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Black holes with a single Killing vector field Black holes with a single Killing vector field Jorge E. Santos Cambridge University - DAMTP Oxford - 23/02/2016 In collaboration with Oscar J. C. Dias, Ben Niehoff and Benson Way 1 / 20

  1. Black holes with a single Killing vector field Seemingly different instabilities in AdS Superradiance Instability - 2/3: The Kerr-AdS 4 black hole (aka Carter solution - ’68): ∆ r ∆ x d s 2 = − � 2 + � 2 d t − (1 − x 2 )d φ d t − (1 + r 2 )d φ � � r 2 + x 2 r 2 + x 2 � d r 2 ∆ r + d x 2 � + a 2 ( r 2 + x 2 ) , ∆ x where 1 + a 2 1 − a 2 � � − 2 M � � ∆ r = (1 + r 2 ) L 2 r 2 a r , ∆ x = (1 − x 2 ) L 2 x 2 . 2 parameters: ( M, a ) . 6 / 20

  2. Black holes with a single Killing vector field Seemingly different instabilities in AdS Superradiance Instability - 2/3: The Kerr-AdS 4 black hole (aka Carter solution - ’68): ∆ r ∆ x d s 2 = − � 2 + � 2 d t − (1 − x 2 )d φ d t − (1 + r 2 )d φ � � r 2 + x 2 r 2 + x 2 � d r 2 ∆ r + d x 2 � + a 2 ( r 2 + x 2 ) , ∆ x where 1 + a 2 1 − a 2 � � − 2 M � � ∆ r = (1 + r 2 ) L 2 r 2 a r , ∆ x = (1 − x 2 ) L 2 x 2 . 2 parameters: ( M, a ) ⇔ ( R + , Ω H ) . 6 / 20

  3. Black holes with a single Killing vector field Seemingly different instabilities in AdS Superradiance Instability - 2/3: The Kerr-AdS 4 black hole (aka Carter solution - ’68): ∆ r ∆ x d s 2 = − � 2 + � 2 d t − (1 − x 2 )d φ d t − (1 + r 2 )d φ � � r 2 + x 2 r 2 + x 2 � d r 2 ∆ r + d x 2 � + a 2 ( r 2 + x 2 ) , ∆ x where 1 + a 2 1 − a 2 � � − 2 M � � ∆ r = (1 + r 2 ) L 2 r 2 a r , ∆ x = (1 − x 2 ) L 2 x 2 . 2 parameters: ( M, a ) ⇔ ( R + , Ω H ) : � L 4 +4 L 2 R 2 + +3 R 4 + T H ≥ 0 ⇒ | Ω H L | ≤ R + → + ∞ 1 . − → 2 L 2 R 2 + +3 R 4 + 6 / 20

  4. Black holes with a single Killing vector field Seemingly different instabilities in AdS Superradiance Instability - 2/3: The Kerr-AdS 4 black hole (aka Carter solution - ’68): ∆ r ∆ x d s 2 = − � 2 + � 2 d t − (1 − x 2 )d φ d t − (1 + r 2 )d φ � � r 2 + x 2 r 2 + x 2 � d r 2 ∆ r + d x 2 � + a 2 ( r 2 + x 2 ) , ∆ x where 1 + a 2 1 − a 2 � � − 2 M � � ∆ r = (1 + r 2 ) L 2 r 2 a r , ∆ x = (1 − x 2 ) L 2 x 2 . 2 parameters: ( M, a ) ⇔ ( R + , Ω H ) : � L 4 +4 L 2 R 2 + +3 R 4 + T H ≥ 0 ⇒ | Ω H L | ≤ R + → + ∞ 1 . − → 2 L 2 R 2 + +3 R 4 + ∂ t and ∂ φ are commuting Killing fields; decompose perturbations in Fourier modes: e − iωt + imφ . 6 / 20

  5. Black holes with a single Killing vector field Seemingly different instabilities in AdS Superradiance Instability - 2/3: The Kerr-AdS 4 black hole (aka Carter solution - ’68): ∆ r ∆ x d s 2 = − � 2 + � 2 d t − (1 − x 2 )d φ d t − (1 + r 2 )d φ � � r 2 + x 2 r 2 + x 2 � d r 2 ∆ r + d x 2 � + a 2 ( r 2 + x 2 ) , ∆ x where 1 + a 2 1 − a 2 � � − 2 M � � ∆ r = (1 + r 2 ) L 2 r 2 a r , ∆ x = (1 − x 2 ) L 2 x 2 . 2 parameters: ( M, a ) ⇔ ( R + , Ω H ) : � L 4 +4 L 2 R 2 + +3 R 4 + T H ≥ 0 ⇒ | Ω H L | ≤ R + → + ∞ 1 . − → 2 L 2 R 2 + +3 R 4 + ∂ t and ∂ φ are commuting Killing fields; decompose perturbations in Fourier modes: e − iωt + imφ . Unstable if quasi-normal modes with Im( ω ) > 0 exist. 6 / 20

