Jose Luis Blázquez Salcedo
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1 Physical Review Letters 112 (2014) Jose Luis Blázquez Salcedo In collaboration with Jutta Kunz, Eugen Radu and Francisco Navarro Lérida
2 1. Introduction: Ansatz and general properties 2. Near-horizon formalism 3. Numerical results 4. Future work: negative cosmological constant
3
4 We will consider black holes in 5 dimensions in Einstein-Maxwell-Chern-Simons theory Black holes in higher dimensions have some special properties Topologies of stationary black holes can be non-spherical For example black ring solution (Emparan 2002) More than one independent plane of rotation In D dimensions there are N = [(D-1)/2] planes of rotation N independent angular momenta
5 Einstein theory in odd D-dimensions R = curvature scalar Einstein-Maxwell-Chern-Simons Gravity coupled to a U(1) electro-magnetic potential A µ and a Chern-Simons term (D=5)
6 We are interested in the higher dimensional generalization of the Kerr-Newman black holes: Axisymmetric and stationary, Spherical topology of the horizon, Asymptotically flat Electrically charged All angular momenta of the same magnitude: J = J 1 = J 2 = = J N We have enhanced U(N) symmetry Not even with these constraints uniqueness is granted We use numerical methods to obtain global solutions with these properties. We also make use of the near-horizon formalism.
7 Ansatz for the metric: Lewis-Papapetrou coordinates. The radial coordinate r is isotropic. Ansatz for the gauge field:
8 Extremal black holes present non integer exponents in their horizon expansion. EM case EMCS case Special parametrization of the functions in order to numerically solve the problem.
9
10 The space-time outside the event horizon of extremal black holes can be divided in two different regions: Near-horizon geometry Bulk geometry Extracting the NHG from a known analytical solution by a coordinate transformation: 1. Move to a frame comoving with the event horizon 2. Center the radial coordinate on the event horizon 3. Scale parameter Λ in the new radial and temporal coordinates. 4. Series expansion for small Λ First term is scale independent: near-horizon geometry
11 Properties of the near-horizon geometry of extremal black holes. H. K. Kunduri and J. Lucietti, Living Reviews in Relativity 16 (2013) Extremal black holes with spherical topology: near-horizon geometry is the product of two independent spaces. Isometries: static case (sphere) rotation (squashed sphere) This factorization is obtained for all the known examples of topologically spherical black holes
12 Hence we can assume such factorization in our black holes (extremal case) Metric: Gauge potential:
13 Near-horizon geometry: Field equations + Ansatz: algebraic relations for the Ansatz parameters Alternatively: Extremal of entropy functional Global charges can be calculated: (J, Q) Horizon charges: area, horizon angular momentum Parameters related to the asymptotical structure of the global solution cannot be calculated: Mass, angular velocity
14 Global solutions and branch structure λ>2
15 Area vs angular momentum (near-horizon and global solutions)
16 Branch structure in the mass vs angular momentum plot:
17 Branch structure scheme:
18 J=0 non-static solutions
19 J=0 non-static solutions
20 Domain of existence
21 Domain of existence: Extremal solutions
22 Extremal vs Ω H =0
23
24 Black holes in Einstein-Maxwell-Chern-Simons with negative cosmological constant. Black holes no longer asymptotically flat: Anti-de-Sitter Same subset of solutions: equal-magnitude angular momenta, spherical topology Non-extremal black holes in EM where previously studied in Jutta Kunz, Francisco Navarro-Lérida, Eugen Radu, Higher dimensional rotating black holes in Einstein- Maxwell theory with negative cosmological constant, Physics Letters B 649 (2007) Extremal black holes also have non-integer exponents near the horizon The near-horizon formalism gives us some interesting analytical results: richer branch structure Branch structure in global charges similar to the EMCS flat case for a certain range of the parameters of the theory
25 Jose Luis Blazquez-Salcedo, Jutta Kunz, Francisco Navarro Lerida, Eugen Radu, Sequences of Extremal Radially Excited Rotating Black Holes, Physical Review Letters 112 (2014)
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