Decay for the wave equation outside a slowly
rotating Kerr black hole
Pieter Blue
19 August 2010
Philadelphia: SIAM nonlinear waves and coherent structures
Pieter Blue Kerr decay
Hidden symmetries and Kerr wave decay
Joint with Lars Andersson.
Kerr spacetime
parameters: M mass and Ma angular momentum.
rotating black hole, expected end state.
black hole for |a| ≤ M; a = 0 is Schwarzschild.
Wave: α
αψ = 0, decoupled, important equation, model.
Goal: robust tools (hopefully) for Kerr stability.
We consider |a| M, exterior r > r+ .
Result: t −1+|a|C decay for |a| M [arXiv:0908.2265].
Pieter Blue Kerr decay
Kerr wave decay: other results
|a| ≤ M, mode decay: Finster-Kamran-Smoller-Yau
a = 0, integrated decay/ local energy decay/ Morawetz
estimate and decay rate estimates: Laba- Soffer, B- Soffer
(but gap), B- Sterbenz, Dafermos- Rodnianski, Metcalfe-
Marzuola- Tataru- Tohaneanu, Luk.
Donninger- Schlag- Soffer.
a M: Boundedness, integrated decay/ local energy
estimate/ Morawetz estimate, decay rates.
Dafermos-Rodnianski, Tataru-Tohaneanu, Tataru (t −3 )
Pieter Blue Kerr decay
Integrated decay/ local energy estimates/ Morawetz
For some localising weight 1A independent of t,
∞
1A |∂ψ|2 d3 xdt ≤ C |∂ψ|2 d3 x.
0 space space
Pieter Blue Kerr decay
General relativity in 1 slide
M: space-time manifold.
g : Lorentz (-,+,+,+ signature) pseudometric.
Time-like vector: g (v , v ) 0.
Summation convention g (v , v ) = gαβ v α v β .
Curve length: |g (γ, γ)|ds.
˙ ˙
(Vacuum) Einstein equation:
Ric[g ] =0.
Pieter Blue Kerr decay
Energy-momentum tensor
Energy-momentum tensor:
γ
T [ψ]αβ = αψ βψ − gαβ ( γψ ψ).
Given a vector-field X , 4-momentum
(X )
P[ψ]α =T [ψ]αβ X β ,
(X )
EX [ψ](Σ) = P[ψ]α dν α .
Σ
Assume spacetime is foliated by Σt with timelike, future-oriented
normal
EX [ψ](Σt ) =EX [ψ](t) = EX [ψ] = EX (t)
Pieter Blue Kerr decay
Energy-momentum properties
Properties:
1. If T timelike,
then ET ≥ 0.
2. We call S a (generalised) symmetry when
α α
α ψ = 0 =⇒ α Sψ = 0,
If S is a symmetry,
then EX [Sψ] has the same properties as EX [ψ].
3. If X is generates a symmetry,
then EX (t2 ) = EX (t1 ).
Otherwise: EX (t2 ) − EX (t1 ) = T [ψ]αβ (α X β) d4 µ
g.
Pieter Blue Kerr decay
Geometry of Schwarzschild and Kerr
Spherical co-ordinates, (t, r , θ, φ):
2Mr 4Mra sin2 θ Σ
g =− 1− dt 2 − dtdφ + dr 2
Σ Σ ∆
2
sin θ 2
+ Σdθ2 + (r 2 + a2 )2 − a2 ∆ sin2 θ dφ ,
Σ
Σ =r 2 + a2 cos2 θ,
∆ =r 2 − 2Mr + a2 .
√
Exterior: r > r+ = M + M 2 − a2 .
Symmetries: ∂t , ∂φ .
Pieter Blue Kerr decay
Kerr Problems
Problems:
1. No timelike, Killing vector: no positive, conserved energy.
2. ∂t , ∂φ only Killing vectors:
ET [k n u] doesn’t control Sobolev norms,
3. Photon orbits: “trapping” at more than one r .
Can’t prove Morawetz/ local energy estimate using a vector
field.
Pieter Blue Kerr decay
Energy
Use blended energy
Stationary vector field timelike for r large ∂t .
Null generator extension timelike for r near r+ ∂ t + ωH ∂ φ .
For |a| small overlap.
Let
Tχ =∂t + χωH ∂φ .
Timelike in full exterior.
Failure to be conserved controlled by Morawetz (local decay)
estimate.