  6. Black holes with a single Killing vector field Seemingly different instabilities in AdS Superradiance Instability - 3/3: ��� ��� ٠� � ��� ��� ��� ��� ��� ��� ��� ��� � + / � Phase Diagram for Kerr-AdS black holes 7 / 20

  7. Black holes with a single Killing vector field Seemingly different instabilities in AdS Superradiance Instability - 3/3: ��� ��� Ω � � ��� ������ ��� ��� ��� ��� ��� ��� ��� � + / � Kerr-AdS with | Ω H L | ≤ 1 : likely to be stable - Hawking and Reall ’00 . 7 / 20

  8. Black holes with a single Killing vector field Seemingly different instabilities in AdS Superradiance Instability - 3/3: ��� ��� ●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Ω � � ��� ● ● ● ● ● ● ● ● ● � = � ������ ��� ��� ��� ��� ��� ��� ��� � + / � Perturbations with m � = 0 are unstable if Re( ω ) ≤ m Ω H : onset saturates inequality - Cardoso et al. ’14 . 7 / 20

  9. Black holes with a single Killing vector field Seemingly different instabilities in AdS Superradiance Instability - 3/3: ● ● ● ���� ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ������ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ���� ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ���� ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● � / � ���� ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● � = � ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ���� ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ���� ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ���� ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ����� ����� ����� ����� ����� ����� ����� � / � � In the microcanonical ensemble: natural variables are ( J, E ) . 7 / 20

  10. Black holes with a single Killing vector field Seemingly different instabilities in AdS Superradiance Instability - 3/3: ■ ● ● ■ ■ ● ���� ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ■ ■ ● ● ■ ● ● ■ ■ ● ● ■ ■ ● ■ ■ ● ● ������ ■ ■ ● ● ■ ● ● ■ ■ ● ● ■ ■ ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ● ● ■ ■ ● ■ ■ ● ● ���� ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ■ ● ● ���� ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ■ ● ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ■ ● ● ■ ■ ● ● ● ■ ■ ● ● ■ ■ ● ● ■ ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ● ■ ■ ● ● � / � ���� ■ ■ ■ ● ● ■ ■ ● ● ● � = � ■ ■ ● ● ● ■ ■ ● ● ■ ■ ■ ● ● ■ ■ ● ● ● ■ ■ ● ● ■ ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ● ■ � = � ■ ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ● ■ ■ ■ ● ● ■ ■ ● ● ■ ■ ● ● ● ■ ■ ■ ● ● ■ ■ ● ● ■ ■ ■ ● ● ● ■ ● ● ■ ���� ■ ■ ● ● ● ■ ● ■ ■ ● ● ● ■ ■ ■ ■ ■ ● ● ● ■ ■ ● ● ■ ■ ● ● ● ■ ■ ■ ● ● ● ● ■ ■ ■ ● ● ● ■ ■ ■ ■ ● ● ■ ■ ■ ● ● ● ● ■ ■ ● ■ ■ ■ ● ● ● ● ● ■ ■ ● ■ ■ ■ ● ● ■ ■ ● ● ■ ● ■ ● ● ● ● ■ ■ ■ ■ ■ ● ● ● ● ● ■ ■ ■ ● ■ ■ ■ ● ● ���� ■ ● ● ■ ● ● ● ■ ■ ■ ● ● ■ ■ ■ ● ■ ● ● ■ ● ■ ■ ● ■ ■ ■ ● ● ● ● ● ■ ■ ● ● ● ■ ■ ■ ■ ■ ● ● ● ■ ■ ● ● ● ■ ■ ● ■ ■ ● ● ■ ● ■ ■ ● ■ ● ■ ■ ● ● ■ ● ● ■ ● ● ■ ■ ■ ● ● ■ ● ■ ● ■ ■ ● ● ■ ● ● ■ ■ ■ ● ● ● ● ■ ■ ■ ■ ● ■ ■ ■ ■ ■ ● ● ● ● ● ● ■ ���� ■ ■ ● ● ● ■ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ����� ����� ����� ����� ����� ����� ����� � / � � Higher m modes appear closer to Ω H L = 1 : Ω H L = 1 is reached m → + ∞ - Kunduri et. al. ’06 . 7 / 20

