.. Einstein-Maxwell-Chern-Simons Black Holes Jutta Kunz Institute of Physics CvO University Oldenburg 3rd Karl Schwarzschild Meeting Gravity and the Gauge/Gravity Correspondence Frankfurt, July 2017 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 1 / 35
. Outline Outline. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 2 / 35
. Outline Outline. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 2 / 35
. Outline Outline. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 2 / 35
. Outline Outline. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 2 / 35
. Outline Introduction. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 3 / 35
Introduction. Generalization of D = 4 Black Holes: D > 4 Tangherlini (1963) Myers and Perry (1986) mmmm static mmmm mmmn rotating mmmn D = 4 Schwarzschild Kerr (M) (M, J) D > 4 Tangherlini Myers-Perry (M) (M, J 1,..., J N ) Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 3 / 35
Introduction. Generalization of D = 4 Black Holes: D > 4 Tangherlini (1963) Myers and Perry (1986) mmmm static mmmm mmmn rotating mmmn D = 4 Schwarzschild Kerr (M) (M, J) D > 4 Tangherlini Myers-Perry (M) (M, J 1,..., J N ) Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 3 / 35
Introduction. Generalization of D = 4 Black Holes: D > 4 Tangherlini (1963) Myers and Perry (1986) mmmm static mmmm mmmn rotating mmmn D = 4 Schwarzschild Kerr (M) (M, J) D > 4 Tangherlini Myers-Perry (M) (M, J 1,..., J N ) Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 3 / 35
Introduction. Myers-Perry Black Holes Myers and Perry 1986 D: dimension of space-time D 1 N: number of independent angular momenta J i : N 2 N: number of independent planes x 3 ϕ 2 (x 3, x 4 ) x 4 θ (x 1, x 2 ) x 2 ϕ 1 x 1 example: D = 5, N = 2 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 4 / 35
Introduction. Myers-Perry Black Holes metric Frolov and Kubiznak (2007) ds 2 D,MP = dt 2 ΠF N + Π mr 2 ε dr2 + (r 2 + a 2 i )(dµ 2 i + µ 2 i dφ 2 i ) + mr2 ε ΠF F 1 ( i=1 dt i=1 ) 2 N a i µ 2 i dφ i + εr 2 dν 2 i=1 N a 2 N i µ2 i r 2 + a 2, Π = (r 2 + a 2 i ) i i=1 N µ 2 i + εν 2 = 1 i=1 coordinate ν only in even dimensions: odd D : ε = 0 even D : ε = 1 mass M and angular momenta J i : M = m (1 + (D 3)) A(S D 2 ) J i = 2m a i A(S D 2 ), i = 1,..., N Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 5 / 35
Introduction. Myers-Perry Black Holes metric in odd D: a i = a ds 2 D,MP = dt 2 + ΠF Π mr 2 dr2 + + mr2 ΠF ( dt N (r 2 + a 2 )(dµ 2 i + µ 2 i dφ 2 i ) i=1 ) 2 N aµ 2 i dφ i i=1 F 1 N i=1 a 2 µ 2 i r 2 + a 2, N Π = (r 2 + a 2 ) i=1 enhanced symmetry: U(1) N U(N) factorization of angular coordinates Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 6 / 35
Introduction. Myers-Perry Black Holes: Domain of Existence 1.00 0.50 D = 5: domain of existence is bounded j 2 0.00-0.50-1.00-1.0-0.5 0.0 0.5 1.0 j 1 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) EMCS Black Holes Frankfurt, July 2017 7 / 35
Introduction. Myers-Perry Black Holes: Domain of Existence 1.00 0.50 D = 5: domain of existence is bounded j 2 0.00-0.50-1.00-1.0-0.5 0.0 0.5 1.0 j 1 scaled angular momenta D = 6: domain of existence is unbounded on axes: for J 1 = J, J 2 = 0 for J 1 = 0, J 2 = J j 1 = J 1 /M (D 2)/(D 3) j 2 = J 2 /M (D 2)/(D 3) Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 7 / 35
