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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.
6
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
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