  11. Black holes with a single Killing vector field Seemingly different instabilities in AdS The nonlinear stability of AdS 7 / 20

  12. Black holes with a single Killing vector field Seemingly different instabilities in AdS The stability problem for spacetimes in general relativity The question Consider a spacetime ( M , g ) , together with prescribed boundary conditions B if timelike boundary exists. 8 / 20

  13. Black holes with a single Killing vector field Seemingly different instabilities in AdS The stability problem for spacetimes in general relativity The question Consider a spacetime ( M , g ) , together with prescribed boundary conditions B if timelike boundary exists. Take small perturbations (in a suitable sense) on a Cauchy surface S . 8 / 20

  14. Black holes with a single Killing vector field Seemingly different instabilities in AdS The stability problem for spacetimes in general relativity The question Consider a spacetime ( M , g ) , together with prescribed boundary conditions B if timelike boundary exists. Take small perturbations (in a suitable sense) on a Cauchy surface S . ??? ??? Does the solution spacetime ( M , g ′ ) that ??? arises still has the same asymptotic causal B B structure as ( M , g ) ? S 8 / 20

  15. Black holes with a single Killing vector field Seemingly different instabilities in AdS The stability problem for spacetimes in general relativity The question Consider a spacetime ( M , g ) , together with prescribed boundary conditions B if timelike boundary exists. Take small perturbations (in a suitable sense) on a Cauchy surface S . ??? ??? Does the solution spacetime ( M , g ′ ) that ??? arises still has the same asymptotic causal B B structure as ( M , g ) ? If so, can we bound the “difference” between the asymptotic form of g and g ′ in S terms of initial data defined on S ? 8 / 20

  16. Black holes with a single Killing vector field Seemingly different instabilities in AdS The stability problem for spacetimes in general relativity The question Consider a spacetime ( M , g ) , together with prescribed boundary conditions B if timelike boundary exists. Take small perturbations (in a suitable sense) on a Cauchy surface S . ??? ??? Does the solution spacetime ( M , g ′ ) that ??? arises still has the same asymptotic causal B B structure as ( M , g ) ? If so, can we bound the “difference” between the asymptotic form of g and g ′ in S terms of initial data defined on S ? In particular, if a geodesically complete spacetime is perturbed, does it remain “complete”? 8 / 20

  17. Black holes with a single Killing vector field Seemingly different instabilities in AdS Minkowski, dS and AdS spacetimes At the linear level, Anti de-Sitter spacetime appears just as stable as the Minkowski or de-Sitter spacetimes. 9 / 20

  18. Black holes with a single Killing vector field Seemingly different instabilities in AdS Minkowski, dS and AdS spacetimes At the linear level, Anti de-Sitter spacetime appears just as stable as the Minkowski or de-Sitter spacetimes. For the Minkowski and de-Sitter spacetimes, it has been shown that small, but finite, perturbations remain small - D. Christodoulou and S. Klainerman ‘93 and Friedrich ‘86. 9 / 20

  19. Black holes with a single Killing vector field Seemingly different instabilities in AdS Minkowski, dS and AdS spacetimes At the linear level, Anti de-Sitter spacetime appears just as stable as the Minkowski or de-Sitter spacetimes. For the Minkowski and de-Sitter spacetimes, it has been shown that small, but finite, perturbations remain small - D. Christodoulou and S. Klainerman ‘93 and Friedrich ‘86. Why has this not been shown for Anti de-Sitter? 9 / 20

  20. Black holes with a single Killing vector field Seemingly different instabilities in AdS Minkowski, dS and AdS spacetimes At the linear level, Anti de-Sitter spacetime appears just as stable as the Minkowski or de-Sitter spacetimes. For the Minkowski and de-Sitter spacetimes, it has been shown that small, but finite, perturbations remain small - D. Christodoulou and S. Klainerman ‘93 and Friedrich ‘86. Why has this not been shown for Anti de-Sitter? It is just not true! 9 / 20

  21. Black holes with a single Killing vector field Seemingly different instabilities in AdS Minkowski, dS and AdS spacetimes At the linear level, Anti de-Sitter spacetime appears just as stable as the Minkowski or de-Sitter spacetimes. For the Minkowski and de-Sitter spacetimes, it has been shown that small, but finite, perturbations remain small - D. Christodoulou and S. Klainerman ‘93 and Friedrich ‘86. Why has this not been shown for Anti de-Sitter? It is just not true! Claim: Some generic small (but finite) perturbations of AdS become large and eventually form black holes. 9 / 20

