the black hole stability problem · 2011. 6. 16. · background estimates spin remarks references...

Background Estimates Spin Remarks References

The black hole stability problem

Lars Andersson

Albert Einstein Institute

joint work with Steffen Aksteiner, Pieter Blue, Jean-Philippe Nicolas

Lars Andersson (AEI) Black hole stability Paris, June 13, 2011 1 / 42


Outline

1 Background

2 Estimates

3 Higher spin fields

4 Concluding remarks



Kerr

The Kerr solution describes a rotating black holeparameters a,M, 0 ≤ |a| ≤ M; aM = angular momentum.a = 0 ⇒ Schwarzschild.stationary (∂t Killing), axisymmetric (∂φ Killing)

The Kerr solution is (expected to be) the unique stationary,asymptotically flat, vacuum, nondegenerate black hole spacetime(Alexakis, Ionescu, & Klainerman, 2010).

An isolated system in GR is (expected to be) asymptoticallystationary =⇒ Kerr should be the end state of evolution of(vacuum) asymptotically flat initial data.

To establish the relevance of the Kerr solution, we must show it isstable — one of the main open mathematical problemsconcerning the Einstein equations.



Kerr

∆ = r2 − 2Mr + a2,

Σ = r2 + a2 cos2 θ,

Π = (r2 + a2)2 −∆a2 sin2 θ

and let r± denote the roots of ∆,

r± = M ±√

M2 − a2

On the exterior region r ≥ r+, the Kerr metric can be written (Boyer &Lindquist, 1967)

gµνdxµdxν =−(

1 −2MrΣ

)dt2 −

4Mra sin2 θΣ

dtdφ+Σ

∆dr2 +Σdθ2

+Π sin2 θ

Σdφ2



Kerr

Maximally extended KerrLars Andersson (AEI) Black hole stability Paris, June 13, 2011 6 / 42


The black hole stability problem

For data close to exterior Kerr data, show that the maximal globallyhyperbolic Cauchy extension is asymptotic to exterior Kerr

Difficulties:

Gauge choice

Kerr recognition

Exploit cancellations

H I+

I−

i+

i0



Exterior Cauchy problem

If Mt0 contains an outer trapped surface with θ+ ≤ 0, it contains aunique outermost MOTS St0 with θ+ = 0.

The outermost MOTS sweep out the apparent horizon Happfoliated by MOTS St .(L.A. & J. Metzger, 2009; L.A., M. Mars, J. Metzger & W. Simon,2009)

Happ is (weakly) spacelike so is an outflow boundary.

Choose inner boundary on (or inside) MOTS.



Exterior Cauchy problem

Choose gauge so that the inner boundary approaches i+?Boundary at infinity – at i0 or on I+? Choose gauge so that outerboundary approaches i+?

H I+

I−

i+

i0



Kerr recognition

Mars-Simon tensor?

Valiente-Kroon, Bäckdahl: The invariant∫

|∇(ABKCD)|2

vanishes only at Kerr. This is an “elliptic” criterion. Is there aversion which fits with the Cauchy problem?

Curvature concomitants.



Fields on Kerr

A model problem for black hole stability is to understand (test) fields onblack hole backgrounds

Wave equation on Schwarzschild (Blue & Sterbenz, 2006),(Dafermos & Rodnianski, 2005) (Marzuola, Metcalfe, Tataru, &Tohaneanu, 2010)

Linear wave equation on slowly rotating Kerr (L.A. & P. Blue, 2009)(Dafermos & Rodnianski, 2008) (Tataru & Tohaneanu, 2008)(Tataru, 2009) (Tohaneanu, 2009)

Non-linear equations on BH backgrounds (Luk, 2010)

Maxwell on Schwarzschild (Blue, 2008)

Asymptotically Schwarzschild (Holzegel, 2010)



Higher spin fields

Spin-1: Maxwell. (L.A., P. Blue & J.-P. Nicolas, 2011) –spin-lowering gives Fackerell-Ipser wave equation, chargevanishing condition, pseudo-differential Morawetz.

For regular initial data, radiation decays and the solution isasymptotic to a Coloumb state.

Spin-2: Linearized gravity – spin-lowering gives generalizedRegge-Wheeler equation – (L.A. & S. Aksteiner, 2011a),Fackerell’s integral (L.A. & S. Aksteiner, 2011b)

Expect similar behavior as for Maxwell

Coloumb states for non-zero spin fields are analogues of modulidegrees of freedom for the Kerr family.



