Rotating Black Holes in Higher Dimensions
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1 Rotating Black Holes in Higher Dimensions Jutta Kunz Institute of Physics CvO University Oldenburg Models of Gravity in Higher Dimensions Bremen, Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 1 / 55
2 Outline Outline 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 2 / 55
3 Outline Outline 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 2 / 55
4 Outline Outline 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 2 / 55
5 Outline Outline 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 2 / 55
6 Outline Outline 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 2 / 55
7 Outline Introduction 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 3 / 55
8 Introduction Einstein Maxwell Black Holes: D = 4 static spherically symmetric Schwarzschild Reissner-Nordström M M,Q rotating axially symmetric Kerr Kerr Newman M,J M,J,Q uniqueness black holes are uniquely determined by their mass M, angular momentum J, charges Q and P horizon topology black hole horizons have spherical topology staticity stationary black holes with non-rotating horizon are static etc Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 4 / 55
9 Introduction Einstein-Maxwell Black Holes: D > 4 D > 4 static spherically symmetric Schwarzschild Tangherlini Reissner-Nordström Tangherlini M M,Q rotating axially symmetric Kerr Myers-Perry Kerr Newman??? M,J M,J,Q static solutions: Tangherlini, NC 27 (1963) 636 rotating solutions: vacuum Myers and Perry, Ann. Phys. (N.Y.) 172 (1986) 304 rotating solutions: Einstein-Maxwell no exact Einstein-Maxwell generalizations of Myers-Perry solutions Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 5 / 55
10 Introduction Einstein-Maxwell Black Holes: D > 4 D > 4 static spherically symmetric Schwarzschild Tangherlini Reissner-Nordström Tangherlini M M,Q rotating axially symmetric Kerr Myers-Perry Kerr Newman??? M,J M,J,Q numerical rotating Einstein-Maxwell black hole solutions: J. Kunz, F. Navarro-Lérida, and K. Petersen, Phys. Lett. B614 (2005) 104 J. Kunz, F. Navarro-Lérida, and J. Viebahn, Phys. Lett. B639 (2006) 362 analytically known rotating charged black hole solutions: Einstein-Maxwell-Chern-Simons Kaluza-Klein, etc Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 5 / 55
11 In this talk Introduction properties of black holes with spherical horizon topology Ω = 0, J > 0 stationary black holes with static horizon Ω < 0, J > 0 counter-rotating black holes Ω 0, J = 0 black holes with rotating horizon, but vanishing J non-uniqueness of black holes with horizon topology S 3 negative horizon mass Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 6 / 55
12 Introduction Myers-Perry Black Holes Myers and Perry, Ann. Phys. (N.Y.) 172 (1986) 304 D-dimensional spacetime: 1 time coordinate D 1 spatial coordinates odd D: even D: number of spatial coordinates is even: D 1 = 2N N orthogonal spatial planes: N = D 1 2 N independent angular momenta number of spatial coordinates is odd: D 1 = 2N + 1 one extra spatial coordinate [ ] D 1 N independent angular momenta: N = 2 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 7 / 55
13 Introduction Myers-Perry Black Holes Myers and Perry, Ann. Phys. (N.Y.) 172 (1986) 304 D: dimension of spacetime [ ] D 1 N: number of independent angular momenta J i : N 2 example D = 5: 2 independent angular momenta (x 3, x 4 ) θ (x 1, x 2 ) Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 8 / 55
