Formation of Higher-dimensional Topological Black Holes
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1 Formation of Higher-dimensional Topological Black Holes José Natário (based on arxiv: with Filipe Mena and Paul Tod) CAMGSD, Department of Mathematics Instituto Superior Técnico Talk at Granada, 2010 José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
2 Outline 1 Introduction Topological black holes Einstein manifolds Einstein equations 2 Collapse to higher dimensional topological black holes Generalized Kottler solutions Generalized Friedman-Lemaître-Robertson-Walker solutions Matching Global properties 3 Collapse with gravitational wave emission Exterior: Bizoń-Chmaj-Schmidt metric Interiors Matching José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
3 Outline 1 Introduction Topological black holes Einstein manifolds Einstein equations 2 Collapse to higher dimensional topological black holes Generalized Kottler solutions Generalized Friedman-Lemaître-Robertson-Walker solutions Matching Global properties 3 Collapse with gravitational wave emission Exterior: Bizoń-Chmaj-Schmidt metric Interiors Matching José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
4 Outline 1 Introduction Topological black holes Einstein manifolds Einstein equations 2 Collapse to higher dimensional topological black holes Generalized Kottler solutions Generalized Friedman-Lemaître-Robertson-Walker solutions Matching Global properties 3 Collapse with gravitational wave emission Exterior: Bizoń-Chmaj-Schmidt metric Interiors Matching José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
5 Outline Introduction 1 Introduction Topological black holes Einstein manifolds Einstein equations 2 Collapse to higher dimensional topological black holes Generalized Kottler solutions Generalized Friedman-Lemaître-Robertson-Walker solutions Matching Global properties 3 Collapse with gravitational wave emission Exterior: Bizoń-Chmaj-Schmidt metric Interiors Matching José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
6 Introduction Topological black holes Topological black holes 4-dimensional topological black holes have been considered in the literature. In these one replaces the spherically symmetric Kottler solution (i.e. Schwarzschild de Sitter or Schwarzschild anti de Sitter) by its planar or hyperbolic counterparts. Modding out by discrete subgroup of R 2 or SL(2, R) yields black holes with toroidal or higher genus horizons. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
7 Introduction Topological black holes Topological black holes 4-dimensional topological black holes have been considered in the literature. In these one replaces the spherically symmetric Kottler solution (i.e. Schwarzschild de Sitter or Schwarzschild anti de Sitter) by its planar or hyperbolic counterparts. Modding out by discrete subgroup of R 2 or SL(2, R) yields black holes with toroidal or higher genus horizons. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
8 Introduction Topological black holes Topological black holes 4-dimensional topological black holes have been considered in the literature. In these one replaces the spherically symmetric Kottler solution (i.e. Schwarzschild de Sitter or Schwarzschild anti de Sitter) by its planar or hyperbolic counterparts. Modding out by discrete subgroup of R 2 or SL(2, R) yields black holes with toroidal or higher genus horizons. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
9 Introduction Topological black holes Topological black holes Caveat 1: needs negative cosmological constant to work. Caveat 2: I has the same topology (times R). Formation of these black holes was studied in [Mena, Natário and Tod]. Many spherical collapses (e.g. Oppenheimer-Snyder) generalize to this setting. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
