Charged Rotating Black Holes in Higher Dimensions
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1 Charged Rotating Black Holes in Higher Dimensions Francisco Navarro-Lérida1, Jutta Kunz1, Dieter Maison2, Jan Viebahn1 MG11 Meeting, Berlin
2 Outline Introduction Einstein-Maxwell Black Holes Einstein-Maxwell-Dilaton Black Holes Einstein-Maxwell-Chern-Simons Black Holes Conclusions
3 Introduction 4D Einstein-Maxwell (EM) black holes Static Rotating Uncharged Schwarzschild (M) Kerr (M, J) Charged Reissner-Nordström (M, Q, P) Kerr-Newman (M, J, Q, P) D>4 Einstein-Maxwell black holes Static Rotating Uncharged Tangherlini (M) Myers-Perry (M, Ji) Charged Tangherlini (M, Q)?
4 Introduction Aim: Higher dimensional Abelian black holes asymptotically flat and with regular horizon Black rings are allowed for D>4 Emparan&Reall 2002 nothing (pure EM theory) D>4 EM + dilaton (EMD theory) Chern-Simons term (EMCS theory; just for odd D)
5 Einstein-Maxwell Black Holes Einstein-Maxwell action Maxwell field strength tensor Einstein equations with stress-energy tensor Maxwell equations
6 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 D>4) No analytical charged rotating solutions for D>4 Numerical approach: too complicated in the general case Restricted case: odd dimensional black holes with equalmagnitude angular momenta Simplification=field equations reduce to a system of 5 ODE s Kunz, Navarro-Lérida, Viebahn 2006 Similar procedure for EMD and EMCS theories
7 Einstein-Maxwell Black Holes (odd D) Ansätze (D=2N+1)
8 Einstein-Maxwell Black Holes (odd D) Regular horizon at r=rh with f(rh)=0 Killing vector null at the horizon Ω=horizon angular velocity Removing a0 : first integral Mass formula Gauntlett, Myers, Townsend 1999
9 Einstein-Maxwell Black Holes (odd D) Domain of existence (scaled quantities) Mass Angular momentum
10 Einstein-Maxwell Black Holes (odd D) Gyromagnetic ratio g=2 for D=4 but... Perturbative value g=(d-2) Aliev 2006
11 Einstein-Maxwell-Dilaton Black Holes Einstein-Maxwell-Dilaton action (units 16 π GD=1) h=dilaton coupling constant Field equations Analytical solutions!!! Kaluza-Klein black holes Kunz, Maison, Navarro-Lérida, Viebahn 2006
12 Einstein-Maxwell-Dilaton Black Holes (Kaluza-Klein) Myers-Perry solution as seed Fixed dilaton coupling constant
13 Einstein-Maxwell-Dilaton Black Holes (Kaluza-Klein) Some quantities Horizon: Surface gravity: Domains of existance: Extremal solutions κsg=0 Scaled quantities:
14 Einstein-Maxwell-Dilaton Black Holes (Kaluza-Klein)
15 Einstein-Maxwell-Dilaton Black Holes (Kaluza-Klein) Similar pattern for D>6
16 Einstein-Maxwell-Dilaton Black Holes (odd D) Restricted case: odd D, equal-magnitude angular momenta Same ansatz as in EM theory + Φ=Φ(r) No constraint on the dilaton coupling constant
17 Einstein-Maxwell-Chern-Simons Black Holes Just for odd D(=2N+1): Chern-Simons term AFN Einstein-Maxwell-Chern-Simons action Einstein equations Maxwell equations Kunz, Navarro-Lérida 2006
18 Einstein-Maxwell-Chern-Simons Black Holes Black hole solutions: regular horizon r=rh Restricted case: same ansatz as for EM black holes First integral of the system of ODE s Mass formula Scaling
19 Einstein-Maxwell-Chern-Simons Black Holes (D=5) Redefinition: Cases: λ=0: Einstein-Maxwell theory λ=1: bosonic sector of minimal D=5 supergravity λ>1 Analytical solutions: only for λ=1 Breckenridge, Myers, Peet, Vafa 1997 Good for testing the numerical scheme (restricted case: J1 = J2 ) Very high accuracy!!!
20 Einstein-Maxwell-Chern-Simons Black Holes (D=5) Domain of existence (extremal solutions) Extremal λ=1 EMCS (supersymmetric branch) Mass saturates Angular momentum satisfies Vanishing horizon angular velocity Instability beyond λ=1 (up to λ=2) supersymmetry marks a borderline between stability and instability λ=2 is a special case infinite set of extremal black holes with the same charges?
21 Einstein-Maxwell-Chern-Simons Black Holes (D=5) Four types of black holes Type I: Corrotating Ω J 0 and Ω=0, J=0 Type II: Static horizon Ω=0 but non-vanishing J 0 (λ 1 and λ=1 ) extremal) Type III: Counterrotating Ω J < 0 ( λ > 1) Type IV: Rotating horizon Ω 0 but J=0 (λ 2 and λ=2 ) extremal)
22 Einstein-Maxwell-Chern-Simons Black Holes (D=5) The horizon mass may be negative!!! Black holes are not uniquely determined by M, Ji, Q (non-uniqueness even for horizons of spherical topology)
23 Conclusions Abelian higher dimensional charged rotating BH s Restricted case: odd D + equal-magnitude angular momenta )system of ODE s EM theory: non-constant gyromagnetic ratio for D>4 Analytical Kaluza-Klein solutions in EMD theory (Odd-D) EMCS theory: D=5 is a special case λ=1: supersymmetry marks a borderline between stability and instability Four types of black holes (for λ>2) Non-uniqueness (for λ>2)
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