Black Holes in Four and More Dimensions
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1 Black Holes in Four and More Dimensions Jutta Kunz Institute of Physics CvO University Oldenburg International Workshop on In-Medium Effects in Hadronic and Partonic Systems Obergurgl, Austria, February 21-25, 2011 Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
2 Outline Outline 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
3 Outline Outline 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
4 Outline Outline 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
5 Outline Outline 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
6 Outline From Rotating Nuclei to Rotating Black Holes 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
7 Outline From Rotating Nuclei to Rotating Black Holes Rotating Nuclei 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
8 The Institute From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
9 The Institute From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
10 The Institute From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
11 The Institute From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
12 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Collective Rotation of Heavy Nuclei How does a nucleus rotate? Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
13 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Collective Rotation of Heavy Nuclei There are 3 groups worldwide trying to answer this question. USA New Zealand Gießen Gießen (YOU!) should be first! Preliminary answer USA: Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
14 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Collective Rotation of Heavy Nuclei Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
15 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Collective Rotation of Heavy Nuclei Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
16 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Collective Rotation of Heavy Nuclei Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
17 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Collective Rotation of Heavy Nuclei Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
18 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Collective Motion of Heavy Nuclei Test of Nuclear Currents in Transverse Electron Scattering, E. Wüst, U. Mosel, J. Kunz, H.G. Andresen and M. Müller, Nucl. Phys. A402 (1983) 235 Rotational Motion at Finite Temperature, J. Kunz and U. Mosel, Nucl. Phys. A406 (1983) 269 Test of Vibrational Current Distributions in Transverse Electron Scattering, E. Wüst, U. Mosel, J. Kunz and A. Schuh, Nucl. Phys. A406 (1983) 285 Flow Patterns for Collective Quadrupole Vibrations in Heavy Nuclei, A. Schuh, J. Kunz and U. Mosel, Nucl. Phys. A412 (1984) 34 The Generator of the Low-Lying Quadrupole Vibration, T. Reitz, J. Kunz, U. Mosel and E. Wüst, Nucl. Phys. A456 (1986) 1 Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
19 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Collective Motion of Heavy Nuclei Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
20 Outline From Rotating Nuclei to Rotating Black Holes Rotating Black Holes 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
21 Flat Space Time From Rotating Nuclei to Rotating Black Holes Rotating Black Holes metric of Minkowski space-time ds 2 = dt 2 + dx 2 + dy 2 + dz 2 Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
22 From Rotating Nuclei to Rotating Black Holes Curved Space Time Rotating Black Holes metric of curved space-time ds 2 = g µν dx µ dx ν Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
23 From Rotating Nuclei to Rotating Black Holes Einstein Equations Rotating Black Holes Einstein equations: matter tells space how to curve G µν = 8πG c 4 T µν Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
24 From Rotating Nuclei to Rotating Black Holes Schwarzschild Space-Time Rotating Black Holes Schwarzschild 1916 Schwarzschild space-time: D = 4 ds 2 = N(r)dt N(r) dr2 +r 2 dω 2 2 Karl Schwarzschild N(r) = 1 2M r static spherically symmetric black hole with mass M Schwarzschild radius r H N(r H) = 0 : r H = 2M event horizon: sphere S 2 Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
25 From Rotating Nuclei to Rotating Black Holes Rotating Black Holes Kerr Space-Time: Rotating Black Holes Kerr 1965 rotating generalizations of Schwarzschild black holes: astrophysical black holes Kerr metric in Boyer Lindquist coordinates dt dϕ cross term: rotation Roy Kerr *1934 ds 2 = ρ 2 ( dt a sin 2 θdφ ) 2 + sin 2 θ ρ 2 ( adt ρ 2 0 dφ ) 2 + ρ 2 dr2 + ρ 2 dθ 2 ρ 2 = r 2 + a 2 cos 2 θ, ρ 2 0 = r2 + a 2, = r 2 2Mr + a 2 specific angular momentum: a = J M a = 0: Schwarzschild Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
26 From Rotating Nuclei to Rotating Black Holes Kerr Space-Time: Horizons Rotating Black Holes horizons = 1 2M r + a2 r 2 = 0 horizon radial coordinate r H r H = M ± M 2 a 2 a < M +: event horizon: spherical topology S 2 : inner horizon a = M: maximal a extremal black hole a > M: naked singularity (cosmic censorship) Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