10 Introduction Topological black holes Topological black holes Caveat 1: needs negative cosmological constant to work. Caveat 2: I has the same topology (times R). Formation of these black holes was studied in [Mena, Natário and Tod]. Many spherical collapses (e.g. Oppenheimer-Snyder) generalize to this setting. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
11 Introduction Topological black holes Topological black holes Caveat 1: needs negative cosmological constant to work. Caveat 2: I has the same topology (times R). Formation of these black holes was studied in [Mena, Natário and Tod]. Many spherical collapses (e.g. Oppenheimer-Snyder) generalize to this setting. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
12 Introduction Einstein manifolds Einstein manifolds (N, dσ 2 ) is a n-dimensional Einstein manifold with Ricci = λ dσ 2. Examples: S n, T n, H n, CP n 2, Taub-NUT, Eguchi-Hanson, Calabi-Yau. Can be used to construct (n + 1)-dimensional Einstein metrics with Ricci = k dσ 2. dσ 2 = dρ 2 + (f (ρ)) 2 dσ 2 If k > 0 then k = ν 2 n, λ = ν 2 (n 1), f = sin(νρ) for some ν > 0. If k = 0 then λ = n 1, f = ρ or λ = 0, f = 1. If k < 0 then k = ν 2 n, λ = ν 2 (n 1), f = sinh(νρ) or k = ν 2 n, λ = 0, f = e ±νρ or k = ν 2 n, λ = ν 2 (n 1), f = cosh(νρ) for some ν > 0. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
13 Introduction Einstein manifolds Einstein manifolds (N, dσ 2 ) is a n-dimensional Einstein manifold with Ricci = λ dσ 2. Examples: S n, T n, H n, CP n 2, Taub-NUT, Eguchi-Hanson, Calabi-Yau. Can be used to construct (n + 1)-dimensional Einstein metrics with Ricci = k dσ 2. dσ 2 = dρ 2 + (f (ρ)) 2 dσ 2 If k > 0 then k = ν 2 n, λ = ν 2 (n 1), f = sin(νρ) for some ν > 0. If k = 0 then λ = n 1, f = ρ or λ = 0, f = 1. If k < 0 then k = ν 2 n, λ = ν 2 (n 1), f = sinh(νρ) or k = ν 2 n, λ = 0, f = e ±νρ or k = ν 2 n, λ = ν 2 (n 1), f = cosh(νρ) for some ν > 0. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
14 Introduction Einstein manifolds Einstein manifolds (N, dσ 2 ) is a n-dimensional Einstein manifold with Ricci = λ dσ 2. Examples: S n, T n, H n, CP n 2, Taub-NUT, Eguchi-Hanson, Calabi-Yau. Can be used to construct (n + 1)-dimensional Einstein metrics with Ricci = k dσ 2. dσ 2 = dρ 2 + (f (ρ)) 2 dσ 2 If k > 0 then k = ν 2 n, λ = ν 2 (n 1), f = sin(νρ) for some ν > 0. If k = 0 then λ = n 1, f = ρ or λ = 0, f = 1. If k < 0 then k = ν 2 n, λ = ν 2 (n 1), f = sinh(νρ) or k = ν 2 n, λ = 0, f = e ±νρ or k = ν 2 n, λ = ν 2 (n 1), f = cosh(νρ) for some ν > 0. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
15 Introduction Einstein manifolds Einstein manifolds (N, dσ 2 ) is a n-dimensional Einstein manifold with Ricci = λ dσ 2. Examples: S n, T n, H n, CP n 2, Taub-NUT, Eguchi-Hanson, Calabi-Yau. Can be used to construct (n + 1)-dimensional Einstein metrics with Ricci = k dσ 2. dσ 2 = dρ 2 + (f (ρ)) 2 dσ 2 If k > 0 then k = ν 2 n, λ = ν 2 (n 1), f = sin(νρ) for some ν > 0. If k = 0 then λ = n 1, f = ρ or λ = 0, f = 1. If k < 0 then k = ν 2 n, λ = ν 2 (n 1), f = sinh(νρ) or k = ν 2 n, λ = 0, f = e ±νρ or k = ν 2 n, λ = ν 2 (n 1), f = cosh(νρ) for some ν > 0. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
16 Einstein manifolds Introduction Einstein manifolds dσ 2 typically has a singularity at ρ = 0: the Kretschmann scalar K is related to the square C 2 of the Weyl tensor of dσ 2 by K = C 2 f 4 + const. This can be avoided for k < 0 with f = e ±νρ, when the metric has an internal infinity (cusp), or f = cosh(νρ), when the metric has a minimal surface and a second asymptotic region (wormhole). José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
