Stability of black holes and solitons in AdS. Sitter space-time

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1 Stability of black holes and solitons in Anti-de Sitter space-time UFES Vitória, Brasil & Jacobs University Bremen, Germany 24 Janeiro 2014

2 Funding and Collaborations Research Training Group Graduiertenkolleg Models of Gravity Work funded by German Research Foundation (DFG) under Research Training Group Models of gravity Work done in collaboration with: Yves Brihaye - Université de Mons, Belgique Sardor Tojiev - Jacobs University Bremen, Germany

3 References Y. Brihaye and B. Hartmann, Phys. Rev. D 81 (2010) Y.Brihaye and B. Hartmann, Phys.Rev. D 84 (2011) Y. Brihaye and B. Hartmann, JHEP 1203 (2012) 050 Y. Brihaye and B. Hartmann, Phys. Rev. D 85 (2012) Y. Brihaye, B. Hartmann and S. Tojiev, Phys. Rev. D 87 (2013) Y. Brihaye, B. Hartmann and S. Tojiev, CQG 30 (2013)

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6 Solitons localized, finite energy, stable, regular solutions of non-linear equations (particle-like) Topological solitons carry topological charge (non-trivial vacuum manifold) In theories with spontaneous symmetry breaking Examples: (Cosmic) Strings, Monopoles, Domain walls... Non-topological solitons carry globally conserved Noether charge In theories with continuous symmetry Examples: Q-balls, boson stars...

7 Black holes exact solutions of Einstein s equations (and its generalizations) solutions of 4-dimensional Einstein-Maxwell equations uniquely characterized by mass M, angular momentum J and charge Q ( No hair conjecture ) Black hole thermodynamics temperature T = κ 2πk B c, entropy S = k Bc 3 A 4G κ: surface gravity, A: horizon area, G: Newton s constant c: speed of light, k B : Boltzmann s constant, : Planck s constant Information encoded in surface Holographic principle

8 Holographic principle (Gauge/gravity duality) d-dim String Theory (d-1)-dim SU(N) gauge theory (Maldacena; Witten; Gubser, Klebanov & Polyakov (1998)) Couplings λ = (l/l s ) 4 = g 2 N, g s g 2 λ: t Hooft coupling N: rank of gauge group l: bulk scale/ads radius l s : string scale g: gauge coupling g s : string coupling classical gravity limit: g s 0 l s /l 0 dual to strongly coupled QFT with N

9 Application: high temperature superconductivity Bismuth-strontium-calcium-copper-oxide Bi 2 Sr 2 Ca Cu 2 O 8+x superconductivity associated to CuO 2 -planes T c 95 K energy gap ω g /T c = 7.9 ± 0.5 (Gomes et al., 2007) compare to BCS value: ω g /T c = 3.5 Strongly coupled electrons in high-t c superconductors? Taken from wikipedia.org

10 Holographic phase transitions T : temperature, µ: chemical potential AdS black hole at T T c, 0 Holographic conductor/ fluid T large AdS soliton at 0 c, T 0 Holographic insulator phase transition scalar hair formation phase transition scalar hair formation phase transition AdS black hole at T T c, 0 Holographic superconductor/ superfluid AdS soliton at c,t 0 Holographic superconductor/ superfluid

11 Instability of black holes in d-dimensional AdS d (radius L) with respect to condensation of scalar field related to Breitenlohner-Freedman bound on mass m of scalar field (Breitenlohner & Freedman, 1982) m 2 mbf,d 2 (d 1)2 = 4L 2 AdS d stable

12 Two type of instabilities uncharged scalar field near-horizon geometry of extremal black holes given by AdS 2 M d 2 (Robinson, 1959; Bertotti, 1959; Bardeen & Horowitz, 1999) if m 2 BF,2 > m2 > m 2 BF,d asymptotic AdS d stable, but (near-)extremal black hole forms scalar hair close to horizon (Gubser, 2008) scalar field charged under U(1), charge e m 2 eff = m 2 e 2 g tt A 2 t if m 2 m 2 BF,d : asymptotic AdS d stable e 2 g tt large close to horizon of black hole m 2 eff < m 2 BF,d close to horizon black hole forms scalar hair