Kerr: ergoregion

Ergoregion ⇒ no globally timelike Killing field ⇒ no positive definiteconserved energy

K2 K1 0 1 2

K2

K1

0

1

2



Kerr: Hidden symmetry

Minkowski:Killing fields: Poincaré Lie algebra so(3, 1) ⋉R4,Conformal symmetries: dilation, K

Schwarzschild: ∂t , so(3) — conserved quantities E ,Lx ,Ly ,LzKerr: ∂t , ∂φ — conserved quantities E ,Lz ,Q = Qαβ γ̇αγ̇β.

The presence of the Carter constant Q allows the geodesicequations on Kerr to be separated.



Kerr: Hidden symmetry

Qαβ is a Killing tensor, Qαβ = Q(αβ), ∇(αQβγ) = 0.

Qαβ is related toKilling-Yano 2-form Kαβ = K[αβ], ∇(αKβ)γ = 0Killing spinor KAB , ∇A(A′KBC) = 0.

A Killing field ξ is a symmetry of the wave operator [Lξ,�g] = 0Q,K are related to symmetry operators:

Q = ∇αQαβ∇β with [�g,Q] = 0,K = iγ5γµ(Kµν∇ν − 16γ

νγλ∇λKµν), with [D,K ]+ = 0

The Carter Killing tensor is not generated by Killing fields ⇒ Q is ahidden symmetry



Kerr: photon region

Photon region ⇒ trapping for null geodesics and waves

rph−

rc

rph+

r+

0

−rph−

−rc

−rph+

−r+

m 2m 3m 4m 5m

K

Location of photon orbits depend on the conserved quantities viaLz/E ,Q/E

2



The Kerr wave operator

� = Σ�g = ∂r∆∂r +1∆R

where �g = ∇α∇α, and R = R(r , E ,Lz ,Q) = R(r , ∂t , ∂φ,Q) is

R = −(r2 + a2)2∂2t − 4aMr∂t∂φ +∆Q + (∆− a2)∂2φ (1)

Here Q is the (modified) Carter operator:

Q =(

1sin θ

∂θ sin θ∂θ

)+ cot2 θ∂2φ + a

2 sin2 θ∂2t

See from this: ∂t , ∂φ,Q commute with �



Energy

∂t fails to be timelike in the ergoregion

Use a time-like “almost Killing field”

Tχ = ∂t + χ∂φ

where χ is a cutoff function. Want Tχ is timelike for r > r+, andtangent to the horizon.

H Bad signr = 3M



Conformal energy

Tortoise coordinate dx =(r2 + a2)

∆dr , x |r=3M = 0

K vector field

K =12(t2 + x2 + 1)T⊥ + txÑ

2∂x

where T⊥ ∼= ∂t + ω∂φ, ω = 2aMrΠ , Ñ2 = (r

2+a2)2

Π

K bulk term has bad sign in a region r0 ≤ r ≤ r1

bad sign



Energy estimates

Momenta Pa = T abX b + qu∇au −∇aqu2/2

Bulk

∇aPa = ∇c∇cu∇buξb + Tab∇

aX b

+ q∇cu∇cu + qu∇c∇cu −∇c∇cqu2/2 (2)

Designer vector fields:Blended energy Tχ = ∂t + χ∂φConformal vector field K ∼ u2+∂u+ + u

2−

∂u−Morawetz A = F∂rRedshift Y

Error terms from Tχ,K ,Y are controlled by the positive bulk fromA.



Trapping in KerrR is the potential for radialmotion of null geodesics

The photon sphere is given by

R = 0,∂R

∂r= 0

F ∂R

∂r

Bulk term for A must be positive. Contains terms of the form FR′,

need −F∂R

∂r≥ 0 ⇒ F changes sign at photon orbits

But R = R(r , E ,Lz ,Q) ⇒

location of photon orbits depends on E ,Lz ,Q

⇒ A must be generalized vector field

A = F(r , ∂t , ∂φ,Q)∂r



Generalized momentum

S2 = {∂2t , ∂t∂φ, ∂

2φ,Q} — 2

nd order symmetry operators.

If �ψ = 0, then �Saψ = 0 for Sa ∈ S2Given T [u]αβ , by polarization define T [ψ1, ψ2]αβ.

Generalized vector field X abβ = X (ab)β

Generalized momentum

Pα[ψ,X ] = T αβ[Saψ,Sbψ]Xabβ

+ qab(Saψ)α(Sbψ)−12(qab)α(Saψ)(Sbψ)



Generalized Morawetz

Positivity of the A bulk reduces to showing that the ODE

v ′′ +9x2 − 34x − 26x2(x + 2)2

v = 0

has a positive solution on x > 0. This is Gauss’ hypergeometricequation of the first type!

