What happens at the horizon of an extreme black...
Transcript of What happens at the horizon of an extreme black...
What happens at the horizon of an extreme blackhole?
Harvey Reall
DAMTP, Cambridge University
Lucietti and HSR arXiv:1208.1437Lucietti, Murata, HSR and Tanahashi arXiv:1212.2557
Murata, HSR and Tanahashi, to appear
Introduction
I Extreme black hole: zero Hawking temperature (surfacegravity)
I e.g. M = |Q| Reissner-Nordstrom, M =√|J| Kerr
I Supersymmetric black holes necessarily extreme
I Are extreme black holes classically stable? Does a small initialperturbation remain small?
Supersymmetry vs stability
I Supergravity BPS bound: M ≥ |Q|, supersymmetric (BPS)solutions saturate this
I Minimum energy ⇒ stability?
I Maybe for field theory in flat spacetime
I Not with dynamical gravity e.g. nonlinear instability of AdSBizon & Rostworowski 2011
I Not even for linear perturbations of a fixed black holespacetime
Stability of black holes
I Consider Kerr solution
I Initial surface Σ extending from future event horizon H+ toinfinity
I Kerr solution arises from initial data on Σ
I Perturb this data: expect small enough perturbation todisperse and spacetime will ”settle down” to new Kerrsolution (with perturbed parameters)
I No proof, even for linearized perturbations
I Best result: no exponentially growing ”modes”∼ e−iωtR(r)Θ(θ)e imφ (Whiting 1989)
Black hole stability Dafermos & Rodnianski 2008-2010
I Schwarzschild or non-extreme Kerr black hole
I Toy model for linearized gravitational perturbations: masslessscalar field
ψ = 0
I Prescribe initial data for ψ on spacelike surface intersectingfuture event horizon H+ (ψ → 0 at infinity)
I ψ and all its derivatives decay outside H+ and in aneighbourhood of H+
Killing Energy: Schwarzschild
I Timelike Killing field ka gives conserved energy-momentumcurrent Ja = −T a
bkb
I Killing energy of ψ on Σ: E =∫
Σ JadΣa, (non-negative,non-increasing in time)
I Try to use E to control ψ
I Problem: outgoing photons in H+ have zero Killing energy ↔energy density degenerates at H+ (doesn’t control derivativeof ψ transverse to H+)
Horizon redshift effect
I Horizon redshift effect: energy of photons in H+ measured byinfalling observer redshifts as e−κv (κ = surface gravity, v =Killing time along H+)
I Wave analogue used to prove decay of problematic derivativeof ψ near H+
I Extreme black hole: κ = 0 so horizon redshift effect is absent
I Energy of outgoing photons at H+ does not decay
I Can’t prove decay of transverse derivative of ψ at H+
Extreme RN: stability Aretakis 2011
I Massless scalar field ψ = 0 in extreme Reissner-Nordstrom
I Stability result: ψ decays on, and outside H+
Extreme RN: conserved quantity Aretakis 2011
I Extreme RN: use ingoing Eddington-Finkelstein coordinates:regular at H+
I Assume spherical symmetry, wave eq. ψ = 0 becomes(M = 1)
2∂v∂r (rψ) + ∂r
((r − 1)2∂rψ
)= 0
I Evaluate at r = 1: ∂v∂r (rψ)|r=1 = 0
I So we have a conserved quantity on H+:
H0[ψ] ≡ ∂r (rψ)|r=1
Extreme RN: non-decay Aretakis 2011
I H0[ψ] = (∂rψ + ψ)r=1 conserved
I ψ → 0 as v →∞ ⇒ ∂rψ generically does not decay at H+
I Trr = (∂rψ)2 ⇒ energy-momentum tensor at H+ does notdecay
I Summary: absence of redshift effect ⇒ outgoing waves at H+
do not decay
Extreme RN: instability Aretakis 2011
I r -derivative of wave eq. ⇒
∂v
[∂2
r (rψ)]r=1
= −(∂rψ)r=1 → −H0
I Hence[∂2
r (rψ)]r=1∼ −H0v as v →∞
I Similarly ∂kr ψ ∼ H0vk−1
I Second and higher transverse derivatives of ψ at H+
generically blow-up at late time: instability
I Interpretation: ∂rψ decays outside H+ but not on H+ hence∂2
r ψ becomes large at late time on H+
I Polynomial, not exponential, time-dependence
I (Numerical results)
Higher partial waves Aretakis 2011
I `th partial wave ψ`: conserved quantity H` = ∂`r [r∂r (rψ`)]r=1
I ⇒ ∂`+1r ψ` generically does not decay at H+, ∂`+2
r ψ`generically blows up at late time on H+
I s-wave instability is strongest (involves fewest derivatives)
Instability in a supersymmetric theory
I Extreme RN is BPS solution of minimal N = 2 supergravitybut this has no scalar field
I Type II supergravity compactified on T 6 has 4-charge BPSblack hole solutions
I These reduce to extreme RN for equal charges
I Moduli fields constant in background: fluctuations aremassless scalars
I Aretakis instability can be embedded in supersymmetric theory
Extreme Kerr instability Aretakis 2011-2012
I Restrict to axisymmetric massless scalar ψ - no superradiance
I Stability result: ψ decays on, and outside H+
I Extreme Kerr not spherically symmetric yet evaluatingψ = 0 at H+ and projecting onto spherical harmonics givesinfinite set of conserved quantities analogous to H`[ψ]
I Transverse derivative of ψ at H+ generically does not decay
I Second and higher transverse derivatives of ψ at H+
generically blow up at late time: instability
General extreme black hole Lucietti & HSR 2012
I ψ = 0 in arbitrary extreme black hole (H+ has compactcross-sections)
I Use ”improved” Gaussian null coordinates near horizon
I ∃ Conserved quantity analogous to Aretakis’ H0
I Generic non-decay of transverse derivative of ψ at H+
I Blow-up of second transverse derivative assuming black holehas an AdS2 in near-horizon geometry (true for all knownextreme black holes)
AdS2 calculation
I Extreme RN has AdS2 × S2 near-horizon geometry:
ds2 = −r 2dv 2 + 2dvdr + dΩ2
I Aretakis argument applies here too - instability?