17 Einstein manifolds Introduction Einstein manifolds dσ 2 typically has a singularity at ρ = 0: the Kretschmann scalar K is related to the square C 2 of the Weyl tensor of dσ 2 by K = C 2 f 4 + const. This can be avoided for k < 0 with f = e ±νρ, when the metric has an internal infinity (cusp), or f = cosh(νρ), when the metric has a minimal surface and a second asymptotic region (wormhole). José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
18 Einstein equations Introduction Einstein equations We consider the (n + 2)-dimensional Einstein equations R ab = Λg ab + κ(t ab 1 n Tg ab), or R ab 1 2 Rg ab + nλ 2 g ab = κt ab. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
19 Outline Collapse to higher dimensional topological black holes 1 Introduction Topological black holes Einstein manifolds Einstein equations 2 Collapse to higher dimensional topological black holes Generalized Kottler solutions Generalized Friedman-Lemaître-Robertson-Walker solutions Matching Global properties 3 Collapse with gravitational wave emission Exterior: Bizoń-Chmaj-Schmidt metric Interiors Matching José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
20 Collapse to higher dimensional topological black holes Generalized Kottler solutions Generalized Kottler solutions The generalized Kottler solutions are vacuum (T ab = 0) solutions given by ds 2 = V (r)dt 2 + (V (r)) 1 dr 2 + r 2 dσ 2, where V (r) = λ n 1 2m Λr 2 r n 1 n + 1. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
21 Collapse to higher dimensional topological black holes Generalized Friedman-Lemaître-Robertson-Walker solutions Generalized Friedman-Lemaître-Robertson-Walker solutions The Generalized Friedman-Lemaître-Robertson-Walker solutions are dust (T ab = µ u a u b ) solutions given by ds 2 = dτ 2 + R 2 (τ) dσ 2, where dσ 2 is any (n + 1)-dimensional Riemannian Einstein metric with Ricci = k dσ 2 and R(τ), µ(τ) satisfy the conservation equation µr n+1 = µ 0 and the generalized Friedman equation Ṙ 2 R 2 + k nr 2 = 2κµ n(n + 1) + Λ n + 1. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
22 Collapse to higher dimensional topological black holes Matching Matching Can match generalized Kottler to generalized FLRW at ρ = ρ 0 provided that f (ρ 0 ) > 0 and m = κµ 0(f (ρ 0 )) n+1. n(n + 1) Similar results for generalized Lemaître-Tolman-Bondi. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
23 Collapse to higher dimensional topological black holes Matching Matching Can match generalized Kottler to generalized FLRW at ρ = ρ 0 provided that f (ρ 0 ) > 0 and m = κµ 0(f (ρ 0 )) n+1. n(n + 1) Similar results for generalized Lemaître-Tolman-Bondi. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
24 Collapse to higher dimensional topological black holes Global properties Global properties If Λ = 0 (hence λ > 0) and (N, dσ 2 ) is not an n-sphere then the locally naked singularity is always visible from I + for k 0, but can be hidden if k > 0 and n 4 (cf. [Ghosh and Beesham]). José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
25 Collapse to higher dimensional topological black holes Global properties Global properties If Λ > 0 (hence λ > 0) and (N, dσ 2 ) is not an n-sphere then the locally naked singularity can be always be hidden except if the FLRW universe is recollapsing (hence k > 0) and n < 4. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
26 Collapse to higher dimensional topological black holes Global properties Global properties For Λ < 0 we have: If λ > 0 and (N, dσ 2 ) is not an n-sphere then the locally naked singularity can always be hidden. If λ = 0 then the cusp singularity is not locally naked. If λ < 0 then no causal curve can cross the wormhole from one I to the other (cf. [Galloway]). José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