Introduction. Myers-Perry Black Holes: ds/ads Gibbons, Lu, Page and Pope (2004) ( )2 N 2µ ai ρ2i dϕi 2 2 2 ds = W (1 + Λ r )dt + W dt U Ξi i=1 ( ) ) N ( 2 r + a2i U + dr2 + ϵ r2 dy 2 + (dρ2i + ρ2i dϕ2i ) Z 2µ Ξ i i=1 (N ( )2 r 2 + a2 ) Λ i 2 ρi dρi + ϵr ydy Ξi W (1 + Λ r2 ) i=1 Λ = 2Λ, Ξi = 1 Λ a2i (D 1)(D 2) N (1 + Λ r2 ) 2 (r + a2i ) r2 ϵ i=1 ) ( N Z a2i ρ2i U= 1 r2 + a2i 1 + Λ r2 i=1 W = ϵy 2 + N ρ2i, Ξ i=1 i Jutta Kunz (Universita t Oldenburg) Z= EMCS Black Holes Frankfurt, July 2017 8 / 35
. Outline Asymptotically Flat EMCS Black Holes. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 9 / 35
Asymptotically Flat EMCS Black Holes. D = 5 Einstein-Maxwell-Chern-Simons Theory Jutta Kunz (Universita t Oldenburg) EMCS Black Holes Frankfurt, July 2017 9 / 35
Asymptotically Flat EMCS 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 5 3 3 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) EMCS Black Holes Frankfurt, July 2017 9 / 35
Asymptotically Flat EMCS Black Holes. D = 5 Einstein-Maxwell-Chern-Simons Theory coupled set of field equations Einstein equations G µν = 2 (F µρ F ρν 14 ) F ρσf ρσ Maxwell Chern-Simons equations unchanged w.r.t. Einstein-Maxwell ν F µν + λ 2 3 ϵµναβγ F να F βγ = 0. breaking of Q Q symmetry if λ 0 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 9 / 35
Asymptotically Flat EMCS Black Holes. D = 5 Einstein-Maxwell-Chern-Simons Theory Ansätze: J 1 = J 2 = J (cohomogeneity-1) metric ds 2 = F 1 (r)dr 2 + 1 4 F 2(r)(σ 2 1 + σ 2 2) + 1 4 F 3(r) ( σ 3 2ω(r)dt ) 2 F0 (r)dt 2, gauge potential A = a 0 (r)dt + a φ (r) 1 2 σ 3, angular dependence left-invariant 1-forms σ i on S 3 σ 1 = cos ψdθ + sin ψ sin θdϕ σ 2 = sin ψdθ + cos ψ sin θdϕ σ 3 = dψ + cos θdϕ θ, ϕ and ψ: Euler angles on S 3 (0 θ π, 0 ϕ 2π, 0 ψ 4π) Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 9 / 35
. Outline Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 10 / 35
Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory. Einstein-Maxwell Black Holes higher dimensional black holes? no closed form solutions except for Tangherlini perturbative solutions small J small Q extremal near horizon solutions odd D, equal J numerical solutions odd D, equal J a single J Jutta Kunz (Universita t Oldenburg) EMCS Black Holes Frankfurt, July 2017 11 / 35
Asymptotically Flat EMCS Black Holes. Einstein-Maxwell Black Holes D = 5 Einstein-Maxwell Theory Gauntlett, Myers, Townsend (1999) first law mass formula dm = T ds + 2ΩdJ + ΦdQ D 3 D 2 M = 2κA H + N i=1 Ω i J i + D 3 D 2 Φ HQ M: mass T : surface gravity S: area Ω: angular velocity J: angular momentum Φ: horizon potential Q: charge Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 11 / 35
Asymptotically Flat EMCS Black Holes. Einstein-Maxwell Black Holes gyromagnetic ratio g: D = 5 Einstein-Maxwell Theory µ mag = g QJ 2M Aliev and Frolov (2004), Aliev (2006) perturbative result in lowest order: g = D 2 Navarro-Lérida (2010) perturbative result in higher order: g D 2 g/(d-2) 1.05 1.04 1.03 1.02 5D 7D 9D 3rd order extremal black holes g D 2 = 1+ 1 ( ) 2 Q +... 16 M 1.01 1.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Q/M Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 11 / 35
Asymptotically Flat EMCS Black Holes. Near Horizon Solutions D = 5 Einstein-Maxwell Theory Kunduri, Lucietti (2013) Blázquez-Salcedo, J.K., Navarro-Lérida (2013) near-horizon geometry of extremal black holes: AdS 2 S 3 metric gauge potential ds 2 = v 1 ( dr2 r 2 r2 dt 2 ) + v 2 [ σ 2 1 + σ 2 2 + v 3 (σ 3 krdt) 2] A µ dx µ = q 1 rdt + q 2 (σ 3 krdt) σ 1 = cos ψdθ + sin ψ sin θdϕ, σ 2 = sin ψdθ + cos ψ sin θdϕ, σ 3 = dψ + cos θdϕ Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 12 / 35