I But massless scalar in AdS2 is stable!
I Here the ”instability” is a coordinate effect
Massive scalar field Lucietti, Murata, HSR & Tanahashi 2012
I ψ −m2ψ = 0 in extreme RN, spherical symmetry
I If m2 = n(n + 1) then can defined conserved quantitiesanalogous to H` with ` = n ⇒ non-decay of ∂n+1
r ψ at H+ etc
I Instability for other values of m confirmed numerically
I Massive scalar is more stable
Extreme RN: gravitational and electromagneticperturbations Lucietti, Murata, HSR & Tanahashi 2012
I Instability of massless scalar suggests possible instability oflinearized gravitational/electromagnetic perturbations
I Gravitational and electromagnetic perturbations coupled
I Spherical harmonics ` = 1, 2, . . . (”non-extreme” perturbationhas ` = 0: non-dynamical)
I Can decouple equations, construct conserved quantities
I ` = 2: non-decay of gauge-invariant quantity at H+ involving3 derivatives of metric/Maxwell potential perturbations
I Expect blow-up at late time on H+ of quantity with 4derivatives
Extreme Kerr: linearized gravitational perturbationsLucietti & HSR 2012
I Null tetrad `, n,m, mI Weyl tensor components: complex Newman-Penrose scalars
ΨA, A = 0, . . . , 4
I Ψ0 = Cabcd`amb`cmd , Ψ4 = Cabcdnambncmd ,
I Perturb Kerr: δΨ0 and δΨ4 invariant under infinitesimalcoordinate transformations and infinitesimal basistransformations
I Each satisfies Teukolsky equation with spin s = 2,−2
I Variation of parameters within Kerr family has δΨ0 = δΨ4 = 0
Teukolsky equation
I Restrict to axisymmetric perturbations
I Evaluate (derivatives of) Teukolsky eq. at H+, project ontospin-weighted spherical harmonics sYj(m=0), j ≥ |s| (eventhough Kerr not spherically symmetric!)
I Obtain infinite set of conserved quantities labelled by (s, j) ⇒non-decay at H+ of quantities involving sufficiently manyderivatives of δΨ0, δΨ4
I j = 2 = −s: non-decay of derivative of δΨ4 on H+
I Expect blow-up of second derivative of δΨ4 at late time onH+
I δΨ0 exhibits much weaker instability
Backreaction
I Is Aretakis instability present in nonlinear theory?
I What is ”endpoint” of instability?
Nonlinear evolution (work in progress)Murata, HSR & Tanahashi
I Model: Einstein-Maxwell theory coupled to massless scalar ψassuming spherical symmetry
I Spherically symmetric metric in double null coordinates:
ds2 = −f (U,V )dUdV + r(U,V )2dΩ2
I Maxwell field ?F = QdΩ (Q is charge: conserved)
I Scalar field ψ(U,V )
Initial data
Figure 1: The initial surface for numerical calculations. On the surface, we give a small scalarfield perturbation which have a compact support Uout < U < Uin.
where!1 = (U, V )|(U ! U0, V = V0) , !2 = (U, V )|(U = U0, V ! V0) . (17)
In our numerical calculations, we set V0 = 0 and U0 = "5.1. From Eq.(15), we can see that,if the constraint equations are satisfied on !, they are also satisfied in whole spacetime.
3.2 Quasi-local mass
Poisson and Israel mass defined the quasi-local mass function as [1]
MH(U, V ) =r
2(1 +
4r,Ur,V
f+
Q2
r2) . (18)
This definition of the dynamical mass coincide with the renormalized Hawking mass in [2].It is known that the mass function is asymptotic to the Bondi mass MB(U) at future nullinfinity: M(U, V ) # MB(U) for V # $. De"erenciating MH with respect to V and U , weobtain
!V MH = "r2r,U
2f(",V )2 , !UMH = "r2r,V
2f(",U)2 , (19)
where we eliminated r,UV , r,V V and r,UU using Eqs.(11), (13) and (14). Above equations implythat, in the region of " = 0, the mass function is constant.