27 Collapse to higher dimensional topological black holes Global properties Global properties Global properties are much more diverse in the generalized Lemaître-Tolman-Bondi case. For instance, one can easily find examples of black hole formation with wormholes inside the matter with positive λ and Λ = 0 (previously seen in 4 dimensions [Hellaby]). José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
28 Outline Collapse with gravitational wave emission 1 Introduction Topological black holes Einstein manifolds Einstein equations 2 Collapse to higher dimensional topological black holes Generalized Kottler solutions Generalized Friedman-Lemaître-Robertson-Walker solutions Matching Global properties 3 Collapse with gravitational wave emission Exterior: Bizoń-Chmaj-Schmidt metric Interiors Matching José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
29 Collapse with gravitational wave emission Exterior: Bizoń-Chmaj-Schmidt metric Exterior: Bizoń-Chmaj-Schmidt metric The Bizoń-Chmaj-Schmidt metric is a vacuum (T ab = 0) solution of the Einstein equations with Λ = 0 given by ds 2+ = Ae 2δ dt 2 + A 1 dr 2 + r 2 4 e2b (σ σ 2 2) + r 2 4 e 4B σ 2 3, where σ i are the standard left-invariant 1-forms on S 3 and A, δ and B are functions of t and r satisfying r A = 2A r + 1 3r (8e 2B 2e 8B ) 2r(e 2δ A 1 ( t B) 2 + A( r B) 2 ); t A = 4rA( t B)( r B); r δ = 2r(e 2δ A 2 ( t B) 2 + ( r B) 2 ); t (e δ A 1 r 3 ( t B)) r (e δ Ar 3 ( r B)) e δ r(e 2B e 8B ) = 0. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
30 Collapse with gravitational wave emission Interiors Interiors Eguchi-Hanson (S 3 /Z 2 ): dσ 2 = ) 1 (1 a4 ρ 4 dρ 2 + ρ2 4 (σ2 1 + σ2) 2 + ρ2 4 k-eguchi-hanson (S 3 /Z p ): where k < 0, p 3, = 1 a4 k-taub-nut: dσ 2 = 1 dρ 2 + ρ2 4 (σ2 1 + σ 2 2) + ρ2 4 σ2 3, ρ 4 k 6 ρ2 and a 4 = 4 3k 2 (p 2)2 (p + 1). ) (1 a4 ρ 4 σ3. 2 dσ 2 = 1 4 Σ 1 dρ (ρ2 L 2 )(σ σ 2 2) + L 2 Σσ 2 3, where Σ = (ρ L)(1 12 k (ρ L)(ρ+3L)). ρ+l José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
31 Collapse with gravitational wave emission Interiors Interiors Eguchi-Hanson (S 3 /Z 2 ): dσ 2 = ) 1 (1 a4 ρ 4 dρ 2 + ρ2 4 (σ2 1 + σ2) 2 + ρ2 4 k-eguchi-hanson (S 3 /Z p ): where k < 0, p 3, = 1 a4 k-taub-nut: dσ 2 = 1 dρ 2 + ρ2 4 (σ2 1 + σ 2 2) + ρ2 4 σ2 3, ρ 4 k 6 ρ2 and a 4 = 4 3k 2 (p 2)2 (p + 1). ) (1 a4 ρ 4 σ3. 2 dσ 2 = 1 4 Σ 1 dρ (ρ2 L 2 )(σ σ 2 2) + L 2 Σσ 2 3, where Σ = (ρ L)(1 12 k (ρ L)(ρ+3L)). ρ+l José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
32 Collapse with gravitational wave emission Interiors Interiors Eguchi-Hanson (S 3 /Z 2 ): dσ 2 = ) 1 (1 a4 ρ 4 dρ 2 + ρ2 4 (σ2 1 + σ2) 2 + ρ2 4 k-eguchi-hanson (S 3 /Z p ): where k < 0, p 3, = 1 a4 k-taub-nut: dσ 2 = 1 dρ 2 + ρ2 4 (σ2 1 + σ 2 2) + ρ2 4 σ2 3, ρ 4 k 6 ρ2 and a 4 = 4 3k 2 (p 2)2 (p + 1). ) (1 a4 ρ 4 σ3. 2 dσ 2 = 1 4 Σ 1 dρ (ρ2 L 2 )(σ σ 2 2) + L 2 Σσ 2 3, where Σ = (ρ L)(1 12 k (ρ L)(ρ+3L)). ρ+l José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
33 Matching Collapse with gravitational wave emission Matching In each case, the interior metric gives consistent data for the Bizoń-Chmaj-Schmidt metric at a comoving timelike hypersurface. Local existence of the radiating exterior in the neighbourhood of the matching surface is then guaranteed. In the case of Eguchi-Hanson (asymptotically locally Euclidean) and k-taub-nut with k < 0 (includes hyperbolic space), the data can be chosen to be close to the data for the Schwarzschild solution. Since this solution is known to be stable [Dafermos and Holzegel], it is reasonable to expect that the exterior will settle down to the Schwarzschild solution. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