14 Introduction Myers-Perry Black Holes Myers and Perry, Ann. Phys. (N.Y.) 172 (1986) 304 D: dimension of spacetime [ ] D 1 N: number of independent angular momenta J i : N 2 example D = 5: 2 independent angular momenta (x 3, x 4 ) θ (x 1, x 2 ) x 2 φ 1 x 1 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 8 / 55
15 Introduction Myers-Perry Black Holes Myers and Perry, Ann. Phys. (N.Y.) 172 (1986) 304 D: dimension of spacetime [ ] D 1 N: number of independent angular momenta J i : N 2 example D = 5: 2 independent angular momenta x 3 φ 2 (x 3, x 4 ) x 4 θ (x 1, x 2 ) x 2 φ 1 x 1 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 8 / 55
16 Introduction Myers-Perry Black Holes metric Frolov and Kubiznak, PRL 98 (2007) ds 2 D,MP = dt 2 ΠF N + Π mr 2 ε dr2 + (r 2 + a 2 i)(dµ 2 i + µ 2 idϕ 2 i) constraint + mr2 ε ΠF F 1 ( N i=1 dt a 2 i µ2 i i=1 ) 2 N a i µ 2 idϕ i + εr 2 dν 2 i=1 r 2 + a 2 i, Π = N µ 2 i + εν 2 = 1 i=1 N (r 2 + a 2 i) coordinate ν enters only in even dimensions: odd D : ε = 0 i=1 even D : ε = 1 mass M and angular momenta J i : M = m(1 + (D 3)) A(S D 2 ) J i = 2ma i A(S D 2 ), i = 1,...,N Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 9 / 55
17 Introduction Myers-Perry Black Holes metric Frolov and Kubiznak, PRL 98 (2007) ds 2 D,MP = dt 2 ΠF N + Π mr 2 ε dr2 + (r 2 + a 2 i)(dµ 2 i + µ 2 idϕ 2 i) constraint + mr2 ε ΠF F 1 ( N i=1 symmetry enhancement: N odd dimensions: µ 2 i = 1 i=1 dt a 2 i µ2 i i=1 ) 2 N a i µ 2 idϕ i + εr 2 dν 2 i=1 r 2 + a 2 i, Π = N µ 2 i + εν 2 = 1 i=1 equal magnitude angular momenta: a i = a N (r 2 + a 2 i) Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 10 / 55 i=1
18 Introduction Myers-Perry Black Holes: Domain of Existence D = 5: domain of existence is bounded solutions at vertices: extremal solutions with vanishing horizon area scaled angular momenta j 1 = J 1 /M (D 2)/(D 3) j 2 = J 2 /M (D 2)/(D 3) Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 11 / 55
19 Introduction Myers-Perry Black Holes: Domain of Existence scaled angular momenta j 1 = J 1 /M (D 2)/(D 3) j 2 = J 2 /M (D 2)/(D 3) D = 5: domain of existence is bounded solutions at vertices: extremal solutions with vanishing horizon area D = 6: domain of existence is unbounded on axes vertices have moved to infinity both angular momenta finite: extremal solutions Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 11 / 55
20 Introduction Myers-Perry Black Holes: Domain of Existence extremal solutions do not exist, when two angular momenta vanish Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 12 / 55
21 Introduction Einstein-Maxwell-Dilaton Black Holes Kaluza-Klein construction: embedding of the D-dimensional MP metric in (D + 1) spacetime with extra coordinate U ds 2 D+1 = du 2 + ds 2 D,MP boost in the t U plane with the 2 2 matrix ( ) cosh α sinhα L = sinhα cosh α (D + 1)-dimensional metric ds 2 D+1 = e 2ιΦ g ρσ dx ρ dx σ + e 2(D 2)ιΦ (du + A ρ dx ρ ) 2 charged rotating black holes for the KK dilaton coupling constant h h = D 1 2(D 1)(D 2) = (D 1)ι Gerard Clement later at this conference... Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 13 / 55
22 Introduction Einstein-Maxwell-Dilaton Black Holes asymptotic expansion and global charges M 1 g tt = (D 2)A rd 3 M g tϕi = J i 1 2A µ2 i +... rd 3 J Q 1 A t = +... (D 3)A rd 3 A ϕi = M i 1 (D 3)A µ2 i +... rd 3 M Σ 1 Φ = (D 3)A(S D ) rd 3 Σ = m( 1 + (D 3)cosh 2 α ) A i = 2ma i cosh αa Q = (D 3)m sinhαcosh αa i = (D 3)ma i sinhαa = (D 3)m sinh2 α A 2(D 2)ι where A := A(S D 2 ) Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 14 / 55