27 From Rotating Nuclei to Rotating Black Holes Kerr Space-Time: Horizons Rotating Black Holes horizons = 1 2M r + a2 r 2 = 0 phase diagram horizon radial coordinate r H r H = M ± M 2 a 2 a < M +: event horizon: spherical topology S 2 : inner horizon a = M: maximal a extremal black hole a > M: naked singularity (cosmic censorship) Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
28 From Rotating Nuclei to Rotating Black Holes Kerr Space-Time: Rotation Rotating Black Holes effects of rotation the rotating space-time drags the orbital plane of a spacecraft along inside the ergosphere everything is inexorably dragged along Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
29 From Rotating Nuclei to Rotating Black Holes Rotating Black Holes Vacuum Black Holes: D = 4 Theorems static spherically symmetric rotating axially symmetric Schwarzschild Kerr M M, J uniqueness black holes are uniquely determined by their global charges Israel s theorem static black holes are spherically symmetric staticity stationary black holes with non-rotating horizon are static horizon topology black hole horizons have spherical topology etc Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
30 From Rotating Nuclei to Rotating Black Holes Rotating Black Holes Black Holes with Matter Fields? standard model QED: photons Abelian gauge field QCD: gluons a = 1,..., 8 WS: W ±, Z 0 QCD: gluons a = 1,..., 8 non-abelian gauge fields non-linearity in field strength tensor Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
31 From Rotating Nuclei to Rotating Black Holes Rotating Black Holes Black Holes with Matter Fields? Subatomic particle zoo: BOSON 5-PACK: Higgs Boson, Z Boson, W Boson, Gluon, Photon what are the consequences of the presence of such fields? Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
32 From Rotating Nuclei to Rotating Black Holes Rotating Black Holes Black Holes in Higher Dimensions? Theodor Kaluza 1921, Oskar Klein 1926 unification of forces: gravity and electromagnetism in 5 dimensions Theodor Kaluza th dimension is small and therefore not observable String Theory Oskar Klein Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
33 Outline Black Holes in Four Dimensions 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
34 Outline Black Holes in Four Dimensions Black Holes with Gauge Fields 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
35 Black Holes in Four Dimensions Black Holes with Gauge Fields Einstein Maxwell Black Holes: D = 4 Theorems 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 global charges Israel s theorem static black holes are spherically symmetric staticity stationary black holes with non-rotating horizon are static horizon topology black hole horizons have spherical topology Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
36 Black Holes in Four Dimensions Uniqueness: Counterexamples Black Holes with Gauge Fields Volkov, Gal tsov 1989, et al. static spherically symmetric Einstein-Yang-Mills black holes: mass M counterexample: EYM Einstein-Yang-Mills n = 0: Schwarzschild 0 < n < : EYM n = : extremal RN no uniqueness further counterexamples Einstein-Yang-Mills- Higgs (doublet) Einstein-Yang-Mills- Higgs (triplet) etc. Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
37 Black Holes in Four Dimensions Black Holes with Gauge Fields Israel s Theorem: Counterexamples Kleihaus, Kunz 1997 static axially symmetric asymptotically flat no uniqueness deformed horizon EYM, EYMH, ES, etc. ǫ = ǫ = ǫ = ǫ = Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
38 Black Holes in Four Dimensions Black Holes with Gauge Fields Israel s Theorem: Counterexamples Ridgway, Weinberg 1995 black holes with only discrete symmetries? perturbative solutions non-perturbative solutions? platonic black holes? Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
39 Outline Black Holes in Four Dimensions Black Holes with Dilatons 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
40 Black Holes in Four Dimensions Staticity: Counterexamples Black Holes with Dilatons Rasheed 1995 Einstein-Maxwell-dilaton black holes: γ = 3 (Kaluza-Klein) J /M S vertical wall W: stationary Ω = 0 solutions ( ) 2 ( ) 2 P 3 Q = 2 3 M M 0.2 W J PQ Q /M P /M surfaces of extremal solutions in Kaluza-Klein theory J increases, M = const Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
41 Black Holes in Four Dimensions Staticity: Counterexamples Black Holes with Dilatons Rasheed 1995 Einstein-Maxwell-dilaton black holes: γ = 3 (Kaluza-Klein) J /M S W Q /M P /M surfaces of extremal solutions in Kaluza-Klein theory a negative fraction of J resides behind the horizon: J H < 0 frame dragging effects allow for a vanishing horizon angular velocity Ω = 0 effect of rotation on the horizon: prolate deformation Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
42 Black Holes in Four Dimensions Staticity: Counterexamples Black Holes with Dilatons Kleihaus, Kunz, Navarro-Lérida 2004 J /M 2 P = Q γ=0 0.6 γ 2 =3 0.4 Ω= P /M what is the shaded region? γ = 0: EM γ = 3: KK γ > 3: EMD? stationary: Ω = 0 γ > 3 extremal: P = Q stationary: Ω = 0, J = PQ Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
43 Black Holes in Four Dimensions Staticity: Counterexamples Black Holes with Dilatons Kleihaus, Kunz, Navarro-Lérida 2004 J /M 2 P = Q γ=0 γ 2 γ=3 = Ω= P /M what is the shaded region? γ = 0: EM γ = 3: KK γ > 3: EMD stationary: Ω = 0 γ > 3 extremal: P = Q stationary: Ω = 0, J = PQ Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
