving decoupled equations for the spin-1 and spin-2 fields. For gravitational perturbations of Kerr one only uses the spin-2 decomposition for T [2](S[+2]

mℓ ) = 0, so this problem does not arise.

In Kerr-Newman instead, the interaction between spin-2 and spin-1 prevents the separability in modes: T [1](S[+1]

mℓ ) = δ(S[+2] mℓ )

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

19/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Reissner-Nordstr¨

• m, Kerr, Kerr-Newman

In a spherically symmetric background, as in Reissner-Nordstr¨

• m, the fact that

the spherical harmonics of different spins are simply related implies that the decomposition in modes passes through: T [1](Y [+1]

mℓ ) = δ(Y [+2] mℓ ) = λY [+1] mℓ

T [2](Y [+2]

mℓ ) = δ(Y [+1] mℓ ) = −λY [+2] mℓ

giving decoupled equations for the spin-1 and spin-2 fields. For gravitational perturbations of Kerr one only uses the spin-2 decomposition for T [2](S[+2]

mℓ ) = 0, so this problem does not arise.

In Kerr-Newman instead, the interaction between spin-2 and spin-1 prevents the separability in modes: T [1](S[+1]

mℓ ) = δ(S[+2] mℓ ) =?

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

19/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Reissner-Nordstr¨

• m, Kerr, Kerr-Newman

In a spherically symmetric background, as in Reissner-Nordstr¨

• m, the fact that

the spherical harmonics of different spins are simply related implies that the decomposition in modes passes through: T [1](Y [+1]

mℓ ) = δ(Y [+2] mℓ ) = λY [+1] mℓ

T [2](Y [+2]

mℓ ) = δ(Y [+1] mℓ ) = −λY [+2] mℓ

giving decoupled equations for the spin-1 and spin-2 fields. For gravitational perturbations of Kerr one only uses the spin-2 decomposition for T [2](S[+2]

mℓ ) = 0, so this problem does not arise.

In Kerr-Newman instead, the interaction between spin-2 and spin-1 prevents the separability in modes: T [1](S[+1]

mℓ ) = δ(S[+2] mℓ ) =?

T [2](S[+2]

mℓ ) = δ(S[+1] mℓ ) =?

and the two equations do not decouple.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

20/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The mode decomposition is not your friend

For the electromagnetic-gravitational perturbations of Kerr-Newman spacetime, the decomposition in modes, done to simplify the analysis of the equations, makes them unsolvable. Our approach to solve this issue is to abandon the decomposition in modes, and perform a physical-space analysis, taking advantage of the tremendous progress in the analysis of the wave equation in black hole backgrounds in the last fifteen years. If one can prove boundedness of a general solution through physical space analysis, it will then in particular imply the absence of exponentially growing mode, therefore mode stability.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

20/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The mode decomposition is not your friend

For the electromagnetic-gravitational perturbations of Kerr-Newman spacetime, the decomposition in modes, done to simplify the analysis of the equations, makes them unsolvable. Our approach to solve this issue is to abandon the decomposition in modes, and perform a physical-space analysis, taking advantage of the tremendous progress in the analysis of the wave equation in black hole backgrounds in the last fifteen years. If one can prove boundedness of a general solution through physical space analysis, it will then in particular imply the absence of exponentially growing mode, therefore mode stability. Mathematician’s comment It makes a mathematician very happy to know that her proof of boundedness of solutions through analytical mathematical tools (which is stronger than mode stability) is indeed needed to prove mode stability.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

21/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The gauge invariant quantities in Kerr-Newman

The first issue to solve to treat electromagnetic-gravitational perturbations of Kerr-Newman is to find what are the Teukolsky variables which represent electromagnetic and gravitational radiations respectively. Those variables should be somewhat “gauge invariant”, i.e. invariant under infinitesimal tetrad transformations:

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

21/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The gauge invariant quantities in Kerr-Newman

The first issue to solve to treat electromagnetic-gravitational perturbations of Kerr-Newman is to find what are the Teukolsky variables which represent electromagnetic and gravitational radiations respectively. Those variables should be somewhat “gauge invariant”, i.e. invariant under infinitesimal tetrad transformations: The curvature spin-2 components Ψ0, Ψ4 are (quadratically) invariant:

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

21/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The gauge invariant quantities in Kerr-Newman

The first issue to solve to treat electromagnetic-gravitational perturbations of Kerr-Newman is to find what are the Teukolsky variables which represent electromagnetic and gravitational radiations respectively. Those variables should be somewhat “gauge invariant”, i.e. invariant under infinitesimal tetrad transformations: The curvature spin-2 components Ψ0, Ψ4 are (quadratically) invariant: but are not enough to describe the full perturbation in Kerr-Newman.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

22/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The electromagnetic spin-1 components φ0 and φ2 are not invariant under infinitesimal rotations since φ1 = 0 in the background: We then have to come up with a different definition for the spin-1 field.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

22/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The electromagnetic spin-1 components φ0 and φ2 are not invariant under infinitesimal rotations since φ1 = 0 in the background: We then have to come up with a different definition for the spin-1 field. The following coupled electromagnetic-curvature components: F := −δφ0 + (2β + 3τ)φ0 − 2σφ1

• f spin +2

B := 3φ0Ψ2 − 2φ1Ψ1

• f spin +1

and their respective negative spin versions happen to be gauge-invariant. Observe that in Chandrasekhar’s book, the use of the phantom gauge φ0 = φ2 = 0 prevented to detect such gauge invariant quantities.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

23/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

In Kerr-Newman the spin is not the only thing that matters

What matters is also the conformal type of the Teukolsky variables, i.e. how they change under a rotation of class III: The Teukolsky variables get rotated as Ψ0 → A−2 Ψ0, F → A−1 F, B → A−1 B We say that Ψ0 is of conformal type 2, and F and B are of conformal type 1.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

23/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

In Kerr-Newman the spin is not the only thing that matters

What matters is also the conformal type of the Teukolsky variables, i.e. how they change under a rotation of class III: The Teukolsky variables get rotated as Ψ0 → A−2 Ψ0, F → A−1 F, B → A−1 B We say that Ψ0 is of conformal type 2, and F and B are of conformal type 1. For a Teukolsky variable we define s to be the spin c to be the conformal type spin type s conformal type c Ψ0 2 2 F 2 1 B 1 1

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

24/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Teukolsky equations in Kerr-Newman

The Teukolsky equation of spin type s and conformal type c in Kerr-Newman is T [s,c](ψ[s,c]) := gM,a,Q ψ[s,c] + 2c ρ2 (r − M)∂rψ[s,c] + 2 ρ2

• c a(r − M)

∆ + si cos θ sin2 θ

• ∂φψ[s,c]

+ 2 ρ2

• c

M(r2 − a2) − Q2r ∆ − r

• − sia cos θ
• ∂tψ[s,c]

+ 1 ρ2 (s − s2 cot2 θ)ψ[s,c]

1arXiv:2002.07228 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

24/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Teukolsky equations in Kerr-Newman

The Teukolsky equation of spin type s and conformal type c in Kerr-Newman is T [s,c](ψ[s,c]) := gM,a,Q ψ[s,c] + 2c ρ2 (r − M)∂rψ[s,c] + 2 ρ2

• c a(r − M)

∆ + si cos θ sin2 θ

• ∂φψ[s,c]

+ 2 ρ2

• c

M(r2 − a2) − Q2r ∆ − r

• − sia cos θ
• ∂tψ[s,c]

+ 1 ρ2 (s − s2 cot2 θ)ψ[s,c] The Teukolsky equations governing electromagnetic-gravitational perturbations are1 T [1,1](B) = δ(F) T [2,1](F) = ∆(Ψ0) T [2,2](Ψ0) = D(F) Notice that the last equation reduces to the Teukolsky equation in Kerr.