23 Introduction Einstein-Maxwell-Dilaton Black Holes P. M. Llatas, PLB 397 (1997) 63 J. Kunz, D. Maison, F. Navarro-Lérida, J. Viebahn, PLB 639 (2006) 95 Domain of existence of EMD black holes: scaled charge q = Q M vs scaled angular momenta j J i i = M (D 2)/(D 3) Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 15 / 55
24 Outline Einstein-Maxwell Black Holes 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 16 / 55
25 Einstein-Maxwell Black Holes Einstein-Maxwell Black Holes static spherically symmetric Schwarzschild Tangherlini Reissner-Nordström Tangherlini M M,Q rotating axially symmetric Kerr Myers-Perry Kerr Newman??? M,J M,J,Q perturbative solutions D = 5: Aliev and Frolov, PRD 69 (2004) numerical solutions D = 5: J 1 = J 2 = J or J 1 = J, J 2 = 0 Kunz, Navarro-Lérida, and Petersen, PLB 614 (2005) 104 numerical solutions odd D 5: J i = J Kunz, Navarro-Lérida, and Viebahn, PLB 639 (2006) 362 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 17 / 55
26 Einstein-Maxwell Black Holes Einstein-Maxwell Action D-dimensional Einstein-Maxwell action Einstein equations stress-energy tensor L = 1 16πG D g(r Fµν F µν ) G µν = R µν 1 2 g µνr = 2T µν T µν = F µρ F ν ρ 1 4 g µνf ρσ F ρσ gauge field equations µ F µν = 0 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 18 / 55
27 Einstein-Maxwell Black Holes Ansatz: Black Holes in Odd Dimensions odd dimensions D: if all N = (D 1)/2 angular momenta J i have equal magnitude the symmetry is strongly enhanced ds 2 = fdt 2 + m N 1 i 1 dr 2 + r 2 cos 2 θ j dθi 2 f + n f r2 N k=1 + m n f r 2 ( k 1 l=0 k=1 cos 2 θ l ) l=0 i=1 j=0 sin 2 θ k ( ε k dϕ k ω r dt ) 2 ( N k 1 ) [ N ( k 1 cos 2 θ l sin 2 θ k dϕ 2 k A µ dx µ = a 0 dt + a ϕ N k=1 ( k 1 l=0 θ 0 0, θ i [0,π/2], i = 1,...,N 1, θ N π/2 ϕ k [0,2π], k = 1,...,N and ε k = ±1 k=1 l=0 cos 2 θ l ) cos 2 θ l ) sin 2 θ k ε k dϕ k ] 2 sin 2 θ k ε k dϕ k Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 19 / 55
28 Einstein-Maxwell Black Holes Black Holes Properties: Global Charges Komar expressions for mass M and angular momenta J i M = 1 D 2 1 α, J (k) = 16πG D D 3 16πG D S D 2 α µ1...µ D 2 ǫ µ1...µ D 2ρσ ρ ξ σ, S D 2 β (k) β (k)µ1...µ D 2 ǫ µ1...µ D 2ρσ ρ η σ (k) equal-magnitude angular momenta J (k) = J electric charge Q = 1 F µ1...µ 8πG D 2 ǫ µ1...µ D 2ρσF ρσ D asymptotic expansion f = 1 ˆM +..., ω = Ĵ rd 3 S D 2 r D , a 0 = ˆQ r D , a ϕ = ˆµ mag r D Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 20 / 55
29 Einstein-Maxwell Black Holes Horizon Properties and Mass Formula horizon is located at r H : f(r H ) = 0 Killing vector χ = ξ + Ω electrostatic potential Φ H is constant N ε k η (k) k=1 χ µ A µ r=rh = Φ H = (a 0 + Ωa ϕ ) r=rh surface gravity κ κ 2 = 1 2 ( µχ ν )( µ χ ν ) Gauntlett, Myers, Townsend, CQG 16 (1999) 1 Smarr mass formula D 3 D 2 M = κa H 8πG D + NΩJ + D 3 D 2 Φ HQ Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 21 / 55
30 Einstein-Maxwell Black Holes Numerical Results: Domain of Existence Kunz, Navarro-Lérida, and Viebahn, PLB 639 (2006) 362 system of ODEs f(r), m(r), n(r), ω(r) a 0 (r), a ϕ (r) a 0 (r) can be removed (first integral) Domain of existence of EM black holes: J Q scaled angular momentum vs scaled charge M (D 2)/(D 3) M Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 22 / 55
31 Einstein-Maxwell Black Holes Numerical Results: Global Charges Kunz, Navarro-Lérida, and Viebahn, PLB 639 (2006) 362 mass angular momentum Sets of non-extremal EM black holes: mass and angular momentum vs horizon angular velocity Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 23 / 55