Asymptotically Flat EMCS Black Holes. Near Horizon Solutions D = 5 Einstein-Maxwell Theory angular momentum area relation for EM black holes vs scaled charge 1.0 0.8 MP branch, 5D RN branch, 5D MP branch, 7D RN branch, 7D MP branch, 9D RN branch, 9D MP branch, 11D RN branch, 11D equal magnitude angular momenta EM in D dimensions 2 solutions 1st solution: A H /J 0.6 J = 2(D 3)A H 0.4 0.2 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Q/J (D-3)/(D-2) MP branch 2nd solution: J = (D 1)J H RN branch Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 12 / 35
Asymptotically Flat EMCS Black Holes. Near Horizon Solutions D = 5 Einstein-Maxwell Theory angular momentum area relation for EM black holes vs scaled charge 1.00 0.75 MP branch, 5D RN branch, 5D MP branch, 7D RN branch, 7D MP branch, 9D RN branch, 9D MP branch, 11D RN branch, 11D equal magnitude angular momenta EM in D dimensions 2 solutions 1st solution: J H /J 0.50 J = 2(D 3)A H 0.25 0.00 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Q/J (D-3)/(D-2) MP branch 2nd solution: J = (D 1)J H RN branch Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 12 / 35
Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory. Near Horizon and Global Solutions A H Q scaled area of EM black holes vs scaled charge M (D 2)/(D 3) M A H /M (D-2)/(D-3) 0.05 0.04 0.03 0.02 0.01 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Q/M Extremal D=5 Extremal D=7 Extremal D=9 Static D=5 Static D=7 Static D=9 equal magnitude angular momenta D dimensions globally realized: MP branch: small Q/M RN branch: large Q/M switch: matching point Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 13 / 35
Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory. Near Horizon and Global Solutions J scaled angular momenta of EM black holes vs scaled charge M (D 2)/(D 3) J/M (D-2)/(D-3) 0.08 0.06 0.04 0.02 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Q/M Extremal D=5 Extremal D=7 Extremal D=9 Static D=5 Static D=7 Static D=9 equal magnitude angular momenta D dimensions globally realized: MP branch: small Q/M RN branch: large Q/M switch: matching point Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 13 / 35
Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory. Near Horizon and Global Solutions D = 5 extremal black holes in EM theory: λ = 0 J /M 3/2 J 1 = J 2, λ=0 0.40 λ=0 0.30 0.20 0.10 0.00-1.0-0.5 0.0 0.5 1.0 3/2 Q/M boundary of domain of existence non-extremal bhs inside naked singularities outside J = 0: static black holes RN black holes extremal RN on the boundary λ = 0 symmetry q q Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 13 / 35
. Outline Asymptotically Flat EMCS Black Holes D = 5 minimal supergravity. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 14 / 35
Asymptotically Flat EMCS Black Holes D = 5 minimal supergravity. Einstein-Maxwell-Chern-Simons Black Holes: λ = 1 Breckenridge, Myers, Peet and Vafa (1997) Chong, Cveti c, Lü and Pope (2005) extremal λ = 1 EMCS black holes: J /M 3/2 0.40 0.30 0.20 0.10 J 1 = J 2, λ=0, 1 λ=0 λ=1 0.00-1.0-0.5 0.0 0.5 1.0 3/2 Q/M boundary of domain of existence non-extremal bhs inside naked singularities outside J = 0: static black holes RN black holes extremal RN on the boundary λ 0 symmetry q q broken Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 15 / 35
