Mode stability on the real axis

Transcription

1 Mode stability on the real axis Siyuan Ma joint work with: Lars Andersson, Claudio Paganini, Bernard F. Whiting ArXiv Albert Einstein Institute, Potsdam, Germany Department of Physics, University of Florida, Gainesville, USA AARG, Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 1 / Unive 36

2 Outline 1 Background 2 Teukolsky Equation 3 Main Theorem 4 ODE Analysis of RTE 5 Partial Proof by Scattering 6 Integral Transformation 7 Proof of Main Theorem Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 2 / Unive 36

3 Introduction The full nonlinear stability of Kerr black hole is one of the main open mathematical problems in general relativity. Important steps on the way to proving Kerr black hole stability are, to obtain dispersive estimates for various test fields in Kerr exterior spacetime, including spin-2 field for the linearized Einstein vacuum equations. Whiting s (1989) proof of mode stability for Kerr was an important step towards tackling Kerr black hole stability. It states that under the condition of no incoming radiation, no non-trivial exponentially growing mode can exist. We extend this result to the analogous case of real frequencies, i.e., assuming the no incoming radiation condition, there exists no non-trivial mode with real frequency. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 3 / Unive 36

4 Kerr Spacetime In Boyer-Lindquist (B-L) coordinates (t,r,θ,φ) ( ds 2 = 1 2Mr ) dt 2 2Mar sin2 (θ) (dtdφ + dφdt) Σ Σ + Σ dr 2 + Σdθ 2 + sin2 (θ) [ (r 2 + a 2 ) 2 a 2 sin 2 (θ) ] dφ 2 (1) Σ here 0 a < M, (r) = r 2 2Mr + a 2 has two zeros r ± = M ± M 2 a 2, and Σ(r, θ) = r 2 + a 2 cos 2 (θ). The Kerr solution describes a rotating black hole 1 two parameters a, M; M= mass, am = angular momentum. 2 a = 0 Schwarzschild. 3 stationary ( t Killing), axisymmetric ( φ Killing). Kerr Uniqueness Conjecture: The Kerr solution is the unique stationary, asymptotically flat, vacuum spacetime. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 4 / Unive 36

5 Existed Results (Partial List) Schwarzschild wave Uniform Bound: Wald, Kay-Wald Schwarzschild wave Morawetz: Blue-Soffer, Blue-Sterbenz, Dafermos-Rodnianski, Luk, Donninger-Schlag-Soffer, Tataru. Schwarzschild Maxwell Morawetz: Blue, Andersson-Bäckdahl-Blue Schwarzschild Linearized gravity Morawetz: Dafermos-Rodnianski-Holzegel Kerr( a M) wave Morawetz: Dafermos-Rodnianski, Andersson-Blue, Tataru-Tohaneanu Kerr( a < M) wave Morawetz: Dafermos-Rodnianski-Shlapentokh-Rothman Scattering and modes: Whiting, Wald-Dimock, Dimock-Kay, Bachelot, Nicolas, Häfner, Finster-Kamran-Smoller-Yau, Shlapentokh-Rothman, Finster-Smoller. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 5 / Unive 36

6 TME Teukolsky Master Equation (TME) (Teukolsky, 1973) governs the perturbations of scalar components with extreme spin weights s = ±s on a fixed Kerr background: 0 = LΦ s = ( r r 1 { (r 2 + a 2 ) t + a φ (r M)s } 2 4s(r + ia cos θ) t + 1 sin θ θ sin θ θ (2) + 1 { a sin 2 sin 2 θ t + φ + is cos θ } 2 )Φs θ here (t, r, θ, φ) are B-L coordinates. s = 0, 1/2, 1, 3/2, 2 correspond to scalar wave, Dirac-Weyl, Maxwell, Rarita-Schwinger, and linearized gravity, respectively. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 6 / Unive 36

