Norihiro Tanahashi (Kavli IPMU)

Transcription

1 Norihiro Tanahashi (Kavli IPMU) in collaboration with James Lucietti, Keiju Murata, Harvey S. Reall based on arxiv:

2 Classical stability of 4D Black holes Stable (mostly) Power-law decay of perturbations Extreme black holes in 4D Novel instability on the horizon Scalar fields EM & Gravitational perturbations 2

3 BH Stability & Price s law Instability of extreme horizons Generalizations Backreaction 3

4 BH Stability: [Regge & Wheeler Stability of a Schwarzschild singularity 1957]

5 Price s law (1972): Scalar field on BH decays by power law: ψ l (t, r = r 0 ) t -(2l+3) Due to backscattering H + I + i 0 5

6 Price s law (1972): Scalar field on BH decays by power law: ψ l (t, r = r 0 ) t -(2l+3) By heuristic arguments Confirmations Generalizations (Sch. Kerr, RN, ) Also on H + and I + [Gundlach+ 1993] Quasi-normal modes... 6

7 Rigorous results: Uniform boundedness [Wald 1979, Kay & Wald 1987] Regular initial data ψ (t 0, r) on Schwarzschild ψ(t,r) < C [ψ(t 0 ) ] everywhere including H + Energy estimate H + Σ(t) I + Σ(t 0 ) 7

8 Rigorous results: Decay laws: Scalar field on stationary asym. flat spacetime ψ(t,r) < C[ψ(t 0 )] t -3 Spherically sym. Scalar on Schwarzschild ψ(t) + t ψ(t) < C[ψ(t 0 )] t -3 including backreaction Many others [Tataru 2010] [Dafermos & Rodnianski 2004] 8

9 Non-extreme BHs are stable Extreme BH? κ = 0 New phenomena on the horizon Aretakis results: ψ(t,r) and its derivatives decay around BH Non-decay & blow-up on horizon 9

10 Aretakis argument: Massless scalar on extreme Reissner-Nordström H + I + Σ(t) i 0 10

11 Aretakis argument: Massless scalar on extreme Reissner-Nordström For Pointwise decay: ψ(v,r) < C[ψ(v 0 )] v -3/5 [Aretakis 2010, 2011] Non-decay & Blow-up on horizon: r ψ(v,r) H [ψ(v 0)] k r ψ(v,r) > H [ψ(v 0 )] v k-1 ( k 2 ) 11

12 Aretakis argument: Massless scalar on extreme Reissner-Nordström For Pointwise decay: ψ(v,r) < C[ψ(v 0 )] v -3/5 r n l ψ(v,r) < C[ψ(v0 )] v -1/4 Non-decay & Blow-up on horizon: r l+1 ψ(v,r) Hl [ψ(v 0)] r l+k ψ(v,r) > H l [ψ(v 0 )] v k-1 ( k 2 ) [Aretakis 2010, 2011] 12

13 Aretakis argument: $ Conserved quantities on extreme horizon w/ Ł 13

14 Aretakis argument: $ Conserved quantities on extreme horizon r = M & l = 0 w/ 14

15 Aretakis argument: $ Conserved quantities on extreme horizon r r = M & l = 0 w/ 15

16 Aretakis argument: $ Conserved quantities on extreme horizon For " l, w/ 16

17 Aretakis argument: Massless scalar on extreme Reissner-Nordström For Pointwise decay: ψ(v,r) < C[ψ(v 0 )] v -3/5 r n l ψ(v,r) < C[ψ(v0 )] v -1/4 Non-decay & Blow-up on horizon: r l+1 ψ(v,r) Hl [ψ(v 0)] r l+k ψ(v,r) > H l [ψ(v 0 )] v k-1 ( k 2 ) [Aretakis 2010, 2011] 17

18 Aretakis argument: $ Conserved quantities on extreme horizons Massless scalar on extreme RN & Kerr Instability is ubiquitous Gravitational perturbations on extreme Kerr Grav. perturbations on high-d extreme horizons H l = 0 case Massive scalar Coupled EM & Grav. perturbations [Aretakis 2011, 2012] [Lucietti & Reall 2012] [Murata 2012] [Lucietti, Murata, Reall, NT 2012] 18