Asymptotically Flat EMCS Black Holes D = 5 minimal supergravity. Einstein-Maxwell-Chern-Simons Black Holes: λ = 1 Breckenridge, Myers, Peet and Vafa (1997) Chong, Cveti c, Lü and Pope (2005) extremal λ = 1 EMCS black holes: J /M 3/2 0.40 0.30 0.20 0.10 J 1 = J 2, λ=0, 1 λ=0 λ=1 0.00-1.0-0.5 0.0 0.5 1.0 3/2 Q/M vertical wall BMPV solutions Q is kept fixed J increases M remains constant first law dm = T ds + Ω i dj i + ΦdQ extremal: T = 0 Q fixed: dq = 0 M fixed: dm = 0??? Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 15 / 35
Asymptotically Flat EMCS Black Holes D = 5 minimal supergravity. Einstein-Maxwell-Chern-Simons Black Holes: λ = 1 Breckenridge, Myers, Peet and Vafa (1997) Chong, Cveti c, Lü and Pope (2005) extremal λ = 1 EMCS black holes: J /M 3/2 0.40 0.30 0.20 0.10 J 1 = J 2, λ=0, 1 λ=0 λ=1 0.00-1.0-0.5 0.0 0.5 1.0 3/2 Q/M horizon angular velocites vanish: Ω i = 0, J = 0 angular momentum is stored in the Maxwell field negative fraction of the angular momentum is stored behind the horizon the effect of rotation is to deform the horizon into a squashed 3-sphere Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 15 / 35
. Outline Asymptotically Flat EMCS Black Holes D = 5 EMCS theory: λ 1. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 16 / 35
Asymptotically Flat EMCS Black Holes D = 5 EMCS theory: λ 1. Einstein-Maxwell-Chern-Simons Black Holes: λ > 1 J.K., Navarro-Lérida (2006) 0.40 0.30 λ=1.5 λ=1 J 1 = J 2 Ω=0 0.40 0.30 λ=2 J 1 = J 2, λ=0, 1, 2 λ=1 Ω=0 J /M 3/2 0.20 0.10 λ=0 J /M 3/2 0.20 0.10 λ=0 0.00-1.5-1.0-0.5 0.0 0.5 1.0 3/2 Q/M black holes with Ω = 0, J 0 black holes with Ω < 0, J > 0 0.00-1.5-1.0-0.5 0.0 0.5 1.0 3/2 Q/M non-extremal counter-rotating (shaded area) Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 17 / 35
Asymptotically Flat EMCS Black Holes D = 5 EMCS theory: λ 1. Einstein-Maxwell-Chern-Simons Black Holes: λ > 1 λ > 2 EMCS black holes 0.3 0.2 r H =0.1 r H =0.5 J 1 = J 2, λ=3, Q=-1 non-extremal black holes black holes are not uniquely determined by M, J i, Q J 0.1 0.0-0.1-0.2-0.3 0.5 0.6 0.7 0.8 0.9 1.0 M non-uniqueness of 5D black holes with horizon topology of a sphere S 3 extremal black holes angular momentum vs mass Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 17 / 35
Asymptotically Flat EMCS Black Holes D = 5 EMCS theory: λ 1. Extremal EMCS Black Holes: λ > 2 j Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2014) 0.04 0.02 0-0.02-0.04 domain of existence: λ = 5 0.008 0.004 0.000-0.004 λ=5 extremal λ=5 extremal, static -0.008 0.56 0.58 0.60 extremal black holes: boundary of domain of existence inside the domain of existence many branches J = 0 non-static black holes sequence of pairs q ns > q s limit -0.6-0.3 0 0.3 0.6 0.9 1.2 q extremal RN: q s Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 18 / 35
Asymptotically Flat EMCS Black Holes D = 5 EMCS theory: λ 1. Extremal EMCS Black Holes: λ > 2 M 160 120 80 Q>0 global Q<0 global branches for fixed Q: λ = 5 C [1,2] C [1,2] C [3,4] C [3,4] (B [2,3], B * [2,3] ) (B [2,3], B* [2,3] ) n=3 n>6 (B [1,2], B * [1,2] ) n=2 (B [1,2], B* [1,2] ) n=1-50 0 50 100 J n=6 n=5 n=4 0 extremal black holes: J = 0 non-static black holes sequence of pairs labelled by integer n limit n : extremal RN branchpoints B and cusps C nonuniqueness Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 19 / 35
Asymptotically Flat EMCS Black Holes D = 5 EMCS theory: λ 1. Extremal EMCS Black Holes: λ > 2 A H 45 30 15 horizon area: λ = 5, Q fixed C nh Q>0 global Q<0 global C [1,2] C [1,2] C [3,4] C [3,4] B [1,2] B [1,2] n=1, 2,... B * [1,2] B * [1,2] Q>0 near horizon Q<0 near horizon 0-100 -50 0 50 100 J C nh extremal black holes: J = 0 non-static black holes same area extremal RN isolated near horizon vs global solutions zero global bh one global bh many global bh Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 19 / 35