Pieter Blue Kerr decay
Hidden symmetries
Hidden symmetry from Carter Killing 2-tensor
1 cos2 θ 2
Q= ∂θ sin θ∂θ + 2
∂φ + a2 sin2 θ∂t .
sin θ sin2 θ
Symmetry algebra
2 2
S2 ={Sa }a = {∂t , ∂t ∂φ , ∂φ , Q},
2
|∆S 2 u|2 ≤ |Qu|2 + |∂φ u|2 + |∂t u|2 ,
2
ET [∆S 2 u] ≤ ET [Sa u].
a
Pieter Blue Kerr decay
Morawetz (local energy) estimate idea
Wave equation
1 1
0= ∂r ∆∂r + R ψ= ∂r ∆∂r + R(r )a Sa ψ.
∆ ∆
Illustrate method by integration by parts (roughly):
A =F∂r
2
0 =(F∂r ψ)(∂r ψ + Rψ)
1
=(∂r ψ) (F )(∂r ψ) + ψ(−F) (∂r R) ψ
2
+ l.o.t.s
+ ∂t (Fψ ∂t ψ) + ∂r (∆(terms)).
∆F bounded =⇒ Σt |∆F(∂r ψ)∂t ψ|d3 µ ≤ ET .
Pieter Blue Kerr decay
Higher energies and momenta for S2 vectors
Let
n
EX ,n+1 [ψ] = EX [Sψ],
i=0 S∈Si
n
|ψ|2 =
n |Sψ|2 .
i=0 S∈Si
Let
T [ψ1 , ψ2 ]αβ =(1/4) (T [ψ1 + ψ2 ]αβ − T [ψ1 − ψ2 ]αβ ) ,
T [ψ1 ]abαβ =T [Sa ψ, Sb ψ]αβ .
Given S2 vector X ab , let
(X ab )
P[ψ]α =T [ψ1 ]abαβ X abβ ,
(X ab )
EX ab [ψ] = P[ψ]α dν α .
Pieter Blue Kerr decay
Morawetz estimate
Morawetz S2 vector field
z a
Aab = w ∂r R Lb ∂r ,
∆
2 2
L = ∂t + ∂φ + Q.
Get Tabαβ α Aabβ (plus corrections) like
z 1/2 z
∆3/2 z 1/2 ∂r w
1/2
−∂r Ra (∂r Sa ψ)(∂r Sb ψ)
∆ ∆
z z
+ w ∂r Ra ∂r Rb Lαβ (∂α Sa ψ)(∂β Sb ψ)
∆ ∆
1 z
+ ∂r ∆∂r z ∂r w ∂r Rb Lb (Sa ψ)(Sb ψ).
4 ∆
Pieter Blue Kerr decay
Bounded energy argument and local energy decay
ETχ ,3 (t2 )+ETχ ,3 (t1 )
1 1 1
≥C |∂ ψ|2 + 1r ∼3M 3 |∂t ψ|2 + |
2 r 2 2 ψ|2 +
2 |ψ|2 d4 µg .
r r r4
ETχ ,3 (t2 ) − ETχ ,3 (t1 ) ≤|a|C (localisation)|∂ 3 ψ|2 d4 µg
≤|a|C (ETχ ,3 (t2 ) + ETχ ,3 (t1 )).
1 + |a|C
ETχ ,3 (t2 ) ≤ ET ,3 (t1 )
1 − |a|C χ
Pieter Blue Kerr decay
Theorem
For |a| < a0 , if ψ satisfies the Kerr wave equation then ∃C :
ETχ ,3 (t)
1 1 1
+ |∂ ψ|2 + 1r ∼3M 3 |∂t ψ|2 + |
2 r 2 2 ψ|2 +
2 |ψ|2 d4 µg
r r r4
≤ CETχ ,3 (0).
Furthermore, ∃c such that for
r+ < r1 < r2 < ∞∃C : ∀t ∈ R, r ∈ (r1 , r2 ), (θ, φ) ∈ S 2
|ψ(t, r , θ, φ)| ≤ C t −1+C |a|
ETχ ,9 (0) + EK,7 (0) + En,3 (0) .
Similarly for r → r+ and r → ∞.
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Decay rate estimate
(r 2 +a2 )2 −a2 ∆ sin2 θ
2
Use vector field K = (t 2 + r∗ )∂t + (r 2 +a2 )2
2tr∗ ∂r∗
to cancel top-order terms.
Still a high-order term with coefficient a to be controlled: lose
C |a| in decay rate.
Use hyperbolic, instead of null, surfaces to get decay near null
infinity.
Use Sterbenz form (∂+ ψ)du+ , Stokes’ theorem, and
Morawetz to extend decay to event horizon.
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Pieter Blue Kerr decay
Pieter Blue Kerr decay