44 Black Holes in Four Dimensions Staticity: Counterexamples Black Holes with Dilatons Kleihaus, Kunz, Navarro-Lérida 2004 J /M 2 P = Q γ=0 γ 2 γ=3 = Ω= P /M counterrotating black holes! γ = 0: EM γ = 3: KK γ > 3: EMD stationary: Ω = 0 γ > 3 extremal: P = Q stationary: Ω = 0, J = PQ Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
45 Black Holes in Four Dimensions Black Holes with Dilatons Einstein-Gauß-Bonnet-Dilaton Black Holes Kleihaus, Kunz, Radu 2011 String Theory: Modified Einstein equations with higher curvature terms Einstein-Gauß-Bonnet-dilaton theory s/ Ω H α 1/2 Kerr J/M 2 Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
46 Real Black Holes Black Holes in Four Dimensions Black Holes with Dilatons Genzel et al Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
47 Real Black Holes Black Holes in Four Dimensions Black Holes with Dilatons Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
48 Outline Black Holes in Higher Dimensions 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
49 Outline Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
50 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes 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 ) uniqueness? are vacuum black holes uniquely determined by their mass and angular momenta? horizon topology? does the horizon necessarily have the topology of a sphere? Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
51 Black Holes in Higher Dimensions Static Vacuum Black Holes: D 4 Generalization of D = 4 Vacuum Black Holes Schwarzschild space-time: D = 4 ds 2 = N(r)dt N(r) dr2 + r 2 dω 2 2 ( rh ) N(r) = 1 r dω 2 2 = dθ2 + r 2 sin 2 θdφ 2 Tangherlini space-time: D > 4 ds 2 = N(r)dt N(r) dr2 + r 2 dω 2 D 2 ( rh ) D 3 N(r) = 1 r dωd 2 2 : metric on the unit SD 2 sphere Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
52 Black Holes in Higher Dimensions Myers-Perry Black Holes Generalization of D = 4 Vacuum 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, x 4 ) θ (x 1, x 2 ) example: D = 5, N = 2 Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
53 Black Holes in Higher Dimensions Myers-Perry Black Holes Generalization of D = 4 Vacuum 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, x 4 ) θ (x 1, x 2 ) x 2 φ 1 example: D = 5, N = 2 x 1 Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
54 Black Holes in Higher Dimensions Myers-Perry Black Holes Generalization of D = 4 Vacuum 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 example: D = 5, N = 2 x 1 Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
55 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Myers-Perry Black Holes: Domain of Existence D = 5: domain of existence is bounded j 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) Black Holes in Four and More Dimensions Obergurgl, / 54
56 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Myers-Perry Black Holes: Domain of Existence D = 5: domain of existence is bounded j 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) Black Holes in Four and More Dimensions Obergurgl, / 54
57 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Myers-Perry Black Holes: Domain of Existence single angular momentum J 1 = J (J i = 0, i > 1) horizon area A H versus angular momentum J Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
58 Outline Black Holes in Higher Dimensions Black Holes with Maxwell Fields 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
59 Black Holes in Higher Dimensions Black Holes with Maxwell Fields Einstein-Maxwell-Chern-Simons Black Holes Chong, Cveti c, Lü and Pope 2005 Einstein-Maxwell-Chern-Simons AF 2 : λ = 1 2 independent angular momenta q 5D supergravity j j 2 Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
60 Black Holes in Higher Dimensions Black Holes with Maxwell Fields Einstein-Maxwell-Chern-Simons Black Holes Kunz and Navarro-Lérida 2006 Einstein-Maxwell-Chern-Simons AF 2 : λ > 1 equal angular momenta J 1 = J 2, λ=0, 1, Ω=0 λ= λ=2 J /M 3/ λ= /2 Q/M black holes with Ω = 0, J 0: non-extremal black holes with Ω < 0, J > 0: counter-rotating black holes Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
61 Black Holes in Higher Dimensions Black Holes with Maxwell Fields Einstein-Maxwell-Chern-Simons Black Holes Kunz and Navarro-Lérida 2006 Einstein-Maxwell-Chern-Simons AF 2 : λ > 1 equal angular momenta 0.4 J 1 =J 2, λ=3, Q=-1 J/M 3/ r H =0.1 r H =0.5 Tangh /2 Q/M non-uniqueness! black holes not uniquely determined by M, J i, Q Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
62 Outline Black Holes in Higher Dimensions Black Strings and Black Rings 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
63 Black Strings Black Holes in Higher Dimensions Black Strings and Black Rings non-asymptotically flat black holes in D dimensions black string metric take Schwarzschild metric in D 1 dimensions: ds 2 D 1 add one extra dimension with coordinate z ds 2 = ds 2 D 1 + dz 2 translation invariance w.r.t. z horizon topology R S D 3 uniform black string string tension Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
64 Black Rings? Black Holes in Higher Dimensions Black Strings and Black Rings Myers, Perry 1986 construction of black rings: 1. take a piece of black string 2. bend it 3. glue endpoints Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