Asymptotically Flat EMCS Black Holes D = 5 EMCS theory: λ 1. Extremal EMCS Black Holes: λ > 2 how do the solutions on these branches differ? what is the origin of the branch structure? Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 20 / 35
Asymptotically Flat EMCS Black Holes D = 5 EMCS theory: λ 1. Radial Excitations of Extremal EMCS Black Holes 0.50 gauge field function a φ n=7 n=6 n=5 n=4 n=3 n=2 0.25 0.00 a ϕ -0.25-0.50-0.75 λ=5 n=1-2 -1 0 1 1/3 log 10 (r/a H ) Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 21 / 35
Asymptotically Flat EMCS Black Holes D = 5 EMCS theory: λ 1. Radial Excitations of Extremal EMCS Black Holes 0 metric function ω -2-4 log 10 ( ω ) -6 1 node -8 2 nodes 3 nodes 4 nodes 5 nodes 6 nodes -10 7 nodes 8 nodes 9 nodes 10 nodes -12-3 -2.5-2 -1.5-1 -0.5 0 log 10 (x) Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 21 / 35
. Outline EMCS Solutions with AdS Asymptotics. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 22 / 35
EMCS Solutions with AdS Asymptotics. Anti-de Sitter space Jutta Kunz (Universita t Oldenburg) EMCS Black Holes Frankfurt, July 2017 22 / 35
EMCS Solutions with AdS Asymptotics. Adding a negative cosmological constant Λ D = 5 Einstein-Maxwell-Chern-Simons action 1 { ( g S = R + 12 ) 16πG 5 L 2 F µνf µν 2λ } 3 3 εmnpqr A m F np F qr d 5 x cosmological constant Λ = 6/L 2 Chern-Simons coupling constant λ λ = 0: λ = 1: Einstein-Maxwell theory bosonic sector of D = 5 gauged supergravity λ 1 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 23 / 35
. Outline EMCS Solutions with AdS Asymptotics Charged solutions. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 24 / 35
EMCS Solutions with AdS Asymptotics Charged solutions. Charged solutions in gauged supergravity: λ = 1 Chong, Cveti c, Lü and Pope (2005): trial and error 2 independent angular momenta, g = 1/L Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 25 / 35
EMCS Solutions with AdS Asymptotics Charged solutions. Charged EMCS solutions: λ 1 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2017) small λ global (non-extremal) J H global (extremal) + near horizon P 2 Gap near horizon P 1 0 0 horizon angular momentum J H vs angular momentum J J Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 26 / 35
EMCS Solutions with AdS Asymptotics Charged solutions. Charged EMCS solutions: λ 1 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2017) small λ 0.004 Near-horizon, λ=0.1 Global, λ=0.1 Near-horizon, λ=0.025 Global, λ=0.025 Near-horizon, EM-AdS Global, EM-AdS Q>0 0.004 Near-horizon, λ=0.5 Global, λ=0.5 Near-horizon, λ=0.1 Global, λ=0.1 Near-horizon, EM-AdS Global, EM-AdS Q<0 J H 0.002 J H 0.002 Gap Gap 0 0 0 0.004 0.008 0 0.004 0.008 J J Q > 0 Q < 0 dots, triangles: limiting solutions, squares: critical solutions A H = 0 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 26 / 35
EMCS Solutions with AdS Asymptotics Charged solutions. Charged EMCS solutions: λ 1 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2017) large λ 0.2 n=1 n=2 n=3 n=4 n=5 n=6 3 0 AdS boundary a ϕ 0 F 2 horizon -3-0.2-0.4 horizon 0 0.25 0.5 0.75 1 r/(1+r) AdS boundary -6 n=1 n=2 n=3 n=4 n=5 n=6 RN-AdS 0 0.5 1 r/(1+r) Q < 0 Q < 0 magnetic gauge potential a φ invariant F 2 = F µν µν Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 26 / 35
