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Philadelphia

VIEWS: 5 PAGES: 30

									 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,
           null vector: g (v , v ) = 0,
           space-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

								
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