1arXiv:2002.07228 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

24/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Teukolsky equations in Kerr-Newman

The Teukolsky equation of spin type s and conformal type c in Kerr-Newman is T [s,c](ψ[s,c]) := gM,a,Q ψ[s,c] + 2c ρ2 (r − M)∂rψ[s,c] + 2 ρ2

• c a(r − M)

∆ + si cos θ sin2 θ

• ∂φψ[s,c]

+ 2 ρ2

• c

M(r2 − a2) − Q2r ∆ − r

• − sia cos θ
• ∂tψ[s,c]

+ 1 ρ2 (s − s2 cot2 θ)ψ[s,c] The Teukolsky equations governing electromagnetic-gravitational perturbations are1 T [1,1](B) = δ(F) T [2,1](F) = ∆(Ψ0) T [2,2](Ψ0) = D(F) Notice that the last equation reduces to the Teukolsky equation in Kerr. Can we perform a physical space analysis of this system?

1arXiv:2002.07228 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

24/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Teukolsky equations in Kerr-Newman

The Teukolsky equation of spin type s and conformal type c in Kerr-Newman is T [s,c](ψ[s,c]) := gM,a,Q ψ[s,c] + 2c ρ2 (r − M)∂rψ[s,c] + 2 ρ2

• c a(r − M)

∆ + si cos θ sin2 θ

• ∂φψ[s,c]

+ 2 ρ2

• c

M(r2 − a2) − Q2r ∆ − r

• − sia cos θ
• ∂tψ[s,c]

+ 1 ρ2 (s − s2 cot2 θ)ψ[s,c] The Teukolsky equations governing electromagnetic-gravitational perturbations are1 T [1,1](B) = δ(F) T [2,1](F) = ∆(Ψ0) T [2,2](Ψ0) = D(F) Notice that the last equation reduces to the Teukolsky equation in Kerr. Can we perform a physical space analysis of this system? Unfortunately no!

1arXiv:2002.07228 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

25/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The energy estimates in physical space

Recall that boundedness of the energy for ψ = 0 is obtained multiplying by ∂tψ and integrating by parts: 0 = ψ · ∂tψ =

• − ∂2

t ψ + ∂2 x ψ

• · ∂tψ

= −∂2

t ψ · ∂tψ − ∂t∂xψ · ∂xψ + ∂x(∂xψ · ∂tψ)

= − 1 2 ∂t(|∂tψ|2 + |∂xψ|2) + boundary terms

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

25/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The energy estimates in physical space

Recall that boundedness of the energy for ψ = 0 is obtained multiplying by ∂tψ and integrating by parts: 0 = ψ · ∂tψ =

• − ∂2

t ψ + ∂2 x ψ

• · ∂tψ

= −∂2

t ψ · ∂tψ − ∂t∂xψ · ∂xψ + ∂x(∂xψ · ∂tψ)

= − 1 2 ∂t(|∂tψ|2 + |∂xψ|2) + boundary terms Similarly for a Regge-Wheeler equation, =

• ψ − V ψ
• · ∂tψ

= − 1 2 ∂t(|∂tψ|2 + |∂xψ|2) − 1 2 V ∂t(|ψ|2) + boundary terms = − 1 2 ∂t(|∂tψ|2 + |∂xψ|2 + V |ψ|2) + boundary terms

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

25/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The energy estimates in physical space

Recall that boundedness of the energy for ψ = 0 is obtained multiplying by ∂tψ and integrating by parts: 0 = ψ · ∂tψ =

• − ∂2

t ψ + ∂2 x ψ

• · ∂tψ

= −∂2

t ψ · ∂tψ − ∂t∂xψ · ∂xψ + ∂x(∂xψ · ∂tψ)

= − 1 2 ∂t(|∂tψ|2 + |∂xψ|2) + boundary terms Similarly for a Regge-Wheeler equation, =

• ψ − V ψ
• · ∂tψ

= − 1 2 ∂t(|∂tψ|2 + |∂xψ|2) − 1 2 V ∂t(|ψ|2) + boundary terms = − 1 2 ∂t(|∂tψ|2 + |∂xψ|2 + V |ψ|2) + boundary terms For a general Teukolsky equation instead: =

• ψ − V ψ − c1∂rψ − c2∂φψ − c3∂tψ
• · ∂tψ
• ne cannot obtain directly boundedness.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