7 Fields solving TME In Kinnersley principal tetrad, let {φ 0, φ 2 } denote the Newman-Penrose scalars of spin weights {1, 1} for a Maxwell test field; { Ψ 0, Ψ 4 } denote the linearized Weyl scalars of spin weights {2, 2} for a solution of the linearized vacuum Einstein equations, taking ζ = r ia cos θ in B-L coordinates, then the scalar fields Φ s satisfying LΦ s = 0 are defined by Φ 2 = 1 ζ 4 Ψ 4, Φ 1 = 1/2 ζ 2 φ 2, (3a) Φ 1 = 1/2 φ 0, Φ 2 = Ψ 0 (3b) Analog for the fields with spin weights ±1/2 and spin weights ±3/2. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 7 / Unive 36

8 Separability A difficulty of the TME for non-zero spin field is, that it does not admit a real action, hence does not provide a conserved energy, which is an obstacle to proving stability for nonzero spin fields in the Kerr exterior spacetime. TME is a separable, spin-weighted wave equation! Spin-0 case: For the massless scalar field, the separability of the scalar wave equation on Kerr is due to the existence of a symmetry operator related to the Carter tensor Q αβ (Chandrasekhar, 1983); Nonzero spin case: For nonzero spin fields, there also exists a symmetry operator, which makes TME separable. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 8 / Unive 36

9 Mode Solutions The separability of TME gives solutions of the separated form which are called mode solutions. Φ s (r, t, θ, φ) = e iωt e imφ S s (θ)r s (r), (4) Insert the mode ansatz (4) into TME (2), making the substitutions t iω, φ im, then L = R + S Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 9 / Unive 36

10 Separated Equations The radial Teukolsky equation(rte) and angular Teukolsky equation(ate) are RR s = 0 SS s = 0 (5a) (5b) R = r r + V 0,s = ( r r + K 2 2iK(r M)s (r M) 2 s 2 ) + 4sirω Λ S = 1 sin θ θ sin θ θ m2 sin 2 θ + a2 cos θ 2 ω 2 2aωs cos θ 2ms cos θ sin 2 s 2 cot 2 θ + Λ + 2aωm a 2 ω 2, (8) θ Here K = (r 2 + a 2 )ω am, and Λ is the separation constant. (6) (7) Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 10 / Unive 36

11 ATE S is formally self-adjoint on [0, π] with respect to sin θdθ. Requiring that the solutions correspond to regular spin-weighted functions fixes the boundary conditions at θ = 0, π and ATE (5b) becomes a Sturm-Liouville problem. The solutions are called spin-weighted spheroidal harmonic functions. There is a discrete, infinite real spectrum Λ for real ω. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 11 / Unive 36

12 Mode with Complex Frequency Φ s (r, t, θ, φ) = e iωt e imφ S s (θ)r s (r) Whiting(1989) showed the mode stability in the upper half plane, under the assumption of no incoming radiation; There do exist mode solutions with no incoming radiation for certain frequencies with negative imaginary part Quasi-normal modes; Teukolsky-Press and Hartle-Wilkins showed that exponentially growing modes must arise by quasi-normal frequencies passing from the lower half plane through the real axis into the upper half plane as a is changed from zero. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 12 / Unive 36

13 Main Theorem Theorem Let Φ s be a separated mode solution to the TME for a sub-extreme Kerr black hole. Assume that Φ s has purely ingoing radiation at the horizon and purely outgoing radiation at infinity, then Φ s = 0. Remark ω = 0 time independent solution of TME. RTE becomes a hypergeometric equation with three regular singular points r, r +,. This equation does not have solutions which are well-behaved at event horizon and at infinity. For ω 0, the substitution (ω, m, s) ( ω, m, s) maps solutions of TME to solutions. Hence we restrict to ω > 0. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 13 / Unive 36

14 Role of Theorem The theorem yields that the radial Teukolsky equation has two fundamental solutions R hor and R out which are ingoing at the horizon, and outgoing at infinity, respectively, and are linearly independent, with non-vanishing Wronskian. This fact plays a central role in the proof of boundedness and decay for scalar waves on full sub-extreme Kerr exterior spacetimes (Dafermos, Rodnianski, and Shlapentokh-Rothman, 2014). In particular it is used to treat the superradiant range of frequencies. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 14 / Unive 36