Generalized Morawetz estimate

Ek(t1) + Ek (t0) ≥∫

[t0,t1]×M

∆2

r4|∂r u|2k +

1r 6h3Mr

|∇t,ωu|2k +1r2

|u|2k



Higher spin fields

NP, GHP formalism: Use tetrad components in null tetradla,na,ma, m̄a, lana = 1, mam̄a = −1. Field equations encoded inGHP operators þ,þ ′,ð,ð ′,spin coefficients ρ, σ, . . .GHP “symmetry operations”

′ ∗ ¯

reduce complexity of calculationsWeyl and Maxwell scalars Ψ0, . . . ,Ψ4, φ0, φ1, φ2.Petrov type D spacetimes (in particular Kerr) in a principal nulltetrad has only Ψ2 6= 0, only ρ, τ non-zero spin-coefficients (up to′).

Ψ2 = −M/(r − ia cos θ)3

in Boyer-Lindquist coordinates.



Higher spin fields

spin-0: scalar waves�ψ = 0

spin-1: Maxwell

Fαβ = F[αβ], ∇αFαβ = 0, ∇[αFβγ] = 0

NP Maxwell scalars φ0, φ1, φ2spin-2: Gravity: Rαβ = 0 implies

∇αWαβγδ = 0

NP Weyl scalars: Ψ0, . . . ,Ψ4.



Higher spin fields

Linearized gravity:

∇backgroundW′ = Γ′Wbackground, R

′ = 0

not the spin-2 field equation if Wbackground 6= 0!

Linearized NP scalars: ΨiB, i = 0, . . . ,4



Killing spinors

Petrov Type D spacetimes (eg. Kerr) admit a Killing spinor

KAB = Ψ−1/32 o(AιB)

satisfying∇A

′

(AKBC) = 0

Hermitian Killing spinor −→ Killing-Yano tensor

Yab = Y[ab], Ya(b;c) = 0

−→ Killing tensor Kab = YacYcb, K(ab;c) = 0

Killing tensor ↔ symmetry operator for wave equation

Killing-Yano tensor ↔ symmetry operator for higher spin equations



Spin lowering

Massless spin-s field φABC···D = φ(ABC···D),

∇A′AφABC···D = 0

If Wαβγδ = 0 and KAB is a Killing spinor, then

φ̂C···D = φABC···DKAB

is a massless spin-(s − 1) field

∇A′Aφ̂ABC···D = 0

For type D backgrounds, spin lowering still leads to interestingresults.



Spin weight 0 potentials

Maxwell:The spin weight zero scalar φ1 solves the wave equation (Fackerell& Ipser, 1972) (

�g − 2Ψ2A)(Ψ

−1/32A φ1) = 0

φ1 is a potential for Maxwell.

Linearized gravity:The spin weight zero scalar Ψ2B (not gauge invariant) satisfies theseparated equation (L.A. & S. Aksteiner, 2011a)

(�g − 8Ψ2A

)(Ψ

−2/32A Ψ2B) = 3�BΨ

1/32A

�BΨ1/32A = 0 is a modified harmonic (wave coordinates) gauge.

Ψ2B is a potential for linearized gravity.



Non-radiating modes

Example: Maxwell on Black Hole background

Conserved charges∫S2 F ,

∫S2 ⋆F

nontrivial due to topology



Non-radiating modes

Correspond to Coloumb solution (φ0, φ1, φ2) = (0, C(r−ia cos θ)2 ,0)

(L.A., P. Blue & J.-P. Nicolas, 2011) General Maxwell initial dataevolve to asymptotic Coloumb state

Proof involves decay estimates for the Fackerell-Ipser equation

To prove a Morawetz estimate for the Fackerell-Ipser equation,must project out non-radiating modes

For linearized gravity must project out “linearized mass” (ℓ = 0)and “linearized angular momentum” (ℓ = 1) in order to proveMorawetz estimate for generalized Regge-Wheeler equation



Maxwell on Kerr

Charge zero condition∫

S2(r ,t) F =∫

S2(r ,t) ∗F = 0leads to

0 =∫

S22

r2 + a2

∆1/2φ1 − ia sin θ(φ0 − φ2)dµ

(in Carter tetrad)

Controls ℓ = 0 mode of φ1 in terms of φ0, φ2.

Eliminating the ℓ = 0 mode allows to use an inequality of the form∫

S(t,r)|∇ωu|2 ≥ 2

∫

S(t,r)|u|2

which gives extra positivity in the Morawetz bulk, at the cost ofterms involving φ0, φ2 of order a.

These terms must be dominated using the F -Morawetz argument.