32 Einstein-Maxwell Black Holes Numerical Results: Gyromagnetic Ratio gyromagnetic ratio g: µ mag = g QJ 2M Aliev and Frolov, PRD 69 (2004) perturbative result: to lowest order g = D 2 nonperturbative result:? Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 24 / 55
33 Einstein-Maxwell Black Holes Numerical Results: Gyromagnetic Ratio gyromagnetic ratio g: µ mag = g QJ 2M Aliev and Frolov, PRD 69 (2004) perturbative result: to lowest order g = D 2 Kunz, Navarro-Lérida, and Viebahn, PLB 639 (2006) 362 non-perturbative result: in general g D 2 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 24 / 55
34 Einstein-Maxwell Black Holes Numerical Results: Gyromagnetic Ratio gyromagnetic ratio g: µ mag = g QJ 2M Aliev and Frolov, PRD 69 (2004) perturbative result: to lowest order g = D 2 Navarro-Lérida, arxiv: perturbative result in higher order: in general g D 2 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 24 / 55
35 Outline Einstein-Maxwell-Chern-Simons Black Holes 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 25 / 55
36 Outline Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 26 / 55
37 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes D = 5 Einstein-Maxwell-Chern-Simons Theory In odd dimensions D = 2n + 1 the Einstein-Maxwell action may be supplemented by a AF n Chern-Simons term. D = 5 Einstein-Maxwell-Chern-Simons action S = 1 { g (R Fµν F µν ) 2λ 16πG εmnpqr A m F np F qr }{{} Chern Simons } d 5 x Chern-Simons coupling constant λ λ = 0: λ = 1: Einstein-Maxwell theory bosonic sector of minimal D = 5 supergravity λ > 1 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 27 / 55
38 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes EM and EMCS Black Holes with J 1 = J 2 = J Breckenridge, Myers, Peet and Vafa, PLB 391 (1997) 93 Chong, Cveti c, Lü and Pope, PRL 95 (2005) domain of existence: λ = 0 EM: λ = 0 equal angular momenta: J 1 = J 2 = J symmetry w.r.t. Q Q scaled angular momentum vs scaled charge Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 28 / 55
39 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes EM and EMCS Black Holes with J 1 = J 2 = J Breckenridge, Myers, Peet and Vafa, PLB 391 (1997) 93 Chong, Cveti c, Lü and Pope, PRL 95 (2005) domain of existence: λ = 1 EMCS: λ = 1 equal angular momenta: J 1 = J 2 = J no symmetry w.r.t. Q Q scaled angular momentum vs scaled charge two sets of extremal solutions vertical branch remaining branch Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 28 / 55
40 Einstein-Maxwell-Chern-Simons Black Holes λ = 1: Supersymmetric Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Breckenridge, Myers, Peet and Vafa, PLB 391 (1997) 93 Chong, Cveti c, Lü and Pope, PRL 95 (2005) extremal λ = 1 EMCS black holes: mass saturates the bound: M 3 2 Q finite angular momenta: J 1 = J 2 = J angular momenta satisfy the bound: J π Q 3 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 29 / 55
41 Einstein-Maxwell-Chern-Simons Black Holes λ = 1: Supersymmetric Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Breckenridge, Myers, Peet and Vafa, PLB 391 (1997) 93 Chong, Cveti c, Lü and Pope, PRL 95 (2005) extremal λ = 1 EMCS black holes: first law: dm = κ 8πG 5 da + Φ H dq + 2ΩdJ J increases, while M and Q remain constant: dj > 0 dm = 0 dq = 0 κ = 0: extremal 0 = 2ΩdJ horizon angular velocity Ω must vanish: Ω = 0 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 29 / 55
