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ブラ 石橋 明浩 Cosmophysics Group IPNS KEK Talk at Osaka City U. 6 June ’08 石橋 明浩 Cosmophysics Group IPNS KEK

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    Cosmophysics Group IPNS KEK

    Talk at Osaka City U. 6 June 08

    Cosmophysics Group IPNS KEK

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    /

    4

    e.g., Black hole

    LHC Black holeHawking

    Cosmophysics Group IPNS KEK

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    Cosmophysics Group IPNS KEK

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    Outline

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    4

    Cosmophysics Group IPNS KEK

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    (Event horizon)

    vs

    (Killing horizon)

    Cosmophysics Group IPNS KEK

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    Black Hole

    Cosmophysics Group IPNS KEK

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    =

    An isolated object

    " Star "

    Observer's world-line

    "Time-like Infinity"

    "STAR" Observer

    Null InfinityN

    ull I

    nfin

    ity

    (Penrose)

    Cosmophysics Group IPNS KEK

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    )

    =

    nullinfinity

    time-like infinity

    null infin

    ity null infinity

    observer's world-line

    An isolated object

    null infinity

    Cosmophysics Group IPNS KEK

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    = ( I+ : null infinity )

    null infinity

    even

    t hor

    izon

    singularity

    black hole

    observer's world-line

    I+ = Black Hole

    Cosmophysics Group IPNS KEK

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    : Black Hole = I+

    Event Horizon H= Black Hole

    Remarks:

    H

    H

    Cosmophysics Group IPNS KEK

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    : (Hawking 71)

    Black Hole

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    A: A > 0

    Remark: 2: S : S > 0 (Bekenstein 73)

    Question:()

    : ()

    Killing vector

    Cosmophysics Group IPNS KEK

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    :

    Killing vector ta

    atb +bta = 0

    i.e., ta

    Cosmophysics Group IPNS KEK

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    Black Hole

    :

    ds2 = (1 rH/r)dt2 + dr2

    1 rH/r + r2d2 : rH = const

    ta = (/t)a : tccta = (r)(/r)a: (r)

    N r rH

    tccta = (rH)ta on N

    Killing vector ta N = (rH): N =

    Cosmophysics Group IPNS KEK

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    Killing Horizons

    :

    N () : Killing vector Ka Ka N

    N : N a(KbKb) = 2Ka ()

    Remarks:

    - Eq. () KbbKa = Ka- N H

    Cosmophysics Group IPNS KEK

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    Minkowski spacetime with its top removed:

    null infinity

    null infinity

    even

    t hor

    izon

    cut here

    Minkowski spacetime

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    Rindler horizon

    "black hole"

    The Rindler horizon is a Killing horizon, but the event horizon is not

    Cosmophysics Group IPNS KEK

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    Multi extreme charged black holes in de Sitter space(Kastor-Traschen 93)

    ds2 = 1U2

    dt2 + e2HtU2dx2 , U :=

    i

    (1 +

    MieHt|x xi|

    )

    The event horizon of two extremal black holes in de Sitterspace is not a Killing horizon

    Cosmophysics Group IPNS KEK

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    Apparent Horizon

    C.F.,

    Event Horizon

    Cosmophysics Group IPNS KEK

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    Stationary Black Holes in D = 4 General Relativity

    Cosmophysics Group IPNS KEK

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    Black Hole

    BH Killing

    Blak Hole (Bardeen, Carter & Hawking 73)

    Killing

    = const. , M =18

    A + HJ

    M : , : , H : , J :

    0 1T = const. , E = TS + work term

    T = /2 (Hawking 75) Cosmophysics Group IPNS KEK

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    Black Hole Black Hole

    No Hair: (Israel-Carter-Robinson-Mazur-Bunting-Chrusciel)

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    . ..

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    Black HoleM Q J

    Black Hole Kerr Black Hole Kerr

    Cosmophysics Group IPNS KEK

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    Cosmophysics Group IPNS KEK

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    Black Hole

    : Uniquness /No Hair Theorem

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    . ..

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    Black HoleM Q J

    Black Hole Kerr Black Hole Kerr

    (Rigidity)

    Cosmophysics Group IPNS KEK

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    No hair

    : (Hawking 73 Chrusciel & Wald 94)

    .

    .

    . ...

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    2

    Cosmophysics Group IPNS KEK

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    Black Ring ?

    Cosmophysics Group IPNS KEK

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    No hairBlack Hole

    : (Hawking 73)

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    . ..