EMCS Solutions with AdS Asymptotics Charged solutions. Charged EMCS solutions: λ 1 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2017) horizon area A H vs charge Q: temperature T H plot Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 26 / 35
. Outline EMCS Solutions with AdS Asymptotics Magnetized solutions. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 27 / 35
EMCS Solutions with AdS Asymptotics Magnetized solutions. Static purely magnetic solutions: λ = 0 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2016) r : a φ c m Φ m = 1 F = 1 4π 2 c m S 2 100 3 0 D=7 D=5 50 D=7-3 D=9 M µ 0 D=5-6 -50-9 D=9-100 0 1 2 3 c m -12 0 1 2 3 c m mass M vs flux parameter c m magnetic moment µ vs c m Solitons Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 28 / 35
EMCS Solutions with AdS Asymptotics Magnetized solutions. Static purely magnetic solutions: λ = 0 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2016) Black holes in D = 5 0.5 0 soliton limit 1 c m =0 c m =1.5 c m =2 c m =2.5 c m =3 Type I Type II M A H 0.5-0.5 c m =0 c m =1.5 c m =2 singular limit c m =2.5 c m =3 Type I Type II -1 0 0.5 1 T H singular limit soliton limit 0 0 0.25 0.5 0.75 T H mass M vs temperature T H horizon area A H vs T H type I: M, A H increase with T H type II: M, A H decrease with T H type I: for all c m type II: for c m < c m Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 28 / 35
EMCS Solutions with AdS Asymptotics Magnetized solutions. Static purely magnetic solutions: λ = 0 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2016) 10 domain of existence: D = 5 Type I black holes 0 M Type II black holes -10 singular limit solitons -20 0 1 2 3 c m Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 28 / 35
EMCS Solutions with AdS Asymptotics Magnetized solutions. Rotating magnetized solutions: λ = 1 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2017) solitons: mass M, angular momentum J and charge Q (R) vs c m 40 30 20 10 0 L=1 3Q (R) M -J c m single parameter no upper bound on c m no radial excitations for λ = 1 radial excitations for λ > 1? 0 1 2 3 4 5 c m 1 3 Q = Q(R) = 1 2 S 3 ( F + 2λ ) 3 3 A F, J = λπ 3 3 c3 m = Φ m Q (R) Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 29 / 35
EMCS Solutions with AdS Asymptotics Magnetized solutions. Rotating magnetized solutions: λ = 1 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2017) black holes: mass M vs temperature T H M 1 0.8 0.6 0.4 0.2 c m = 0 c m = 0.2 c m = -0.2 c m = 0.33 c m = -0.33 c m = 0.66 c m = -0.66 3 parameters: J, Q (R), c m c m 0: relevant for small T H only large c m : black holes thermodynamically stable rotating J = 0 black holes 0 0 0.2 0.4 0.6 0.8 T H J = 0.003, Q (R) = 0.044 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 29 / 35
EMCS Solutions with AdS Asymptotics Magnetized solutions. Rotating magnetized solutions: λ = 1 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2017) black holes: area A H vs temperature T H A H 100 10 1 0.1 c m = 0 c m = 0.2 c m = -0.2 c m = 0.33 c m = -0.33 c m = 0.66 c m = -0.66 3 parameters: J, Q (R), c m c m 0: relevant for small T H only large c m : black holes thermodynamically stable rotating J = 0 black holes 0 0.2 0.4 0.6 0.8 T H J = 0.003, Q (R) = 0.044 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 29 / 35
EMCS Solutions with AdS Asymptotics Magnetized solutions. Rotating magnetized solutions: λ = 1 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2017) T H, M, c m T H, c m, A H stationary J = 0 black holes L = 1, Q (R) = 0.044 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 29 / 35