15 Analysis of Singular Point All these asymptotic expansions are considered in the whole complex r-plane. The point r = is a rank 1 irregular singular point. The two normal solutions near r = have the asymptotic forms (The Thomé solutions) R s e ±iωr r ±2iMω r s 1 (9) The Stokes phenomenon implies these asymptotic series do not represent any solution in a whole vicinity of the irregular point. However, in some particular regions, they do represent the asymptotic expansions of the solutions. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 15 / Unive 36

16 Analysis of Singular Point The Stokes line is the real line Ir = 0 in the complex r-plane. It decomposes the complex r-plane into two Stokes sectors. Considering a sectorial region S = { r > A, 0 α < Arg r < β π} contained in one Stokes sector, the asymptotic expansions of the two normal solutions hold uniformly for r S. In each Stokes sector, one of the two normal solutions is exponentially increasing dominant solution, and one is exponentially decreasing recessive solution. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 16 / Unive 36

17 Analysis of Singular Point r + Regular singular point r + and r : The characteristic exponents {ρ j } j=1,2 are {ξ s/2, ξ + s/2} and {η s/2, η + s/2}, respectively: ξ = i(am 2Mr +ω), η = i(am 2Mr ω) (10) r + r r + r At r +, we assume Rρ 1 Rρ 2. In the non-resonant case (ρ 1 ρ 2 not an integer), the Frobenius solutions at r + are of the form R j (r) = (r r + ) ρ j R r+,j(r), j = 1, 2, (11) where R r+,j(r) are analytic in a neighborhood of r + of radius r + r ; In the resonant case (ξ = 0 and s an integer), R 2 (r) = (r r + ) ρ 2 R r+,2(r) (12) R 1 (r) = (r r + ) ρ 1 R r+,1 + A 2 (r r + ) ρ 2 R r+,2(r) ln(r r + ) (13) Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 17 / Unive 36

18 No Incoming Radiation Condition Waves, which are outgoing at infinity and ingoing at the horizon, are called to satisfy the no incoming radiation condition. Outgoing waves at r = should have positive radial group velocity, and must have an asymptotic form compatible with peeling: ϕ s = O(r 3 s ) Φ s = O(r 1 s ), as r. (14) As seen by a physically well-behaved observer (Hawking-Hartle tetrad and E-F coordinates), ingoing waves at the horizon must be non-special (i.e. neither vanishing nor singular at the horizon) and should have negative radial group velocity. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 18 / Unive 36

19 No Incoming Radiation Condition Definition (No Incoming Radiation Condition) Let R s be a solution of RTE. Then we shall say that R s has no incoming radiation provided R s (r) { e iωr r 2iMω r s 1, at r =, (r r + ) ξ s/2, at r = r +. (15) In particular, we shall require that R s (r) is equal to the Frobenius solution with characteristic exponent ξ s/2 at r = r + and equal near infinity r = to the normal solution which is recessive in the upper half plane. Remark The theorem can be rephrased as saying that if there is any solution R s (r) to RTE satisfying (15), then R s (r) = 0. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 19 / Unive 36

20 Outline of Partial Proof by Scattering Transform RTE to a Schrödinger form; Due to some property of the imaginary part of the potential, there exists a conserved mixed Wronskian; Take the scattering ansatz and evaluate the conserved mixed Wronskian at two boundaries to get an identity involving two extreme components; Employ TSI to get an identity involving only one extreme component; Analyze this identity to prove mode stability for some cases. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 20 / Unive 36

21 Transform to Schrödinger form Perform υ s = (r 2 + a 2 ) 1/2 R s and dr equation of Schrödinger form: d 2 dr 2 dr = r 2 +a 2, then RTE becomes an υ s + V s υ s = 0 (16) with V resc = 1 (r 2 + a 2 ) d 2 dr 2 V s = (r 2 + a 2 ) 2 V 0,s V resc (r 2 + a 2 ) 1/2 = (r 2 + a 2 ) 4 ( a 2 + 2Mr(r 2 a 2 ) ). (17) Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 21 / Unive 36