  22. Black holes with a single Killing vector field Seemingly different instabilities in AdS Minkowski, dS and AdS spacetimes At the linear level, Anti de-Sitter spacetime appears just as stable as the Minkowski or de-Sitter spacetimes. For the Minkowski and de-Sitter spacetimes, it has been shown that small, but finite, perturbations remain small - D. Christodoulou and S. Klainerman ‘93 and Friedrich ‘86. Why has this not been shown for Anti de-Sitter? It is just not true! Claim: Some generic small (but finite) perturbations of AdS become large and eventually form black holes. The energy cascades from low to high frequency modes in a manner reminiscent of the onset of turbulence. 9 / 20

  23. Black holes with a single Killing vector field Seemingly different instabilities in AdS Heuristics AdS acts like a confining finite box. Any generic finite excitation which is added to this box might be expected to explore all configurations consistent with the conserved charges of AdS - including small black holes. 10 / 20

  24. Black holes with a single Killing vector field Seemingly different instabilities in AdS Heuristics AdS acts like a confining finite box. Any generic finite excitation which is added to this box might be expected to explore all configurations consistent with the conserved charges of AdS - including small black holes. Special (fine tuned) solutions need not lead to the formation of black holes. 10 / 20

  25. Black holes with a single Killing vector field Seemingly different instabilities in AdS Heuristics AdS acts like a confining finite box. Any generic finite excitation which is added to this box might be expected to explore all configurations consistent with the conserved charges of AdS - including small black holes. Special (fine tuned) solutions need not lead to the formation of black holes. Some linearized gravitational modes will have corresponding nonlinear solutions - Geons - Dias, Horowitz and JES. 10 / 20

  26. Black holes with a single Killing vector field Seemingly different instabilities in AdS Heuristics AdS acts like a confining finite box. Any generic finite excitation which is added to this box might be expected to explore all configurations consistent with the conserved charges of AdS - including small black holes. Special (fine tuned) solutions need not lead to the formation of black holes. Some linearized gravitational modes will have corresponding nonlinear solutions - Geons - Dias, Horowitz and JES. These solutions are special since they are exactly periodic in time and invariant under a single continuous symmetry. 10 / 20

  27. Black holes with a single Killing vector field Seemingly different instabilities in AdS Heuristics AdS acts like a confining finite box. Any generic finite excitation which is added to this box might be expected to explore all configurations consistent with the conserved charges of AdS - including small black holes. Special (fine tuned) solutions need not lead to the formation of black holes. Some linearized gravitational modes will have corresponding nonlinear solutions - Geons - Dias, Horowitz and JES. These solutions are special since they are exactly periodic in time and invariant under a single continuous symmetry. Geons are analogous to nonlinear gravitational plane waves. 10 / 20

  28. Black holes with a single Killing vector field Seemingly different instabilities in AdS Heuristics AdS acts like a confining finite box. Any generic finite excitation which is added to this box might be expected to explore all configurations consistent with the conserved charges of AdS - including small black holes. Special (fine tuned) solutions need not lead to the formation of black holes. Some linearized gravitational modes will have corresponding nonlinear solutions - Geons - Dias, Horowitz and JES. These solutions are special since they are exactly periodic in time and invariant under a single continuous symmetry. Geons are analogous to nonlinear gravitational plane waves. This Heuristic argument has been observed numerically for certain types of initial data, but fails for other types. 10 / 20

  29. Black holes with a single Killing vector field Seemingly different instabilities in AdS What has been observed: Spherical scalar field collapse in AdS - Bizon and Rostworowski . 11 / 20

  30. Black holes with a single Killing vector field Seemingly different instabilities in AdS What has been observed: Spherical scalar field collapse in AdS - Bizon and Rostworowski . No matter how small the initial energy, the curvature at the origin grows and eventually forms a black hole. 0.04 0.035 0.03 0.025 x H 0.02 0.015 0.01 0.005 0 20 25 30 35 40 45 ε 11 / 20

  31. Black holes with a single Killing vector field Seemingly different instabilities in AdS What has been observed: Spherical scalar field collapse in AdS - Bizon and Rostworowski . No matter how small the initial energy, the curvature at the origin grows and eventually forms a black hole. 0.04 0.035 0.03 0.025 x H 0.02 0.015 0.01 0.005 0 20 25 30 35 40 45 ε Black holes form: ∆ t ∝ ε − 2 , matches na¨ ıve KAM intuition and 3 rd order calculation - Dias, Horowitz and JES. 11 / 20