EMCS Solutions with AdS Asymptotics Magnetized solutions. Rotating magnetized solutions: λ = 1 Blázquez-Salcedo, J.K., Navarro-Lérida, Radu (2017) T H, Ω H, c m T H, R(r H ), c m stationary J = 0 black holes L = 1, Q (R) = 0.044 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 29 / 35
. Outline EMCS Solutions with AdS Asymptotics Magnetized squashed solutions. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 30 / 35
EMCS Solutions with AdS Asymptotics. Squashed solitons: λ = 1 Cassani and Martelli (2014) Magnetized squashed solutions boundary metric: squashed S 3 sphere ds 2 (bdry) = L2 dω 2 (v) dt2, dω 2 (v) = 1 ( dθ 2 + sin 2 θdϕ 2 + (dψ + v cos θdϕ) 2) 4 v: control parameter M 10 3 10 2 10 1 10 0 10-1 susy black holes susy solitons -1 J 0 1 0 Gutowski-Reall black holes 2.5 Q squashed susy solitons ( 5 M = πl 2 288 + 2 27v 2 7 36 v2 + 89 ) 864 v4 5 J = πl3 27 Q = 2πL2 9 3 ( v 2 1 ) 3 ( v 2 1 ) 2 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 31 / 35
EMCS Solutions with AdS Asymptotics Magnetized squashed solutions. Magnetized squashed solutions: λ = 1 work in progress boundary metric: squashed S 3 sphere ds 2 (bdry) = L2 dω 2 (v) dt2, dω 2 (v) = 1 ( dθ 2 + sin 2 θdϕ 2 + (d 4 ψ + v cos θdϕ) 2) v: control parameter M 10 3 10 2 10 1 10 0 10-1 susy black holes susy solitons -1 J 0 1 0 Gutowski-Reall black holes 2.5 Q 5 nonextremal black holes event horizon of spherical topology, no pathologies characterization: M, Q, J, c m r H 0: squashed spinning charged solitons J = Φ m Q. Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 32 / 35
EMCS Solutions with AdS Asymptotics Magnetized squashed solutions. Magnetized squashed solutions: λ = 1 work in progress boundary metric: squashed S 3 sphere ds 2 (bdry) = L2 dω 2 (v) dt2, dω 2 (v) = 1 ( dθ 2 + sin 2 θdϕ 2 + (d 4 ψ + v cos θdϕ) 2) v: control parameter 3 1.2 log 10 A H 1.5 0 J=1.07, Q=3.49 J=0.8, Q=2.3 J=0.8, Q=1.21 J=0.6, Q=1.21 J=0.61, Q=1.21 susy black hole log 10 M 0.6 J=1.07, Q=3.49 J=0.8, Q=2.3 J=0.8, Q=1.21 J=0.6, Q=1.21 J=0.61, Q=1.21 susy black hole susy solitons susy solitons -1.5 0 0 0.5 1 0 0.5 1 T H T H black holes v = 1.65, c = 1, L = 1 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 32 / 35
EMCS Solutions with AdS Asymptotics Magnetized squashed solutions. Magnetized squashed solutions: λ = 1 work in progress boundary metric: squashed S 3 sphere ds 2 (bdry) = L2 dω 2 (v) dt2, dω 2 (v) = 1 ( dθ 2 + sin 2 θdϕ 2 + (d 4 ψ + v cos θdϕ) 2) v: control parameter 2 J=1.07, Q=3.49 J=0.8, Q=2.3 J=0.8, Q=1.21 J=0.6, Q=1.21 J=0.61, Q=1.21 susy black hole 1.6 Ω H 0 susy solitons ε 1.3 J=1.07, Q=3.49 J=0.8, Q=2.3 J=0.8, Q=1.21 J=0.6, Q=1.21 J=0.61, Q=1.21 susy black hole susy solitons -2 1 0 0.5 1 0 0.5 1 T H T H black holes v = 1.65, c = 1, L = 1 Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 32 / 35
EMCS Solutions with AdS Asymptotics Magnetized squashed solutions. Magnetized squashed solutions: λ = 1 work in progress boundary metric: squashed S 3 sphere ) 1( 2 ds2(bdry) = L2 dω2(v) dt2, dω2(v) = dθ + sin2 θdϕ2 + (dψ + v cos θdϕ)2 4 v: control parameter black holes Jutta Kunz (Universita t Oldenburg) EMCS Black Holes Frankfurt, July 2017 32 / 35
EMCS Solutions with AdS Asymptotics Magnetized squashed solutions. Magnetized squashed solutions: λ = 1 work in progress boundary metric: squashed S 3 sphere ds 2 (bdry) = L2 dω 2 (v) dt2, dω 2 (v) = 1 ( dθ 2 + sin 2 θdϕ 2 + (d 4 ψ + v cos θdϕ) 2) v: control parameter black holes Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 32 / 35