22 Conserved Wronskian For s 0, the potential V s is complex. It has some properties: { { limr r+ V s = (iκ + s + k + ) 2 k+ = ω a lim r V s = ω 2 2Mr with + m κ + = r+ r (18) 4Mr + Define the Wronskian ( is the derivative w.r.t. r ) then the Wronskian is conserved: V s = V s. (19) W [υ s, υ s ] = υ sυ s υ s υ s. W [υ s, υ s ] = 0. (20) Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 22 / Unive 36

23 Scattering Ansatz We take a scattering ansatz { Y hole,s s/2 e ik+r, at r = r + R s (r) Y in,s e iωr r s 1 + Y out,s e iωr r s 1, at r =. (21) then evaluating the Wronskian at r + and, and using the conservation of Wronskian: 4Mr + (ik + + sκ + )Y hole,s Y hole, s = 2iωY in,s Y in, s + 2iωY out,s Y out, s. (22) Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 23 / Unive 36

24 Teukolsky-Starobinsky Identities (TSI) Define the differential operators D, D : D =(d/dr ik/ ) D =(d/dr + ik/ ), (23) then TSI: s/2 D 2s s/2 R s = C s R s s/2 D 2s s/2 R s = C s R s (24a) (24b) To determine C s 2 in (24), apply s/2 D 2s s/2 to (24a) and use (24b) to get s/2 D 2s s D 2s s/2 R s = C s 2 R s (25) Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 24 / Unive 36

25 Mode Stability (Partial Proof) Applying TSI (24) to scattering ansatz (21), A s Y hole,s 2 = Y in,s 2 B s Y out,s 2 (26) The reflection and transmission coefficients R s, T s are defined as Y out,s 2 R s = B s Y in,s 2, T Y hole,s 2 s = A s Y in,s 2 we get that T s = 1 R s. For half-integer spins, A s > 0, R s < 1, (26) mode stability; For integer spins, A s changes sign with ωk +. When ω(ω am 2Mr + ) < 0, A s < 0, R s > 1 superradiance and no proof for mode stability. Conclusion: Mode stability holds for all frequency ranges except in the case of superradiance frequencies for integer spin fields. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 25 / Unive 36

26 General Strategy Rescaling RTE to canonical form Confluent Heun equation(che) Perform integral transform to obtain a solution to a CHE with different parameters. Rescale this new equation to Schrödinger form. Show that this Schrödinger equation has a real potential (hence conserved Wronskian) and no superradiance. Employ this conserved Wronskian to conclude that the transformed solution is zero, and show that this is sufficient to proof that the original solution has to be zero. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 26 / Unive 36

27 Heun equation The rescaled radial function g(r) = (r r + ) ξ+s/2 (r r ) η+s/2 e iωr R(r) satisfies a CHE (Canonical form): 0 = T r g(r) = (r r )(r r + ) d 2 g dr 2 + (γ(r r + ) + δ(r r ) + p(r r )(r r + )) dg dr + (αp(r r ) + σ)g with the parameters γ = 2η + 1 s δ = 2ξ + 1 s p = 2iω α = 1 2s σ = Λ 2iω(1 2s)r s Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 27 / Unive 36

28 Kernel for Integral Transform Choose the kernel K(x, r) as K(x, r) = e p (x r )(r r ) r + r. (27) then there exists another CHE operator T x satisfying with the following parameters ( T x T r )K(x, r) = 0. (28) γ := α = 1 2s δ := γ + δ α = 4iMω + 1 p := p = 2iω α := γ = 2η + 1 s σ := σ Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 28 / Unive 36