26/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Chandrasekhar transformation

This issue appears already in Schwarzschild and in Kerr: one would like to pass Teukolsky equation → Regge-Wheeler equation ψ − V ψ = c1∂rψ + c2∂φψ + c3∂tψ → ψ − V ψ = 0

2Acta Mathematica, 222: 1-214 (2019)

• 3Ann. PDE, 5, 2 (2019)

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

26/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Chandrasekhar transformation

This issue appears already in Schwarzschild and in Kerr: one would like to pass Teukolsky equation → Regge-Wheeler equation ψ − V ψ = c1∂rψ + c2∂φψ + c3∂tψ → ψ − V ψ = 0 Chandrasekhar describes a transformation theory from the Newman-Penrose perturbation (Teukolsky) to the metric perturbation (Regge-Wheeler). Dafermos-Holzegel-Rodnianski introduced a physical-space version of the Chandrasekhar transformation, first in Schwarzschild2, then in Kerr3, which allows to prove boundedness for the solutions of the Teukolsky equation, passing through a (generalized) Regge-Wheeler equation.

2Acta Mathematica, 222: 1-214 (2019)

• 3Ann. PDE, 5, 2 (2019)

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

27/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Regge-Wheeler equations in Kerr-Newman

From the Teukolsky equations in Kerr-Newman T [1,1](B) = δ(F) T [2,1](F) = D(Ψ0) T [2,2](Ψ0) = ∆(F) we only need to transform the first two.

4arXiv:2002.07228 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

27/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Regge-Wheeler equations in Kerr-Newman

From the Teukolsky equations in Kerr-Newman T [1,1](B) = δ(F) T [2,1](F) = D(Ψ0) T [2,2](Ψ0) = ∆(F) we only need to transform the first two. Since B and F are both of conformal type c = 1, the Chandrasekhar transformation consists in taking one derivative along the ingoing null direction: p = ∆(B), q = ∆(F)

4arXiv:2002.07228 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

27/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The Regge-Wheeler equations in Kerr-Newman

From the Teukolsky equations in Kerr-Newman T [1,1](B) = δ(F) T [2,1](F) = D(Ψ0) T [2,2](Ψ0) = ∆(F) we only need to transform the first two. Since B and F are both of conformal type c = 1, the Chandrasekhar transformation consists in taking one derivative along the ingoing null direction: p = ∆(B), q = ∆(F) and we obtain4 the following generalized Regge-Wheeler system gM,a,Q p − V1p − i 2a cos θ ρ2 ∂tp = (r − ia cos θ)3 (r2 + a2 cos2 θ)5/2 δ(q) gM,a,Q q − V2q − i 4a cos θ ρ2 ∂tq = − (r + ia cos θ)3 (r2 + a2 cos2 θ)5/2 δ(p) for real positive potentials V1 and V2.

4arXiv:2002.07228 Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

28/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The energy estimates for the Regge-Wheeler equations

Since p and q are complex, we multiply their equations by ∂tp and ∂tq respectively, and take the real part: gp · ∂tp − V1p · ∂tp − i 2a cos θ |ρ|2 ∂tp · ∂tp = z(r, θ) δ(q) · ∂tp gq · ∂tq − V2q · ∂tq − i 4a cos θ |ρ|2 ∂tq · ∂tq = −z(r, θ) δ(p) · ∂tq

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

28/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The energy estimates for the Regge-Wheeler equations

Since p and q are complex, we multiply their equations by ∂tp and ∂tq respectively, and take the real part: gp · ∂tp − V1p · ∂tp − i 2a cos θ |ρ|2 ∂tp · ∂tp = z(r, θ) δ(q) · ∂tp gq · ∂tq − V2q · ∂tq − i 4a cos θ |ρ|2 ∂tq · ∂tq = −z(r, θ) δ(p) · ∂tq The real parts of gp · ∂tp and gq · ∂tq give some positive energy terms, as we have seen before. The same is true for the term involving the real potential: V1p · ∂tp + V 1p · ∂tp = 1 2 V1∂t(|p|2)