34 Bibliography Appendix Filipe Mena, JN and Paul Tod, Gravitational Collapse to Toroidal and Higher Genus asymptotically AdS Black Holes, Advances in Theoretical and Mathematical Physics 12 (2008) , arxiv: [gr-qc]. Filipe Mena, JN and Paul Tod, Formation of Higher-dimensional Topological Black Holes, Annales Henri Poincaré 10 (2010) , arxiv: [gr-qc]. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
35 Bibliography Appendix Filipe Mena, JN and Paul Tod, Gravitational Collapse to Toroidal and Higher Genus asymptotically AdS Black Holes, Advances in Theoretical and Mathematical Physics 12 (2008) , arxiv: [gr-qc]. Filipe Mena, JN and Paul Tod, Formation of Higher-dimensional Topological Black Holes, Annales Henri Poincaré 10 (2010) , arxiv: [gr-qc]. José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
36 Bibliography Appendix S. Ghosh and A. Beesham, Higher-dimensional Inhomogeneous Dust Collapse and Cosmic Censorship, Physical Review D 64 (2001) , arxiv:gr-qc/ G. Galloway, A Finite Infinity Version of Topological Censorship, Classical and Quantum Gravity 13 (1996) C. Hellaby A Kruskal-like model with finite density, Classical and Quantum Gravity 4 (1987) José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
37 Bibliography Appendix S. Ghosh and A. Beesham, Higher-dimensional Inhomogeneous Dust Collapse and Cosmic Censorship, Physical Review D 64 (2001) , arxiv:gr-qc/ G. Galloway, A Finite Infinity Version of Topological Censorship, Classical and Quantum Gravity 13 (1996) C. Hellaby A Kruskal-like model with finite density, Classical and Quantum Gravity 4 (1987) José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
38 Bibliography Appendix S. Ghosh and A. Beesham, Higher-dimensional Inhomogeneous Dust Collapse and Cosmic Censorship, Physical Review D 64 (2001) , arxiv:gr-qc/ G. Galloway, A Finite Infinity Version of Topological Censorship, Classical and Quantum Gravity 13 (1996) C. Hellaby A Kruskal-like model with finite density, Classical and Quantum Gravity 4 (1987) José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
39 Bibliography Appendix P. Bizoń, T. Chmaj and B. Schmidt, Critical Behaviour in Vacuum Gravitational Collapse in dimensions, Physical Review Letters 95 (2005) , arxiv:gr-qc/ H. Pederson, Eguchi-Hanson Metrics With a Cosmological Constant, Classical and Quantum Gravity 2 (1985) M. Dafermos and G. Holzegel, On the Nonlinear Stability of Higher Dimensional Triaxial Bianchi-IX Black Holes, Advances in Theoretical and Mathematical Physics 10 (2006) , arxiv:gr-qc/ José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
40 Bibliography Appendix P. Bizoń, T. Chmaj and B. Schmidt, Critical Behaviour in Vacuum Gravitational Collapse in dimensions, Physical Review Letters 95 (2005) , arxiv:gr-qc/ H. Pederson, Eguchi-Hanson Metrics With a Cosmological Constant, Classical and Quantum Gravity 2 (1985) M. Dafermos and G. Holzegel, On the Nonlinear Stability of Higher Dimensional Triaxial Bianchi-IX Black Holes, Advances in Theoretical and Mathematical Physics 10 (2006) , arxiv:gr-qc/ José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
41 Bibliography Appendix P. Bizoń, T. Chmaj and B. Schmidt, Critical Behaviour in Vacuum Gravitational Collapse in dimensions, Physical Review Letters 95 (2005) , arxiv:gr-qc/ H. Pederson, Eguchi-Hanson Metrics With a Cosmological Constant, Classical and Quantum Gravity 2 (1985) M. Dafermos and G. Holzegel, On the Nonlinear Stability of Higher Dimensional Triaxial Bianchi-IX Black Holes, Advances in Theoretical and Mathematical Physics 10 (2006) , arxiv:gr-qc/ José Natário (IST, Lisbon) Higher-dimensional Topological Black Holes IST / 23
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