  32. Black holes with a single Killing vector field Seemingly different instabilities in AdS What has been observed: Spherical scalar field collapse in AdS - Bizon and Rostworowski . No matter how small the initial energy, the curvature at the origin grows and eventually forms a black hole. 0.04 0.035 0.03 0.025 x H 0.02 0.015 0.01 0.005 0 20 25 30 35 40 45 ε Black holes form: ∆ t ∝ ε − 2 , matches na¨ ıve KAM intuition and 3 rd order calculation - Dias, Horowitz and JES. Certain types of initial data do not do this: do not seem to form black holes at late times! - Balasubramanian et. al. 11 / 20

  33. Black holes with a single Killing vector field Seemingly different instabilities in AdS What has been observed: Spherical scalar field collapse in AdS - Bizon and Rostworowski . No matter how small the initial energy, the curvature at the origin grows and eventually forms a black hole. 0.04 0.035 0.03 0.025 x H 0.02 0.015 0.01 0.005 0 20 25 30 35 40 45 ε Black holes form: ∆ t ∝ ε − 2 , matches na¨ ıve KAM intuition and 3 rd order calculation - Dias, Horowitz and JES. Certain types of initial data do not do this: do not seem to form black holes at late times! - Balasubramanian et. al. Understand why special fine tuned solutions - Geons - exist. 11 / 20

  34. Black holes with a single Killing vector field Geons as special solutions Geons - Horowitz and JES ’14 Geons are time-periodic regular horizonless solutions of the Einstein equation, which do not seem to thermalize. 12 / 20

  35. Black holes with a single Killing vector field Geons as special solutions Geons - Horowitz and JES ’14 Geons are time-periodic regular horizonless solutions of the Einstein equation, which do not seem to thermalize. The boundary stress-tensor contains regions of negative and positive energy density around the equator: 12 / 20

  36. Black holes with a single Killing vector field Geons as special solutions Geons - Horowitz and JES ’14 Geons are time-periodic regular horizonless solutions of the Einstein equation, which do not seem to thermalize. The boundary stress-tensor contains regions of negative and positive energy density around the equator: It is invariant under K = ∂ ∂t + ω ∂ ∂φ, m which is timelike near the poles but spacelike near the equator. 12 / 20

  37. Black holes with a single Killing vector field Geons as special solutions Geons - Horowitz and JES ’14 Geons are time-periodic regular horizonless solutions of the Einstein equation, which do not seem to thermalize. The boundary stress-tensor contains regions of negative and positive energy density around the equator: It is invariant under K = ∂ ∂t + ω ∂ ∂φ, m which is timelike near the poles but spacelike near the equator. It satisfies the first law m d E = ω d J . 12 / 20

  38. Black holes with a single Killing vector field Geons as special solutions Geons - Horowitz and JES ’14 Geons are time-periodic regular horizonless solutions of the Einstein equation, which do not seem to thermalize. The boundary stress-tensor contains regions of negative and positive energy density around the equator: It is invariant under K = ∂ ∂t + ω ∂ ∂φ, m which is timelike near the poles but spacelike near the equator. It satisfies the first law m d E = ω d J . Unclear if they can have the same energy, i.e. coexist, with large AdS black holes! 12 / 20

  39. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them One black hole to interpolate them all and in the darkness bind them 12 / 20

  40. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Surfing the Geon: Since Geons rotate rigidly, one can ask whether small black holes can surf the Geon! 13 / 20

  41. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Surfing the Geon: Since Geons rotate rigidly, one can ask whether small black holes can surf the Geon! This is possible if the black hole rotates rigidly with angular velocity Ω H = ω/m , ensuring zero flux across the horizon. 13 / 20

  42. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Surfing the Geon: Since Geons rotate rigidly, one can ask whether small black holes can surf the Geon! This is possible if the black hole rotates rigidly with angular velocity Ω H = ω/m , ensuring zero flux across the horizon. If such solutions exist, we have a black hole with a single Killing vector field - black resonator - conjectured by Reall ’03 ! 13 / 20

  43. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Surfing the Geon: Since Geons rotate rigidly, one can ask whether small black holes can surf the Geon! This is possible if the black hole rotates rigidly with angular velocity Ω H = ω/m , ensuring zero flux across the horizon. If such solutions exist, we have a black hole with a single Killing vector field - black resonator - conjectured by Reall ’03 ! Evades rigidity theorem because the only Killing field is the horizon generator! 13 / 20