F -Morawetz

Schwarzschild case – robust for small |a|A bulk term for Maxwell

Tab∇aAb = V02(|φ0|2 + |φ2|2) + V1|φ1|2 (3)

where

V02 = ∂rF , V1 = −2∂r (FVL)

VL

Figure: V02 dotted line, V1 solid line.



Complex potential

Ψ2 = −M

(r−ia cos θ)3 is complex

no conserved energyUse energy-momentum tensor for �− 2Mr3Error term in energy estimate ∼

∫[t0,t1]×M

ar3 Im(ū∂tu)

This has support in the photon region ⇒ can’t be controlled using“classical” MorawetzTo improve the Morawetz bulk near the photon region apseudo-differential Morawetz vector field B of the form

B ∼ arctan(|τ |αF)∂r

for suitable α, where τ is the t-Fourier variable.B can be constructed so that the Morawetz bulk from A + Bcontrols terms of the form u∇t,ωu in the photon region



Concluding remarks

Significant progress has been made in numerical and analyticalstudies of the global properties of black hole spacetimesHowever, the main problems are still open:

Cosmic censorshipUniqueness of KerrStability of Kerr

We expect the methods presented here generalize to linearizedgravity on Kerr, and to be useful for analyzing stability of Kerrunder small nonlinear perturbations



References I

Aksteiner, S., & Andersson, L. (2011a, March). Linearized gravity andgauge conditions. Classical and Quantum Gravity, 28(6),065001-+.

Alexakis, S., Ionescu, A. D., & Klainerman, S. (2010, October).Uniqueness of Smooth Stationary Black Holes in Vacuum: SmallPerturbations of the Kerr Spaces. Communications inMathematical Physics, 299, 89-127.

Andersson, L., & Aksteiner, S. (2011b). On Fackerell’s integral.Andersson, L., & Blue, P. (2009, August). Hidden symmetries and

decay for the wave equation on the Kerr spacetime. ArXive-prints.

Andersson, L., Blue, P., & Nicolas, J.-P. (2011). Decay for the Maxwellfield on the Kerr spacetime.



References II

Andersson, L., Mars, M., Metzger, J., & Simon, W. (2009). The timeevolution of marginally trapped surfaces. Classical QuantumGravity, 26(8), 085018, 14. Available fromhttp://dx.doi.org/10.1088/0264-9381/26/8/085018

Andersson, L., & Metzger, J. (2009, September). The Area ofHorizons and the Trapped Region. Communications inMathematical Physics, 290, 941-972.

Blue, P. (2008). Decay of the Maxwell field on the Schwarzschildmanifold. J. Hyperbolic Differ. Equ., 5(4), 807-856.

Blue, P., & Sterbenz, J. (2006). Uniform decay of local energy and thesemi-linear wave equation on Schwarzschild space. Comm.Math. Phys., 268(2), 481–504.

Boyer, R. H., & Lindquist, R. W. (1967). Maximal analytic extension ofthe Kerr metric. J. Mathematical Phys., 8, 265–281.


http://dx.doi.org/10.1088/0264-9381/26/8/085018


References III

Dafermos, M., & Rodnianski, I. (2005). The red-shift effect andradiation decay on black hole spacetimes.(arXiv.org:gr-qc/0512119)

Dafermos, M., & Rodnianski, I. (2008, November). Lectures on blackholes and linear waves. ArXiv e-prints.

Fackerell, E. D., & Ipser, J. R. (1972, May). Weak ElectromagneticFields Around a Rotating Black Hole. Phys. Rev. D, 5,2455-2458.

Holzegel, G. (2010, October). Ultimately SchwarzschildeanSpacetimes and the Black Hole Stability Problem. ArXiv e-prints.

Luk, J. (2010, September). The Null Condition and Global Existencefor Nonlinear Wave Equations on Slowly Rotating KerrSpacetimes. ArXiv e-prints.


http://arxiv.org/abs/gr-qc/0512119


References IV

Marzuola, J., Metcalfe, J., Tataru, D., & Tohaneanu, M. (2010,January). Strichartz Estimates on Schwarzschild Black HoleBackgrounds. Communications in Mathematical Physics, 293,37-83.

Tataru, D. (2009, October). Local decay of waves on asymptotically flatstationary space-times. ArXiv e-prints.

Tataru, D., & Tohaneanu, M. (2008, October). Local energy estimateon Kerr black hole backgrounds. ArXiv e-prints.

Tohaneanu, M. (2009, October). Strichartz estimates on Kerr blackhole backgrounds. ArXiv e-prints.


the black hole stability problem · 2011. 6. 16. · background estimates spin remarks references...

Documents