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