  44. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Surfing the Geon: Since Geons rotate rigidly, one can ask whether small black holes can surf the Geon! This is possible if the black hole rotates rigidly with angular velocity Ω H = ω/m , ensuring zero flux across the horizon. If such solutions exist, we have a black hole with a single Killing vector field - black resonator - conjectured by Reall ’03 ! Evades rigidity theorem because the only Killing field is the horizon generator! We have constructed these solutions: ten coupled 3D nonlinear partial differential equations of Elliptic type. 13 / 20

  45. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Black resonators 1/3: One helical Killing field: ∂ T = ∂ t + Ω H ∂ φ . 14 / 20

  46. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Black resonators 1/3: One helical Killing field: ∂ T = ∂ t + Ω H ∂ φ . Their line element can be adapted to ∂ T : � L 2 − y 2 A ∆ y (d T + y χ 1 d y ) 2 + 4 y 2 + B d y 2 d s 2 = (1 − y 2 ) 2 ∆ y + 4 y 2 + S 1 � 2 � � 2 − x 2 χ 3 d y + y 2 x � d x + yx 2 − x 2 χ 2 d T 2 − x 2 √ � 2 � � dΨ + y 2 Ωd T + x 2 − x 2 χ 4 d x + (1 − x 2 ) 2 y 2 + y χ 5 d y + S 2 1 − x 2 14 / 20

  47. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Black resonators 1/3: One helical Killing field: ∂ T = ∂ t + Ω H ∂ φ . Their line element can be adapted to ∂ T : � L 2 − y 2 A ∆ y (d T + y χ 1 d y ) 2 + 4 y 2 + B d y 2 d s 2 = (1 − y 2 ) 2 ∆ y + 4 y 2 + S 1 � 2 � � 2 − x 2 χ 3 d y + y 2 x � d x + yx 2 − x 2 χ 2 d T 2 − x 2 √ � 2 � � dΨ + y 2 Ωd T + x 2 − x 2 χ 4 d x + (1 − x 2 ) 2 y 2 + y χ 5 d y + S 2 1 − x 2 2D moduli space: � π T ≡ 1 + 3 y 2 + and ε ≡ d φχ 4 (0 , 1 , φ ) sin( m φ ) . 4 πy + 0 14 / 20

  48. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Black resonators 1/3: One helical Killing field: ∂ T = ∂ t + Ω H ∂ φ . Their line element can be adapted to ∂ T : � L 2 − y 2 A ∆ y (d T + y χ 1 d y ) 2 + 4 y 2 + B d y 2 d s 2 = (1 − y 2 ) 2 ∆ y + 4 y 2 + S 1 � 2 � � 2 − x 2 χ 3 d y + y 2 x � d x + yx 2 − x 2 χ 2 d T 2 − x 2 √ � 2 � � dΨ + y 2 Ωd T + x 2 − x 2 χ 4 d x + (1 − x 2 ) 2 y 2 + y χ 5 d y + S 2 1 − x 2 2D moduli space: � π T ≡ 1 + 3 y 2 + and ε ≡ d φχ 4 (0 , 1 , φ ) sin( m φ ) . 4 πy + 0 Bifurcating Killing sphere - Killing horizon generated by ∂ T . 14 / 20

  49. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Black resonators 2/3: ���� ●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ▼ ▼ ���� ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ◆ ◆ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▲▲▲ ▲ ▲ ● ◆ ● ◆ ● ◆ ● ◆ ���� ▲ ▲ ▲ ● ▲ ● ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ■ ● ■ ● ● ■ ● ■ ■ ● ε = ��� ● ■ ● ■ ● ■ ● ■ ■ ε = ������ ● � / � ���� ■ ● ■ ● ■ ● ◆ ε = ������ ■ ● ■ ● ■ ● ● ▲ � + = ���� ● ● ● ���� ● ● � + = ����� ● ● ���� ���� ����� ����� ����� ����� ����� ����� ����� � / � � 15 / 20

  50. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Black resonators 2/3: ���� ●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ▼ ▼ ���� ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ◆ ◆ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▲▲▲ ▲ ▲ ● ◆ ● ◆ ● ◆ ● ◆ ���� ▲ ▲ ▲ ● ● ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ■ ● ■ ● ● ■ ● ■ ■ ● ε = ��� ● ■ ● ■ ● ■ ● ■ ■ ε = ������ ● � / � ���� ■ ● ■ ● ■ ● ◆ ε = ������ ■ ● ■ ● ■ ● ● ▲ � + = ���� ● ● ● ���� ● ● � + = ����� ● ● ���� ���� ����� ����� ����� ����� ����� ����� ����� � / � � Black resonators extend from the onset of superradiance instability to the Geons (‘onset of turbulent instability’). 15 / 20