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

28/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The energy estimates for the Regge-Wheeler equations

Since p and q are complex, we multiply their equations by ∂tp and ∂tq respectively, and take the real part: gp · ∂tp − V1p · ∂tp − i 2a cos θ |ρ|2 ∂tp · ∂tp = z(r, θ) δ(q) · ∂tp gq · ∂tq − V2q · ∂tq − i 4a cos θ |ρ|2 ∂tq · ∂tq = −z(r, θ) δ(p) · ∂tq The real parts of gp · ∂tp and gq · ∂tq give some positive energy terms, as we have seen before. The same is true for the term involving the real potential: V1p · ∂tp + V 1p · ∂tp = 1 2 V1∂t(|p|2) The first order term of the form i∂t cancels out: i 2a cos θ |ρ|2 ∂tp · ∂tp + i 2a cos θ |ρ|2 ∂tp · ∂tp = i 2a cos θ |ρ|2

• ∂tp · ∂tp − ∂tp · ∂tp
• = 0

and similarly for q.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

29/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The energy estimates for the Regge-Wheeler equations

To control the coupling term on the right hand side, we have to add the two equations and obtain: z(r, θ)δ(q) · ∂tp + z(r, θ)δ(q) · ∂tp − z(r, θ)δ(p) · ∂tq − z(r, θ)δ(p) · ∂tq = z(r, θ)

• δ(q) · ∂tp − δ(p) · ∂tq
• + z(r, θ) (δ(q) · ∂tp − δ(p) · ∂tq)

Integrating by parts in ∂t and in δ, we have δ(q) · ∂tp − δ(p) · ∂tq = −δ(∂tq) · p − δ(p) · ∂tq+ boundary terms = ∂tq · δ(p) − δ(p) · ∂tq+ boundary terms = boundary terms

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

29/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

The energy estimates for the Regge-Wheeler equations

To control the coupling term on the right hand side, we have to add the two equations and obtain: z(r, θ)δ(q) · ∂tp + z(r, θ)δ(q) · ∂tp − z(r, θ)δ(p) · ∂tq − z(r, θ)δ(p) · ∂tq = z(r, θ)

• δ(q) · ∂tp − δ(p) · ∂tq
• + z(r, θ) (δ(q) · ∂tp − δ(p) · ∂tq)

Integrating by parts in ∂t and in δ, we have δ(q) · ∂tp − δ(p) · ∂tq = −δ(∂tq) · p − δ(p) · ∂tq+ boundary terms = ∂tq · δ(p) − δ(p) · ∂tq+ boundary terms = boundary terms Finally, one needs to check that the boundary terms created in this way define a positive-definite energy. It is not difficult to see that in Reissner-Nordstr¨

• m they do in

the full sub-extremal range |Q| < M, and so for |a| ≪ M the boundedness of a positive energy can be obtained in Kerr-Newman.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

30/31 Kerr-Newman BH perturbation theory The Teukolsky equations The Regge-Wheeler equations

Conclusions

In order to overcome the issue of “indissolubility of the coupling between spin-1 and spin-2 fields” for electromagnetic-gravitational perturbations of the Kerr-Newman metric gM,a,Q, we derive non-separated equations, and we aim to analyze them in physical space. The equations governing electromagnetic-gravitational perturbations of Kerr-Newman are a system of three coupled Teukolsky equations, depending on the spin and the conformal type, for which physical-space estimates cannot be obtained; a system of two coupled generalized Regge-Wheeler, obtained from the above through the Chandrasekhar transformation, which crucially have:

real potentials and first order terms of the form i∂t, (like in Kerr) coupling terms of the form z(r, θ)δ(q) and −z(r, θ)δ(p) (like in Reissner-Nordstr¨

• m)

Thanks to those favorable properties, boundedness of the energy (and spacetime energy decay estimates) can be obtained for small |a|/M, bypassing the failed attempt to mode stability.

Elena Giorgi Gravity Initiative, Princeton University Electromagnetic-gravitational perturbations of Kerr-Newman black hole

Thank you for your attention!

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