29 Integral Transform Defining g(x) = C K(x, r)(r r ) γ 1 (r r + ) δ 1 e pr g(r)dr, we have that T x g(x) = T x K(x, r)(r r ) γ 1 (r r + ) δ 1 e pr g(r)dr C = T r K(x, r)(r r ) γ 1 (r r + ) δ 1 e pr g(r)dr C ( dk(x, r) = (r r ) γ (r r + ) δ e pr g(r) K(x, r) dg(r) ) dr dr + K(x, r)(r r ) γ 1 (r r + ) δ 1 e pr T r g(r)dr. C Hence T x g(x) = 0 if g(x) is well defined for the chosen contour C and C ( dk(x, r) (r r ) γ (r r + ) δ e pr g(r) K(x, r) dg(r) ) dr dr C = 0 (30) Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 29 / Unive 36

30 Scaling CHE to Schrödinger The rescaled function U(x) = (x 2 + a 2 ) 1 2 (x r ) s (x r + ) 2iMw e iωx g(x) satisfies the Schrödinger equation U + V (x)u = 0 (31) where denotes a derivative w.r.t. x and dx dx = x2 +a 2 x 2 2Mx+a 2. The important fact is that V (x) is real, hence (W [U, Ū]) = 0, and that V x=r+ = (r + r ) 2 ω 2, r 2 + lim V (x) = ω2 x Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 30 / Unive 36

31 Property of Integrand Define h(r) = (r r ) η s/2 (r r + ) ξ+s/2 e iωr R(r), { 1 for r r h(r) + r 2ξ s 1 for r (32) Define In the non-resonant case (ξ 0 or s not a positive integer), h(r) is analytic at r +. h(r) = O(r s 1 ) holds uniformly in S. I (x, 2ξ s) = C 2iω(x r )(r r + ) e r + r (r r + ) 2ξ s h(r)dr (33) we shall calculate the limit lim x x 2ξ s+1 I (x, 2ξ s). This argument is closely related to the proof of Watson s Lemma, which can be used to derive an expansion at x = of this expression. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 31 / Unive 36

32 Contour for s 0 Choose C = {r [r +, i )} in the complex r-plane, see below. C r r + r Figure: Euler Contour Due to the exponential decay of the kernel e 2iω(x r )(r r ) r + r as Ir, the boundary condition (30) is satisfied and g(x) is well-defined. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 32 / Unive 36

33 lim x U(x) for s 0 Using the Euler s integral 0 e x x ζ 1 dx = Γ(ζ), we have lim x 2ξ s+1 I (x, 2ξ s) = x In addition, for large x, ( ) i(r+ r ) 2ξ s+1 Γ(2ξ s + 1)h(r + ). 2ω (34) di (x, 2ξ s) dx = O(1/x)I (x, 2ξ s) (35) lim x U(x) = C s(ω, ξ) lim x r + s/2 R(r) = C s (ω, ξ) R r+,1(r + ) with C s (ω, ξ) > 0 bounded. (36) Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 33 / Unive 36

34 Proof for s 0 I We can write U(x) as U(x) = Z(x)I (ν(x), 2ξ s). where Z(x) = (x 2 + a 2 ) 1/2 (x r ) s (x r + ) 2iMω e iωx e 2iω(x r ) (37) Therefore, W [U, Ū] = W [Z, Z] I 2 + W [I, Ī ] Z 2 (38) Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 34 / Unive 36

35 Proof for s 0 II For x r +, we have This gives Z (x) = 4iMκ + ωz(x) + O(x r + ) W [Z, Z](r + ) = 16iM 2 κ + ωr + (r + r ) 2s Together with lim x r+ di (ν(x),α) dx = 0, we have W [U, Ū](r + ) = W [Z, Z](r + )I (ν(r + ), 2ξ s)ī (ν(r + ), 2ξ s) In particular, iw [U, Ū](r + ) > 0 (39) Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 35 / Unive 36

36 Proof for s 0 III For x, since for large x: Z = (iω + O(1/x))Z(x), (40) together with the fact di (x,2ξ s) dx = O(1/x)I (x, 2ξ s), these imply lim iw [U, Ū](x) = 2ω lim x x U(x) 2 By the conservation of Wronskian, for ω > 0, U( ) = U(r + ) = 0. R r+,1(r + ) = 0, and hence that R(r) 0. Mode( stability Albert Einstein on the real Institute, axis Potsdam, Germany AARG, Department of Physics, 36 / Unive 36