42 Einstein-Maxwell-Chern-Simons Black Holes λ = 1: Supersymmetric Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Breckenridge, Myers, Peet and Vafa, PLB 391 (1997) 93 Chong, Cveti c, Lü and Pope, PRL 95 (2005) extremal λ = 1 EMCS black holes: horizon angular velocities vanish: Ω i = Ω = 0, J 0 angular momentum is stored in the Maxwell field negative fraction of the angular momentum is stored behind the horizon frame dragging effects: cancellation at the horizon Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 29 / 55
43 Einstein-Maxwell-Chern-Simons Black Holes λ = 1: Supersymmetric Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Breckenridge, Myers, Peet and Vafa, PLB 391 (1997) 93 Chong, Cveti c, Lü and Pope, PRL 95 (2005) extremal λ = 1 EMCS black holes: horizon angular velocities vanish: Ω i = Ω = 0, J 0 no generalization of the staticity theorem to EMCS theory the effect of rotation is to deform the horizon into a squashed 3-sphere Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 29 / 55
44 Einstein-Maxwell-Chern-Simons Black Holes λ > 1: Rotating D = 5 Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Kunz and Navarro-Lérida, PRL 96 (2006) black holes with Ω = 0, J 0: non-extremal black holes with Ω < 0, J > 0: counter-rotating black holes Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 30 / 55
45 Einstein-Maxwell-Chern-Simons Black Holes Instability of 5D EMCS Black Holes d 2 M dj 2 for extremal black holes J=0 5D Einstein-Maxwell-Chern-Simons Black Holes instability beyond λ = 1 supersymmetry marks the borderline between stability and instability λ = 2 is special Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 31 / 55
46 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Non-Uniqueness of 5D EMCS Black Holes? λ = 2 EMCS black holes angular momentum and mass vs Ω λ = 2: set of extremal rotating J = 0 solutions appears to be present λ = 2: infinite set of extremal black holes with the same charges Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 32 / 55
47 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Non-Uniqueness of 5D EMCS Black Holes λ > 2 EMCS black holes non-uniqueness of 5D black holes with horizon topology of a sphere S 3 angular momentum vs mass black holes are not uniquely determined by M, J i, Q angular momentum vs area Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 33 / 55
48 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Non-Uniqueness of 5D EMCS Black Holes λ > 2 EMCS black holes non-uniqueness of 5D black holes with horizon topology of a sphere S 3 non-uniqueness of 5D black holes and black rings (S 1 S 2 ) Emparan and Reall, PRL 88 (2002) angular momentum vs mass black holes are not uniquely determined by M, J i, Q A (GM) 3/2 vs 27π 32G J M 3/2 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 33 / 55
49 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Domain of Existence of 5D EMCS Black Holes λ > 2 EMCS black holes domain of existence static extremal black hole is no longer on boundary (almost) extremal black holes J = 0, Ω 0 (type 3) continuous set of black holes Brodbeck, Heusler, Straumann & Volkov, PRL 79 (1997) 4310 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 34 / 55
50 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Negative Horizon Mass of 5D EMCS Black Holes Kunz and Navarro-Lérida, PLB 643 (2006) 55 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 35 / 55
51 Outline Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 36 / 55