  51. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Black resonators 2/3: ���� ●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ▼ ▼ ���� ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ◆ ◆ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▲▲▲ ▲ ▲ ● ◆ ● ◆ ● ◆ ● ◆ ���� ▲ ▲ ▲ ● ● ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ■ ● ■ ● ● ■ ● ■ ■ ● ε = ��� ● ■ ● ■ ● ■ ● ■ ■ ε = ������ ● � / � ���� ■ ● ■ ● ■ ● ◆ ε = ������ ■ ● ■ ● ■ ● ● ▲ � + = ���� ● ● ● ���� ● ● � + = ����� ● ● ���� ���� ����� ����� ����� ����� ����� ����� ����� � / � � Black resonators extend from the onset of superradiance instability to the Geons (‘onset of turbulent instability’). Black resonators exist in regions where the Kerr-AdS solution is beyond extremality. 15 / 20

  52. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Black resonators 3/3: � + = ���� ������� ������� ▲ ▲▲▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ������� ▲ ������� ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ������� ▲ ������� ▲ ▲ ▲ ������� ������� � / � � � / � � ������� ������� ������� ������� ������� ������� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ � / � � / � When black resonators coexist with Kerr-AdS solutions, they have higher entropy - 2 nd order phase transition. 16 / 20

  53. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Black resonators 3/3: When black resonators coexist with Kerr-AdS solutions, they have higher entropy - 2 nd order phase transition. Their horizon is deformed along the φ direction along which they rotate - embedding in 3D spacetime - δZ ≡ Z − ¯ Z . 16 / 20

  54. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 1/3 ? If the dynamics was restricted to specific values of m , say m = 2 , then this would be likely. . . 17 / 20

  55. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 1/3 ? If the dynamics was restricted to specific values of m , say m = 2 , then this would be likely. . . However, recall that m = 2 becomes stable in a region where Kerr-AdS is unstable to perturbations with m > 2 ! 17 / 20

  56. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 1/3 ? If the dynamics was restricted to specific values of m , say m = 2 , then this would be likely. . . However, recall that m = 2 becomes stable in a region where Kerr-AdS is unstable to perturbations with m > 2 ! In addition, the cloud of gravitons - hair - never backreacts very strongly on the geometry - central black hole really looks like Kerr-AdS. 17 / 20

  57. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 1/3 ? If the dynamics was restricted to specific values of m , say m = 2 , then this would be likely. . . However, recall that m = 2 becomes stable in a region where Kerr-AdS is unstable to perturbations with m > 2 ! In addition, the cloud of gravitons - hair - never backreacts very strongly on the geometry - central black hole really looks like Kerr-AdS. Finally, higher m black resonators seem to have increasing entropy with increasing m . 17 / 20

  58. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 1/3 ? If the dynamics was restricted to specific values of m , say m = 2 , then this would be likely. . . However, recall that m = 2 becomes stable in a region where Kerr-AdS is unstable to perturbations with m > 2 ! In addition, the cloud of gravitons - hair - never backreacts very strongly on the geometry - central black hole really looks like Kerr-AdS. Finally, higher m black resonators seem to have increasing entropy with increasing m . From Green et al., it is clear that these solutions are all unstable given that K is spacelike in certain regions! 17 / 20

  59. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 1/3 ? If the dynamics was restricted to specific values of m , say m = 2 , then this would be likely. . . However, recall that m = 2 becomes stable in a region where Kerr-AdS is unstable to perturbations with m > 2 ! In addition, the cloud of gravitons - hair - never backreacts very strongly on the geometry - central black hole really looks like Kerr-AdS. Finally, higher m black resonators seem to have increasing entropy with increasing m . From Green et al., it is clear that these solutions are all unstable given that K is spacelike in certain regions! Conjecture: there is no endpoint - Dias, Horowitz and JES ’11 17 / 20

  60. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 2/3 ? Surprisingly, SUSY can help us ruling out other candidate endpoints. 18 / 20

  61. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 2/3 ? Surprisingly, SUSY can help us ruling out other candidate endpoints. Possible endpoint is a black resonator with Ω H L = 1 , so that the arguments of Green et al. do not apply. 18 / 20