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    Black Hole H Killing N Black Hole

    Remarks:

    Einstein Black Hole Black Hole

    Cosmophysics Group IPNS KEK

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    Summary: Black holes in 4D general relativity

    Asymptotically flat stationary BHs in 4-dimensions

    Exact solutions e.g., Kerr metric

    Stability Stable final state of dynamicsTopology Cross-sections 2-sphereSymmetry Rigid static or axisymmetricUniqueness Vacuum Kerr-metricBH Thermodynamics quantum aspects

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    . ...

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    Which properties of 4D BHs are extended to D > 4?

    Cosmophysics Group IPNS KEK

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    Black Holes in D > 4 General Relativity

    Cosmophysics Group IPNS KEK

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    Asymptotically flat stationary D > 4 BHs

    Exact Solutions much larger variety

    Stability not fully studied yet

    Topology more varieties

    Symmetry This talk

    Uniqueness not unique!

    BH Thermodynamics generalize to D > 4

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Asymptotically flat stationary D > 4 BHs

    Exact Solutions much larger variety

    Stability not fully studied yet

    Topology more varieties

    Symmetry This talk

    Uniqueness not unique!

    BH Thermodynamics generalize to D > 4

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Asymptotically flat stationary D > 4 BHs

    Exact Solutions much larger variety

    Stability not fully studied yet

    Topology more varieties

    Symmetry This talk

    Uniqueness not unique!

    BH Thermodynamics generalize to D > 4

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Asymptotically flat stationary D > 4 BHs

    Exact Solutions much larger variety

    Stability not fully studied yet

    Topology more varieties

    Symmetry This talk

    Uniqueness not unique!

    BH Thermodynamics generalize to D > 4

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Asymptotically flat stationary D > 4 BHs

    Exact Solutions much larger variety

    Stability not fully studied yet

    Topology more varieties

    Symmetry This talk

    Uniqueness not unique!

    BH Thermodynamics generalize to D > 4

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Asymptotically flat stationary D > 4 BHs

    Exact Solutions much larger variety

    Stability not fully studied yet

    Topology more varieties

    Symmetry This talk

    Uniqueness not unique!

    BH Thermodynamics generalize to D > 4

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Exact solutions in D > 4

    Static spherical holes in D > 4 (Tangherlini 63)

    Stationary rotating black holes in D > 4 (Myers-Perry 82) Topology SD2

    [(D 1)/2] independent spins D > 6, No upper-bound on J ultra-spinning

    horizon 0 = grr = i(

    1 +(Ji/M)2

    r2

    ) GM

    rD3

    Cosmophysics Group IPNS KEK

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    Exact solutions in D > 4

    Surprise in D > 4 !

    Rotating black-rings in D = 5 (Emparan-Reall 02)

    Topology of the horizon S1 S2

    3-commuting Killing fields Isom: R SO(2) SO(2)

    not uniquely specified by global charges (M, J1, J2)

    Cosmophysics Group IPNS KEK

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    Phase diagram of 5D Black-Hole and -Rings

    1 black-hole and 2 black-rings with the same (M, J1, J2 = 0)

    .

    .

    . ...

    .

    Uniqueness theorem no longer holds as it stands

    Cosmophysics Group IPNS KEK

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    Exact solutions in D > 4

    Solutions akin to Emparan-Realls ring (M, J1 6= 0, J2 = 0)

    Black-ring w/ two angular momenta (M, J1 6= 0, J2 6= 0)

    (Pomeransky & Senkov 06)

    (Hollands & Yazadjiev 07, Morisawa-Tomizawa & Yasui 07)

    Concentric multi-black solutions:Black di-rings (ring + ring) (Iguchi & Mishima 07)

    Black-Saturn (hole + ring) (Elvang & Figueras 07)

    Orthogonal-di-/Bicycling-Rings (ring + ring)(Izumi 07 Elvang & Rodriguez 07)

    Cosmophysics Group IPNS KEK

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    Cosmophysics Group IPNS KEK

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    Black holes and rings on various manifolds

    MP in AdS or dS e.g., (Gibbons, Lu, Page, Pope 04)

    on Gibbons-Hawking e.g. (BMPV 97, Gauntlett-Gutowski-Hull-Pakis-Reall 03) on 4-Euclid space e.g. (Elvang-Emparan-Mateos-Reall 04)

    on Kaluza-Klein

    on Eguchi-Hanson

    on Taub-NUT

    on Goedele.g., (Ishihara, Kimura, Matsuno, Nakagawa, Tomizawa 06-08)

    Black-branes

    Cosmophysics Group IPNS KEK

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    Black Hole

    .