52 Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Theory odd-d Einstein-Maxwell-Chern-Simons Lagrangian L = 1 { R F µν F µν + 8 λ } F 16πG D D + 1 ǫµ1µ2...µd 2µD 1µD µ1µ 2...F µd 2µ D 1 A µd }{{} Chern Simons Chern-Simons coupling constant λ λ = 0: Einstein-Maxwell theory λ 0: λ dimensionful except for D = 5 scaling transformation: D = 2N + 1 r H γr H, Ω Ω/γ, λ γ N 2 λ, Q γ D 3 Q,... λ Q = λ/ Q (D 5)/2(D 3) Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 37 / 55
53 Einstein-Maxwell-Chern-Simons Black Holes Rotating D = 7 EMCS Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes Kunz and Navarro-Lérida, PLB 643 (2006) 55 domain of existence types of black holes Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 38 / 55
54 Einstein-Maxwell-Chern-Simons Black Holes λ > 1: Rotating D = 7 Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes non-uniqueness negative horizon mass Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 39 / 55
55 Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes λ > 1: Rotating D = 9 EMCS Black Holes Kunz and Navarro-Lérida, PLB 643 (2006) 55 magnetic moment vs angular momentum non-static black holes with J = 0, Ω = 0 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 40 / 55
56 Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes Domain of Existence of 5D EMCS Black Holes Chong, Cveti c, Lü and Pope, PRL 95 (2005) EMCS λ = 1: 2 independent angular momenta Miriam Cveti c later at this conference... Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 41 / 55
57 Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes Domain of Existence of 5D EMCS Black Holes Chong, Cveti c, Lü and Pope, PRL 95 (2005) EMCS λ = 1: 2 independent angular momenta EMCS λ 1: work in progress Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 41 / 55
58 Outline Anti-de Sitter Black Holes 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 42 / 55
59 Anti-de Sitter Black Holes Rotating Einstein Maxwell Black Holes: D > 4, Λ 0 Λ 0 Kerr-(A)dS Myers-Perry-(A)dS?????? exact: vacuum Hawking, Hunter and Taylor-Robinson, PRD 59 (1999) Gibbons, Lü, Page and Pope, PRL 93 (2004) exact: supergravity Chong, Cveti c, Lü and Pope, PRL 95 (2005) , etc Miriam Cveti c later at this conference... Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 43 / 55
60 Anti-de Sitter Black Holes Rotating Einstein Maxwell Black Holes: D > 4, Λ 0 Λ 0 Kerr-(A)dS Myers-Perry-(A)dS?????? numerical: odd D, equal magnitude J Einstein-Maxwell-anti de Sitter Kunz, Navarro-Lérida and Radu, PLB649 (2007) 463 Einstein-Maxwell-de Sitter Brihaye and Delsate, CQG24 (2007) 4691, etc Yves Brihaye later at this conference... Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 43 / 55
61 Anti-de Sitter Black Holes Einstein Maxwell Anti-de Sitter Black Holes cosmological constant (D 1)(D 2) Λ = 2l 2 D-dimensional Einstein-Maxwell action 1 S = d D x [ ] (D 1)(D 2) g R + 16πG D l 2 F µν F µν M static black holes: Reissner-Nordström-AdS-metric ds 2 = V (r)dt 2 + dr2 V (r) + r2 dω 2 D 2 V (r) = 1 m r D 3 + r Claus Lämmerzahl later at this conference... q2 r2 + 2D 6 l 2 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 44 / 55
62 Anti-de Sitter Black Holes EM-AdS: Domain of Existence scaling properties l = λl, J = λ D 2 J, r H = λr H, κ = λ 1 κ, M = λ D 3 M Q = λ D 3 Q Ω = λ 1 Ω etc. scale invariant ratio Scaled angular momentum J M (D 2)/(D 3) vs. scaled charge Q M l Q = l Q 1/(D 3) Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 45 / 55