65 Black Holes in Higher Dimensions Black Rings in D = 5 Black Strings and Black Rings Emparan and Reall 2002 black ring horizon topology S 1 S 2 static ring attraction: gravity/string tension shrink ring repulsion: conical singularity inside: push outside: pull unbalanced ring attraction: gravity/string tension repulsion: rotation along S 1 centrifugal force balanced ring Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
66 Black Holes in Higher Dimensions Black Rings in D = 5 Black Strings and Black Rings Emparan and Reall 2002 phase diagram black holes S 3 maximal J black rings S 1 S 2 minimal J horizon area A H vs. angular momentum J nonuniqueness region with MP black holes thick black rings thin black rings Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
67 Black Holes in Higher Dimensions Black Strings and Black Rings From the Grey Ring to the Black Ring The grey ring: Heinrich-Buff-Ring Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
68 Black Holes in Higher Dimensions Black Strings and Black Rings From the Grey Ring to the Black Ring The black ring: Einstein-Maxwell theory 1 Analytical curve b=3.0 b=5.0 b=8.0 b= q=0 q=1.05 A H J 2 Kleihaus, Kunz, Schnülle 2011 Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
69 Black Holes in Higher Dimensions Black Saturn in D = 5 Black Strings and Black Rings Elvang and Figueras 2007 composite systems Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
70 Black Holes in Higher Dimensions Black Saturn in D = 5 Black Strings and Black Rings Elvang and Figueras 2007 composite systems corotation both angular velocities have the same sign counterrotation the angular velocities have different signs frame dragging e.g. J BH = 0 with Ω BH 0 or J BH 0, Ω BH = 0 Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
71 Black Holes in Higher Dimensions Black Saturn in D = 5 Black Strings and Black Rings Elvang and Figueras 2007 phase diagram: equilibrium (T, Ω) A H further nonuniqueness J Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
72 Black Holes in Higher Dimensions And More Species in D = 5 Black Strings and Black Rings dirings bicycling rings etc.... Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
73 Species in D 6? Black Holes in Higher Dimensions Black Strings and Black Rings Emparan, Harmark, Niarchos, Obers, Rodriguez 2007 perturbative calculations for thin rings Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
74 Outline Black Holes in Higher Dimensions Black Strings and Caged Black Holes 1 From Rotating Nuclei to Rotating Black Holes Rotating Nuclei Rotating Black Holes 2 Black Holes in Four Dimensions with Matter Fields Black Holes with Gauge Fields Black Holes with Dilatons 3 Black Holes in Higher Dimensions Generalization of D = 4 Vacuum Black Holes Black Holes with Maxwell Fields Black Strings and Black Rings Black Strings and Caged Black Holes 4 Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
75 Black Holes in Higher Dimensions Uniform Black Strings Black Strings and Caged Black Holes Gregory, Laflamme 1993 piece of uniform black string thick: stable 2 length scales Schwarzschild horizon r H compact dimension L possiblities thick: r H > L thin: r H < L thin: unstable Gregory-Laflamme instability small mass critical length what happens at M crit? Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
76 Black Holes in Higher Dimensions Nonuniform Black Strings Black Strings and Caged Black Holes no translation invariance w.r.t. z: dependence on the compact coordinate z horizon topology S D 3 S 1 existence above the Gregory-Laflamme instability perturbative solutions Gubser, Kol, Sorkin numerical solutions Wiseman; Kleihaus, Kunz, Radu; Sorkin Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
77 Caged Black Holes Black Holes in Higher Dimensions Black Strings and Caged Black Holes Sorkin, Kol, Piran, Harmark; Kudoh, Wiseman horizon topology S D 2 maximal mass bounded by the size of the compact dimension large masses deformation of the horizon small masses Schwarzschild Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
78 Black Holes in Higher Dimensions Black Strings and Caged Black Holes Black Strings Caged Black Holes 3 D = BH US M/M NBS n/n 0 evidence for a horizon topology changing transition Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
79 Black Holes in Higher Dimensions Black Strings and Caged Black Holes Black Strings Caged Black Holes BH D = 6 extrapolation US M/M NBS n/n 0 evidence for a horizon topology changing transition Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
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82 Outlook Black Holes: Extrapolation for D 6 Emparan, Figueras 2010 asymptotically flat black holes and black rings in D 6? A H Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54 J
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85 Final Remarks Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
86 Final Remarks Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
87 Final Remarks Outlook Jutta Kunz (Universität Oldenburg) Black Holes in Four and More Dimensions Obergurgl, / 54
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