3.3 Construction of initial data
In our anzatz (8), there are residual gauge freedoms as
U # U(U) , V # V (V ) . (20)
We fix the residual gauge by taking initial data as
r(!) = r(!) , (21)
where r(U, V ) is the radial coordinate in background RN spacetime, which is written in doublenull coordinate (U, V ) introduced in Sec.1. On !1, such a initial data is simply written as
3
Initial data uniquely specified by outgoing wavepacket, amplitudeε, and initial Bondi mass Mi .Data is RN except in Uout < U < Uin. Singularity at r = 0 OK ifthere is an event horizon ⇒ Mi ≥ Q.
Initial data
I For given ε, how do we choose Mi?
I For large enough Mi , there are trapped surfaces behind anapparent horizon (trapped surface: ingoing and outgoing nullgeodesics normal to surface are converging)
I Reduce Mi so that data contains no trapped surfaces but stillcontains an apparent horizon: ”degenerate apparent horizon”,must have radius r = Q
I ”Exterior” initial data is non-extreme RN
Results
0.001
0.01
0.1
1
0 50 100 150 200
0.1
1
0 50 100 150 200
Results
I Spacetime eventually settles down to a non-extreme RN blackhole with κ = O(ε)
I For a time V ∼ 1/ε, the evolution is similar to the test field inextreme RN (gauge choice: V ∼ Eddington-Finkelstein)
I Slow decay ∼ e−κV of transverse derivative of field at horizon
I Linear growth of second transverse derivative until timeV ∼ 1/ε, then slow decay
Nonlinear instability
Maximum value of second transverse derivative at horizon is O(1)as ε→ 0: instability!
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0 50 100 150 200
Apparent and event horizons
Evolution of apparent horizon (Q = 1, ε = 0.05)
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
1 2 3 4 5 6-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
1 1.5 2 2.5 3
Figure 2: Functions !(V, r) and "r!(V, r) for fixed V slices. The initial amplitude of scalarfield is # = 0.05. We can see that these functions decay as V increases.
1
1.002
1.004
1.006
1.008
1.01
1.012
1.014
0 10 20 30 40 50
(a) Horizons
1
1.0005
1.001
1.0015
1.002
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5
(b) Bondi mass
Figure 3: The left figure shows the apparent and event horizons in r-coordinate for # = 0.05.They are increasing functions in V and the event horizon is located outside of the apparenthorizon. The right figure shows Bondi mass as the function of U . The Bondi mass is decreasingas U increases. The right end of the curve corresponds to the apparent horizon.
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Position of event horizon is r = Q +O(ε)
Toy model for back reaction
I Linear scalar field ψ = O(ε) in non-extreme RN withM = Q +O(ε2)
I Evaluate wave equation on H+: equation involving ψ and∂r (rψ). Assume |ψ| bounded by its behaviour for extreme RN
I Find ∂r (rψ)|H+ has slow exponential decay ∼ e−κv wheresurface gravity κ = O(ε)
I ∂2r (rψ)|H+ grows linearly to O(1) at time v ∼ 1/ε, slow
exponential decay thereafter
I Agrees with numerical results
Dynamical extreme black holes
I Above initial data: no trapped surfaces but apparent horizonpresent. Trapped surfaces form in time evolution.
I Decrease Mi a little: no apparent horizon in initial data buttrapped surfaces and apparent horizon form in time evolution
I Decrease Mi too much: no event horizon (”naked singularity”)
I Critical value of Mi : event horizon but no trapped surfaces:dynamical extreme BH (definition)
I Third Law (Israel 1986): ”non-extreme BH cannot becomeextreme”; this BH is always extreme
Dynamical formation of extreme black hole
Apparent horizon radius against time (Q = 1, ε = 0.1)
0.998
0.999
1
1.001
1.002
1.003
1.004
1.005
0 10 20 30 40 50
Dynamical formation of extreme black holes
I Preliminary results indicate that solution approaches extremeRN outside H+ but scalar field on H+ behaves just as inlinear theory: ψ → 0, ∂rψ → H0, ∂2
r ψ ∼ H0v
I ”Final state” is extreme RN with ψ = 0 on and outside H+
but ∂rψ = H0 on H+
I Energy-momentum tensor and curvature tensors discontinuousat H+
I H0 is ”hair” on the horizon?
I Entropy is same as for extreme RN
Summary
I Various test fields in extreme black hole spacetimes suffer aninstability
I This instability persists in nonlinear theory, genericallyevolving to a non-extreme black hole
I Extreme black holes formed dynamically exhibit extraparameter(s) on horizon
Open questions
I CFT interpretation of conserved quantities
I Extreme RN/Kerr: infinite set of conserved quantities - forwhich extreme black holes do we have this?
I Interior structure of extreme black holes formed dynamically
I Formation of extreme black holes with charged scalar