63 Anti-de Sitter Black Holes EM-AdS: Global Properties mass M vs. horizon velocity Ω mass Q, r H, l fixed M vs. Ω 2 branches merge and end at Ω max first branch begins at static black hole second branch diverges for Ω 1/l angular momentum Q, r H, l fixed J vs. Ω analogous to M vs. Ω Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 46 / 55
64 Anti-de Sitter Black Holes EM-AdS: Global Properties gyromagnetic ratio properties of g D-dependence D 2 g D 1 perturbative result g = (D 2) + â2 l 2, â 2 l 2 1 gyromagnetic ratio g vs. horizon velocity Ω 2nd branch ends for Ω 1/l properties of g for a single J A. N. Aliev, CQG24 (2007) 4669 perturbative result 2 g D 2 Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 47 / 55
65 Anti-de Sitter Black Holes EM-AdS: Horizon Properties area A vs. horizon velocity Ω 2nd branch ends for Ω 1/l surface gravity κ vs. horizon velocity Ω Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 48 / 55
66 Anti-de Sitter Black Holes EM-AdS: Horizon Properties Kerr-AdS black holes have oblate horizon deformation the presence of charge allows for prolate deformation of the horizon horizon deformation vs. charge Q equatorial horizon circumference L e polar horizon circumference L p ratio L e /L p can ratio of horizon circumferences L e /L p go to zero? Smarr mass formula? Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 49 / 55
67 Anti-de Sitter Black Holes EM-AdS: Thermodynamics entropy S vs. temperature T Q crit and J crit : 3 branches of black hole solutions, when the electric charge Q and the angular momentum J are not very large Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 50 / 55
68 Outline Conclusions and Outlook 1 Introduction 2 Einstein-Maxwell Black Holes in Odd Dimensions 3 Einstein-Maxwell-Chern-Simons Black Holes 5D Einstein-Maxwell-Chern-Simons Black Holes Odd-D Einstein-Maxwell-Chern-Simons Black Holes 4 Einstein-Maxwell Anti-de Sitter Black Holes 5 Conclusions and Outlook Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 51 / 55
69 Conclusions and Outlook Conclusions: EMCS Black Holes D = 5 EM-Chern-Simons Black Holes λ = 1 1 < λ < 2 λ 2 Ω = 0, J > 0 black holes stationary with static horizon prolate horizon Ω < 0, J > 0 black holes counter-rotating black holes Ω 0, J = 0 black holes rotating horizon, but vanishing J non-uniqueness of black holes with horizon topology S 3 negative horizon mass D = 9 EM-Chern-Simons Black Holes Ω = 0, J = 0 black holes stationary and non-static Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 52 / 55
70 Conclusions and Outlook Comparison: D = 4 EMD Black Holes Rasheed, NPB 454 (1995) 379 Kleihaus, Kunz and Navarro-Lérida, PRD 69 (2004) surfaces of extremal solutions in Kaluza-Klein theory coupling constant γ = 3 what are the criteria for Ω = 0 black holes, counter-rotation, etc Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 53 / 55
71 Conclusions and Outlook Outlook: Topology Changing Transitions? Phase Structure of Higher-Dimensional Black Rings and Black Holes a H Emparan, Harmark, Niarchos, Obers, Rodriguez [hep-th] j Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 54 / 55
72 Conclusions and Outlook Comparison: Nonuniform Black Strings static NUBS: topology changing transition? rotating NUBS: topology changing transition? caged black holes nonuniform black strings rotating caged black holes??? Burkhard Kleihaus later at this conference... Jutta Kunz (Universität Oldenburg) Rotating Black Holes 418th WE-Heraeus-Seminar 55 / 55
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