  62. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 2/3 ? Surprisingly, SUSY can help us ruling out other candidate endpoints. Possible endpoint is a black resonator with Ω H L = 1 , so that the arguments of Green et al. do not apply. If the central black hole is small, the system has two components: 18 / 20

  63. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 2/3 ? Surprisingly, SUSY can help us ruling out other candidate endpoints. Possible endpoint is a black resonator with Ω H L = 1 , so that the arguments of Green et al. do not apply. If the central black hole is small, the system has two components: Black resonator ≈ Kerr-AdS + Geon. 18 / 20

  64. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 2/3 ? Surprisingly, SUSY can help us ruling out other candidate endpoints. Possible endpoint is a black resonator with Ω H L = 1 , so that the arguments of Green et al. do not apply. If the central black hole is small, the system has two components: Black resonator ≈ Kerr-AdS + Geon. Search first for a Geon with Ω H L = 1 . 18 / 20

  65. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 2/3 ? Surprisingly, SUSY can help us ruling out other candidate endpoints. Possible endpoint is a black resonator with Ω H L = 1 , so that the arguments of Green et al. do not apply. If the central black hole is small, the system has two components: Black resonator ≈ Kerr-AdS + Geon. Search first for a Geon with Ω H L = 1 . From the first law this means d E = d J/L . 18 / 20

  66. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 2/3 ? Surprisingly, SUSY can help us ruling out other candidate endpoints. Possible endpoint is a black resonator with Ω H L = 1 , so that the arguments of Green et al. do not apply. If the central black hole is small, the system has two components: Black resonator ≈ Kerr-AdS + Geon. Search first for a Geon with Ω H L = 1 . From the first law this means d E = d J/L . Search for purely gravitational solutions with E = J/L ! 18 / 20

  67. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 3/3 ? Recall a central result of four-dimensional gauged SUGRA due to Gary Gibbons and Chris Hull - Positivity of Energy 19 / 20

  68. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 3/3 ? Recall a central result of four-dimensional gauged SUGRA due to Gary Gibbons and Chris Hull - Positivity of Energy Σ ( ˆ ∇ µ ǫ ) † ( ∇ µ ǫ )dΣ 0 ≥ 0 � E − | J | /L = 2 19 / 20

  69. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 3/3 ? Recall a central result of four-dimensional gauged SUGRA due to Gary Gibbons and Chris Hull - Positivity of Energy Σ ( ˆ ∇ µ ǫ ) † ( ∇ µ ǫ )dΣ 0 ≥ 0 � E − | J | /L = 2 Since Σ is arbitrary, we can only have E = J/L if and only if ∇ a ǫ = 0 , i.e. if the solution admits nontrivial Killing spinors - if the solution is SUSY. 19 / 20

  70. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 3/3 ? Recall a central result of four-dimensional gauged SUGRA due to Gary Gibbons and Chris Hull - Positivity of Energy Σ ( ˆ ∇ µ ǫ ) † ( ∇ µ ǫ )dΣ 0 ≥ 0 � E − | J | /L = 2 Since Σ is arbitrary, we can only have E = J/L if and only if ∇ a ǫ = 0 , i.e. if the solution admits nontrivial Killing spinors - if the solution is SUSY. We have investigated whether purely gravitational SUSY solutions exist: 19 / 20

  71. Black holes with a single Killing vector field One black hole to interpolate them all and in the darkness bind them Possible Endpoint of the Superradiance Instability - 3/3 ? Recall a central result of four-dimensional gauged SUGRA due to Gary Gibbons and Chris Hull - Positivity of Energy Σ ( ˆ ∇ µ ǫ ) † ( ∇ µ ǫ )dΣ 0 ≥ 0 � E − | J | /L = 2 Since Σ is arbitrary, we can only have E = J/L if and only if ∇ a ǫ = 0 , i.e. if the solution admits nontrivial Killing spinors - if the solution is SUSY. We have investigated whether purely gravitational SUSY solutions exist: Short answer: No! 19 / 20

  72. Black holes with a single Killing vector field Outlook Conclusions: We have constructed black holes with a single Killing field. They interpolate between superradiance onset and geons. New phase dominates microcanonical ensemble. 20 / 20

  73. Black holes with a single Killing vector field Outlook Conclusions: We have constructed black holes with a single Killing field. They interpolate between superradiance onset and geons. New phase dominates microcanonical ensemble. What to ask me after the talk: Infinite non-uniqueness for Kerr-AdS? How large is it? What is the story in the canonical and grand-canonical ensembles? 20 / 20

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