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    I I Black Hole

    J Black Hole

    Cosmophysics Group IPNS KEK

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    Rigidity of D > 4 black holes

    Cosmophysics Group IPNS KEK

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    Symmetry properties of black holes

    Assertion:

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    . ..

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    (1) The event horizon of a stationary, electro-vacuum BHis a Killing horizon

    (2) If rotating, the BH spacetime must be axisymmetric

    The event horizon is rigidly rotating with respect to null infinity

    Black Hole Rigidity

    Cosmophysics Group IPNS KEK

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    4 HawkingEvent Horizon 2

    D > 4

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    : BH Rigidity Theorem in D > 4Event Horizon

    Cosmophysics Group IPNS KEK

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    Brief sketch of proof of Theorem 1

    Cosmophysics Group IPNS KEK

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    Cosmophysics Group IPNS KEK

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    Sketch of Proof of Step 1

    Killing field Ka

    t

    HS

    K

    Ka

    KaKa = 0 and tKa = 0 on HKgab = 0 (Killing eqn.) on H = const. (KccKa = Ka) on H

    Try this one ! Ka = ta Sa

    Cosmophysics Group IPNS KEK

  • . . . . . .

    .

    .

    . ...

    .

    = const. =: Ka

    t

    HS

    S

    K

    K

    Ka + Sa = ta = Ka + Sa

    .

    .

    . ..

    .

    .

    :

    SaDaf(x) + (x)f(x) = Ka = f(x)Ka

    Cosmophysics Group IPNS KEK

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    4 Hawking 73

    fixed point

    H

    ( )

    p

    p

    4, 2-Sa

    .

    .

    . ..

    .

    .

    Sa

    (x) =1P

    P0

    [s(x)]ds

    P : = const

    Cosmophysics Group IPNS KEK

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    D > 4

    .

    .

    . ..

    .

    .

    Sa Sa

    , 5D Myers-Perry BH w/ 2-rotations (1), (2):

    S3 , ta = Ka + Sa

    Sa = (1)a(1) + (2)

    a(2)

    Killing vector a (1)/(2) Sa

    Cosmophysics Group IPNS KEK

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    D > 4: (Hollands, AI & Wald 07)

    .

    .

    . ...

    .

    Sa Ergodic theorem guarantees the limit (x) exists

    (x) := limT

    1T

    T0

    [s(x)]ds (1)

    Einsteins equations yields (x) is a constant, so that

    limT

    1T

    T0

    [s(x)]ds = =1

    Area()

    (x)d

    time-average space-average

    Cosmophysics Group IPNS KEK

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    H

    U =

    co

    ns

    t. line

    s

    = const. linesH H OGaussian-Null-Coordinates (GNCs)x = (U, , A)

    : null geodesics affine parameter

    Event horizon H: = 0

    Cosmophysics Group IPNS KEK

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    Gaussian-Null-Coordinates (GNCs)

    In GNCs x = (U, , A) the metric takes the form

    .

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    . ..

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    .

    ds2 = 2dUd 2dU2 2AdAdU + ABdAdB components : , A, AB

    The event horizon H is at = 0

    Ka := (/U)a satisfies

    KaKa = O() , KccKa = (x)Ka + O()

    So, if Ka is a Killing field and (x) is constant, will be the surface gravity

    Cosmophysics Group IPNS KEK

  • . . . . . .

    .

    .

    . ...

    .

    t

    HS

    S

    K

    K

    GNCs (U, , A) U = const.-surface

    .

    .

    . ..

    .

    .

    : GNCs (U, , A) = const. = GNCs (U , , A)

    Cosmophysics Group IPNS KEK

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    Ka = F (x) Ka, KccKa = Ka, 6= 0

    .

    .

    . ..

    .

    .

    (1) SF (x) = (x)F (x) + Ka

    (2) SU = 1 FF (x)

    {x}

    Eq. (1)Ergodic theorem Einsteins equation

    (3) DA = 12SA on H

    well-defined

    F (x) =

    0P (x, T )dT , P (x, T ) = exp

    (

    T[s(x)]ds

    )

    T > 0, P (x, T ) < e()T <

    Cosmophysics Group IPNS KEK

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    Solution to Eq. (2):

    Eq. (2)Eq. (1)

    .

    .

    . ..

    .

    .

    S = (x) log F (x) U !