EMCS Solutions with AdS Asymptotics Magnetized squashed solutions. Magnetized squashed solutions: λ = 1 work in progress boundary metric: squashed S 3 sphere ds 2 (bdry) = L2 dω 2 (v) dt2, dω 2 (v) = 1 ( dθ 2 + sin 2 θdϕ 2 + (d 4 ψ + v cos θdϕ) 2) v: control parameter 10 3 susy black holes 10 2 Gutkowski-Real susy black holes (2004) round sphere: v = 1 no magnetization: c m = 0 M 10 1 10 0 10-1 susy solitons -1 J 0 1 0 Gutowski-Reall black holes 2.5 Q 5 M = πl2 216 (3α2 1)(31 + 76α 2 + 64α 4 ) + C J = πl3 216 (1 4α2 ) 2 (7 + 8α 2 ) Q = πl2 12 3 (4α2 1)(5 + 4α 2 ) Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 32 / 35
EMCS Solutions with AdS Asymptotics Magnetized squashed solutions. Magnetized squashed solutions: λ = 1 work in progress boundary metric: squashed S 3 sphere ds 2 (bdry) = L2 dω 2 (v) dt2, dω 2 (v) = 1 ( dθ 2 + sin 2 θdϕ 2 + (d 4 ψ + v cos θdϕ) 2) v: control parameter 10 3 susy black holes 10 2 10 1 Gutowski-Reall black holes M susy solitons 10 0 10-1 -1 J 0 2.5 Q 1 0 new susy black holes ( 7913 M = πl 2 34848 + 33280 35937v 2 7 36 v2 + 89 ) 864 v4 ( 16640 J = πl 3 35937 2795 8712 v2 + 1 9 v4 1 ) 27 v6 5 Q = π 3L 2 1 13068 ( 6449 1936 v 2 + 968 v 4) A H = 7π 2 L 3 455 121 c m = ± L 3 ( 1 v 2 ) Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 32 / 35
. Outline Conclusion and Outlook. 1 Introduction. 2 Asymptotically Flat EMCS Black Holes D = 5 Einstein-Maxwell Theory D = 5 minimal supergravity D = 5 EMCS theory: λ 1. 3 EMCS Solutions with AdS Asymptotics Charged solutions Magnetized solutions Magnetized squashed solutions. 4 Conclusion and Outlook Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 33 / 35
. Conclusions Conclusion and Outlook Einstein-Maxwell-Chern-Simons black holes: D = 5 M 160 120 80 Q>0 global Q<0 global C [1,2] C [1,2] C [3,4] C [3,4] (B [2,3], B * [2,3] ) (B [2,3], B* [2,3] ) n=3 n>6 (B [1,2], B * [1,2] ) n=2 (B [1,2], B* [1,2] ) n=1-50 0 50 100 J mass vs angular momentum n=6 n=5 n=4 0 black holes with surprising properties non-uniqueness Ω = 0, J 0 non-static J = 0 sequences of radially excited extremal black holes... questions in which theories? in D = 4? consequences?... Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 33 / 35
. Conclusions Conclusion and Outlook Einstein-Maxwell-Chern-Simons AdS solutions: D = 5 10 black holes and solitons c m = 0: many properties retained c m 0: new solitons M 0-10 Type I black holes Type II black holes singular limit solitons -20 0 1 2 3 c m mass vs magnetic flux new black holes J = Φ mq squashed magnetized solutions new solitons new black holes... questions domain of existence?... Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 33 / 35
Conclusion and Outlook. Higher Dimensions: D 6 Dias, Figueras, Monteiro, Santos, Emparan (2009) unstable modes of Myers-Perry black holes: D 6 Jutta Kunz (Universita t Oldenburg) EMCS Black Holes Frankfurt, July 2017 34 / 35
Conclusion and Outlook. Higher Dimensions: D 6 Emparan, Figueras (2010) D 6 horizon area vs. angular momentum at fixed mass Jutta Kunz (Universita t Oldenburg) EMCS Black Holes Frankfurt, July 2017 34 / 35
. THANKS Conclusion and Outlook Jose Francisco Eugen Blázquez-Salcedo Navarro-Lérida Radu Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 35 / 35
. THANKS Conclusion and Outlook Thank you very much for your attention Jutta Kunz (Universität Oldenburg) EMCS Black Holes Frankfurt, July 2017 35 / 35