Francisco Navarro-L

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Charged Rotating Black Holes in Higher Dimensions

• Francisco Navarro-Lérida1, Jutta Kunz1, Dieter
  Maison2, Jan Viebahn1

MG11 Meeting, Berlin 25.7.2006
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Outline

• Introduction
• Einstein-Maxwell Black Holes
• Einstein-Maxwell-Dilaton Black Holes
• Einstein-Maxwell-Chern-Simons Black Holes
• Conclusions

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Introduction

• 4D Einstein-Maxwell (EM) black holes
• Dgt4 Einstein-Maxwell black holes

Static Rotating
Uncharged Schwarzschild (M) Kerr (M, J)
Charged Reissner-Nordström (M, Q, P) Kerr-Newman (M, J, Q, P)
Static Rotating
Uncharged Tangherlini (M) Myers-Perry (M, Ji)
Charged Tangherlini (M, Q) ?
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Introduction

• Aim Higher dimensional Abelian black holes
  asymptotically flat and with regular horizon
• Black rings are allowed for Dgt4 EmparanReall
  2002
• Dgt4 EM

nothing (pure EM theory)
dilaton (EMD theory)
Chern-Simons term (EMCS theory just for odd D)
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Einstein-Maxwell Black Holes

• Einstein-Maxwell action
• Maxwell field strength tensor
• Einstein equations
• with stress-energy tensor
• Maxwell equations

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Einstein-Maxwell Black Holes

• General black holes characterized by mass M
  N(D-1)/2 angular momenta Ji and charge Q(no
  magnetic charge for Dgt4)
• No analytical charged rotating solutions for Dgt4
• Numerical approach too complicated in the
  general case
• Restricted case odd dimensional black holes with
  equal-magnitude angular momenta
• Simplificationfield equations reduce to a system
  of 5 ODEs Kunz, Navarro-Lérida, Viebahn 2006
• Similar procedure for EMD and EMCS theories

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Einstein-Maxwell Black Holes (odd D)

• Ansätze (D2N1)

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Einstein-Maxwell Black Holes (odd D)

• Regular horizon at rrH with f(rH)0
• Killing vector null at the horizon?horizon
  angular velocity
• Removing a0 first integral
• Mass formula Gauntlett, Myers, Townsend 1999

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Einstein-Maxwell Black Holes (odd D)
Domain of existence (scaled quantities)
Angular momentum
Mass
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Einstein-Maxwell Black Holes (odd D)

• Gyromagnetic ratio
• g2 for D4 but ...
• Perturbative value g(D-2) Aliev 2006

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Einstein-Maxwell-Dilaton Black Holes

• Einstein-Maxwell-Dilaton action
• (units 16 ? GD1) hdilaton coupling constant
• Field equations
• Analytical solutions!!! Kaluza-Klein black
  holesKunz, Maison, Navarro-Lérida, Viebahn 2006

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Einstein-Maxwell-Dilaton Black Holes
(Kaluza-Klein)

• Myers-Perry solution as seed
• Fixed dilaton coupling constant

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Einstein-Maxwell-Dilaton Black Holes
(Kaluza-Klein)

• Some quantities
• Horizon
• Surface gravity
• Domains of existance Extremal solutions ?sg0
• Scaled quantities

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Einstein-Maxwell-Dilaton Black Holes
(Kaluza-Klein)
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Einstein-Maxwell-Dilaton Black Holes
(Kaluza-Klein)

• Similar pattern for Dgt6

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Einstein-Maxwell-Dilaton Black Holes (odd D)
Einstein-Maxwell-Dilaton Black Holes (odd D)

• Restricted case odd D, equal-magnitude angular
  momenta
• Same ansatz as in EM theory ??(r)
• No constraint on the dilaton coupling constant

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Einstein-Maxwell-Chern-Simons Black Holes

• Just for odd D(2N1) Chern-Simons term AFN
• Einstein-Maxwell-Chern-Simons action
• Einstein equations
• Maxwell equations
• Kunz, Navarro-Lérida 2006

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Einstein-Maxwell-Chern-Simons Black Holes

• Black hole solutions regular horizon rrH
• Restricted case same ansatz as for EM black
  holes
• First integral of the system of ODEs
• Mass formula
• Scaling

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Einstein-Maxwell-Chern-Simons Black Holes (D5)

• Redefinition
• Cases
• Analytical solutions only for ?1Breckenridge,
  Myers, Peet, Vafa 1997
• Good for testing the numerical
  scheme(restricted case J1J2)
• Very high accuracy!!!

?0 Einstein-Maxwell theory ?1 bosonic sector
of minimal D5 supergravity ?gt1
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Einstein-Maxwell-Chern-Simons Black Holes (D5)

• Domain of existence(extremal solutions)
• Extremal ?1 EMCS
• (supersymmetric branch)
• Mass saturates
• Angular momentum satisfies
• Vanishing horizon angular velocity

• Instability beyond ?1 (up to ?2)
  supersymmetry marks a borderline between
  stability and instability
• ?2 is a special case infinite set of
  extremal black holes with the same charges?

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Einstein-Maxwell-Chern-Simons Black Holes (D5)
Four types of black holes

• Type I Corrotating ? J 0 and ?0 , J0
• Type II Static horizon ?0 but non-vanishing J
  ? 0 (? 1 and ?1 ) extremal)
• Type III Counterrotating ? J lt 0 ( ? gt 1)
• Type IV Rotating horizon ? ? 0 but J0 (? 2
  and ?2 ) extremal)

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Einstein-Maxwell-Chern-Simons Black Holes (D5)
The horizon mass may be negative !!!
Black holes are not uniquely determined by M, Ji,
Q (non-uniqueness even for horizons of spherical
topology)
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Conclusions

• Abelian higher dimensional charged rotating BHs
• Restricted case odd D equal-magnitude angular
  momenta ) system of ODEs
• EM theory non-constant gyromagnetic ratio for
  Dgt4
• Analytical Kaluza-Klein solutions in EMD theory
• (Odd-D) EMCS theory D5 is a special case
• ?1 supersymmetry marks a borderline between
  stability and instability
• Four types of black holes (for ?gt2)
• Non-uniqueness (for ?gt2)