    Einsten equation DA = 12SA on H

    S

    (DADA +

    12DAA

    )= 0

    DADA = 12DAA

    U GNCs x = (, U , xA)

    Cosmophysics Group IPNS KEK

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    ( = 0)

    (1) S log F (x) + F (x)

    = (x)

    (2) SU = 1 FF (x)

    = 0Eq. (1)(2):

    S = J(x)

    Eq. (1) Einsteins equation DA = 12SA on H

    Eq. (2) (x)Einsteins equation

    C.f., for 6= 0, := log F (x) U , J := (x)

    Cosmophysics Group IPNS KEK

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    Must a D > 4 Extremal BH be Rigid?

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Towards Rigidity Proof of D > 4 Extremal BHs

    Facts:

    Saprojection of ta on a cross-section generates AbelianLie-group, G, of isometries, S on a compact space

    Closure of G must be in the isometry group of (, AB) andisomorphic to a N -torus U(1)N with N = dim(G)

    Hence there exist Killing vector fields a1 , . . . , aN on (, ab)

    Sa = (1)a(1) + + (j)a(j)for some numbers := (1, . . . , N ) RN

    When the orbits of Sa are not closed, N > 2 and i/j 6= Q forall i 6= j

    Cosmophysics Group IPNS KEK

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    .

    Lemma

    .

    .

    .

    . ..

    .

    .

    Let J be a real analytic function on with the property that

    0 = limT

    1T

    T0

    J (x) d .

    Let = (1, . . . , N ) RN satisfy the following condition:There exists a q > 0 such that

    | m| > || |m|q

    holds for all but finitely many m ZN . Then

    S = J

    has a real analytic solution on .

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Key point of the proof

    Let J(x,m) be the Fourier coefficients of J :

    SJ(x,m) = im J(x,m)

    Then,

    (x) = i

    mZN\0

    J(x,m)m

    formally solvesS = J(x)

    The condition | m| > || |m|q gurantees uniform bound

    |(x)| 6

    mZN\0

    const.ec|m|

    m 6const.||

    mZN\0|m|q ec|m| <

    Therefore the formal solution constructed above makes sense

    Cosmophysics Group IPNS KEK

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    Rigidity of D > 4 Extremal BHs: (Hollands & AI in progress)

    Theorem:

    Consider an analytic solution to the vacuum Einstein equationscontaining a stationary black hole of with extremal horizon = 0.Let = (1, . . . , N ) be the angular velocities associated withprojection Sa of ta onto .

    .

    .

    . ..

    .

    .

    If these satisfy the condition, called Diophantine condition

    | m| > || |m|q

    for some q > 0 and for all but finitely many m ZN , then there existsa Killing field Ka whose orbits are tangent to the null-generators of H.

    Thus, H is a Killing horizonUsing this result, we can also obtain Axi-symmetry Killing fields, ai

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Remarks:

    We need to impose two technical assumptions:

    Analyticity

    Diophantine condition: q > 0 such that |m | > || |m|q

    e.g., q > 0 s.t.,ij

    mjmi

    >1

    mqi

    In mathematics, it is well-known that

    .

    .

    . ..

    .

    .

    The diophantine condition holds for all = (1, . . . , N ),

    except for a set of Lebesgue measure zero

    (i.e., when i/j for i 6= j are Liouville numbers)

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Staticity Theorems

    .

    .

    . ..

    .

    .

    D > 4

    ( = 0 )

    .

    .

    . ...

    .

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Summary

    Cosmophysics Group IPNS KEK

  • . . . . . .

    Summary: Stationary BHs in D > 4 General Relativity

    Exact Solutions much larger varietyRotating Holes (Myers & Perry 86) Rotating Rings (Emparan & Reall 02) & many

    Stability not fully studied yetStatic vacuum stable (AI & Kodama 03)Rotating holes only partial results:

    Special case (e.g., Kunduri-Lucietti-Reall 06, Murata-Soda 08)

    Topology more varieties Some restrictions, (Galloway & Schoen 05)Symmetry rigid (Hollands, AI & Wald 07 Hollands & AI)Uniqueness non-unique Hole and Rings w/ the same (J, M)

    Static holes: e.g., (Gibbons, Ida & Shiromizu 02)

    Uniqueness in 5D rotating holes/rings (Morisawa-Ida 04, Hollands & Yazadjiev 07)

    (Morisawa, Tomizawa & Yasui 07)

    BH Thermodynamics generalize to D > 4 e.g. (Rogatko 07) Cosmophysics Group IPNS KEK