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The black hole stability problem
Lars Andersson
Albert Einstein Institute
joint work with Steffen Aksteiner, Pieter Blue, Jean-Philippe Nicolas
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Outline
1 Background
2 Estimates
3 Higher spin fields
4 Concluding remarks
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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.
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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
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Kerr
Maximally extended KerrLars Andersson (AEI) Black hole stability Paris, June 13, 2011 6 / 42
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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
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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.
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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
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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.
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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)
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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.
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Kerr: ergoregion
Ergoregion ⇒ no globally timelike Killing field ⇒ no positive definiteconserved energy
K2 K1 0 1 2
K2
K1
0
1
2
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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.
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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
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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
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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 �
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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
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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
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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.
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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
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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)
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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
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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.
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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.
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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
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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
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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.
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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.
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Non-radiating modes
Example: Maxwell on Black Hole background
Conserved charges∫S2 F ,
∫S2 ⋆F
nontrivial due to topology
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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
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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.
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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.
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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
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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
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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.
Lars Andersson (AEI) Black hole stability Paris, June 13, 2011 39 / 42
Background Estimates Spin Remarks References
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.
Lars Andersson (AEI) Black hole stability Paris, June 13, 2011 40 / 42
http://dx.doi.org/10.1088/0264-9381/26/8/085018
Background Estimates Spin Remarks References
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.
Lars Andersson (AEI) Black hole stability Paris, June 13, 2011 41 / 42
http://arxiv.org/abs/gr-qc/0512119
Background Estimates Spin Remarks References
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.
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