19 Aretakis argument: $ Conserved quantities on extreme horizons Massless scalar on extreme RN & Kerr Instability is ubiquitous Gravitational perturbations on extreme Kerr Grav. perturbations on high-d extreme horizons H l = 0 case Massive scalar Coupled EM & Grav. perturbations [Aretakis 2011, 2012] [Lucietti & Reall 2012] [Murata 2012] [Lucietti, Murata, Reall, NT 2012] 19

20 Main results: [Lucietti, Murata, Reall, NT 2012] Instability for 1. H l = 0 2. Massive scalar 3. Coupled EM & Grav. perturbations on extreme RN background 20

21 1. Instability for H l = 0 Horizon instability H l What if H l = 0? a. No Instability b. $ Instability, same growth rate c. $ Instability, slower growth rate Confirm numerically 21

22 1. Instability for H l = 0 Horizon instability H l What if H l = 0? a. No Instability b. $ Instability, same growth rate c. $ Instability, slower growth rate Confirm numerically 22

23 1. Instability for H l = 0 Numerics: Massless scalar field on extreme RN Double-null coordinates i + u H + I + v 23

24 H l=0 0 ϕ(r) r 24

25 H l=0 0 r ϕ(r) r 25

26 H l=0 0 r 2 ϕ(r) r 26

27 H l=0 0: r 2 ϕ(r=m) v 27

28 H l=0 = 0: r 2 ϕ(r=m) v 28

29 H l=0 = 0: r 3 ϕ(r=m) v 29

30 1. Instability for H l = 0 Numerical results: H l=0 0: H l=0 = 0: [ Match with Aretakis proof (2012) ] 30

31 2. Instability for massive scalar Conserved quantity? i. Set m 2 = n (n+1) M -2 & l = 0 ii. Apply n r fn (r), then set r = M Ł 31

32 2. Instability for massive scalar iii. Find c k=1,,n to make a k=1,,n = 0 Ł Conserved quantity? i. Set m 2 = n (n+1) M -2 & l = 0 ii. Apply n r fn (r), then set r = M conserved Ł 32

33 m 2 = 2 ϕ(r=m) v 33

34 m 2 = 2 Decay + Oscillation ϕ(r=m) v 34

35 Damped oscillation on RN background: [Koyama & Tomimatsu 2000] 35

36 m 2 = 2 ϕ osci (r=m) v 36

37 m 2 = 2 v 37

38 General m 2 = n(n+1): rk ϕ v -(n+1)+k H 0 (v) ~ r ϕ(v) v 38

39 3. Instability for coupled EM & Grav. Perturbations Einstein + Maxwell + Scalar Moncrief s gauge invariant equations i. Hamiltonian formulation [Moncrief 1974] ii. Expand O( ε 2 ) iii. Take δ(constraint) as canonical variables

40 3. Instability for coupled EM & Grav. Perturbations Ex.) l = 1, odd parity on extreme RN 40

41 3. Instability for coupled EM & Grav. Perturbations 41

42 2012: Linear perturbations Non-decay & Blow-up on extreme horizon Backreaction? Non-decay & Blow-up Singularity etc? Suppression by nonlinearity non-extreme RN? Clarify by solving the full Einstein equations 42

43 Gravity + Maxwell + Scalar: EoMs: Initial data = Extreme RN + Small wave packet 43

44 With backreaction ϕ(r) r 44

45 No backreaction r 2 ϕ(r) r 45

46 With backreaction r 2 ϕ(r) r 46

47 r 2 ϕ AH Blow up linearly v Suppression

48 BH Stability & Price s law 2011 Instability of extreme horizons 2012 Generalizations 2013 Backreaction Linear Instability on extreme horizons is universal Backreaction: Linear blow-up Non-linear suppression Non-extreme Any other final state? Any physical applications? 48

49 49

50 Conformal isometry [Bizon & Friedrich 2012] [Lucietti, Murata, Reall, NT 2012] Extreme RN has discrete conformal isometry Newman Penrose constants on I + NP constants «Aretakis constants by Φ 50

51 AdS 2 analysis [Lucietti, Murata, Reall, NT 2012] Extremal Near-horizon geometry = AdS 2 Massless scalar on AdS2 Massive scalar Global: U V Ł for m 2 = n(n+1) Poincare: u v u = v 51

52 Preliminary results: r ϕ(r) r 52