13 Example (Y. Brihaye & B.Hartmann, Phys.Rev. D81 (2010) ) d=4+1 BF bound m 2 eff AdS unstable AdS stable Pioneering example: RNAdS black hole close to T = 0 unstable to form scalar hair close to horizon (Gubser, 2008)

14 Anti-de Sitter/Conformal fiel theory (AdS/CFT) correspondence applied: use AdS black hole and soliton solutions of classical gravity theory to describe phenomena appearing in strongly coupled Quantum Field Theories values of bulk field on AdS boundary sources for dual theory Applications gravity condensed matter: holographic superconductors/superfluids gravity Quantumchromodynamics (QCD) e.g. holographic description of quark-gluon plasma

15 Higher order curvature corrections Coleman-Mermin-Wagner theorem: no spontaneous symmetry breaking of continuous symmetry in (2+1) dim at finite temperature BUT: holographic superconductors have been constructed using (3+1)-dim black hole solutions of classical gravity (Hartnoll, Herzog & Horowitz (2008) and many more...) higher order curvature corrections on gravity side???

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17 Action Gauss-Bonnet gravity + scalar field ψ + U(1) gauge field A µ 1 S = d 5 x g (16πGL matter 16πG + R 2Λ + α ( R µνλρ R µνλρ 4R µν R µν + R 2)) 4 with matter Lagrangian L matter = 1 4 F MNF MN (D M ψ) D M ψ m 2 ψ ψ, M, N = 0, 1, 2, 3, 4 F MN = M A N N A M field strength tensor D M ψ = M ψ iea M ψ covariant derivative Λ = 6/L 2 : cosmological constant G: Newton s constant, α: Gauss Bonnet coupling e: gauge coupling, m 2 : mass of the scalar field ψ

18 Ansatz for static solutions Metric ds 2 = f (r)a 2 (r)dt f (r) dr 2 + r 2 L 2 dσ2 k,3 with dξ 2 dσ 2 3 for k = 1 hyperbolic k,3 = dx 2 + dy 2 + dz 2 for k = 0 flat dω 2 3 for k = 1 spherical Matter fields A M dx M = φ(r)dt, ψ = ψ(r)

19 Equations of motion f = 2r k f + 2r 2 /L 2 r 2 + 2α(k f ) γ r 3 ( 2e 2 φ 2 ψ 2 + f (2m 2 a 2 ψ 2 + φ 2 ) + 2f 2 a 2 ψ 2 ) 2fa 2 r 2 + 2α(k f )) a = γ r 3 (e 2 φ 2 ψ 2 + a 2 f 2 ψ 2 ) af 2 (r 2 + 2α(k f )) ( ) 3 φ = r a φ + 2 e2 ψ 2 φ a f ( 3 ψ = r + f ) ( + a e ψ 2 φ 2 ) f a f 2 a 2 m2 ψ f where γ = 16πG

20 Conditions on the horizon (Black holes) Regular horizon at r = r h > 0 f (r h ) = 0, a(r h ) finite φ(r h ) = 0, ψ m 2 ψ ( r 2 + 2αk ) (r h ) = 2rk + 4r/L 2 γr ( 3 m 2 ψ 2 + φ 2 /(2a 2 ) ) r=rh

21 Behaviour on the AdS boundary Matter field where φ(r 1) = µ Q r 2, ψ(r 1) = ψ r λ + ψ + r λ+ λ ± = 2 ± 4 + m 2 L 2 eff with effective AdS radius L 2 2α ( ) eff 1 1 4α/L 2 L2 1 α/l 2 + O(α 2 ) Q: charge (k = 1); charge density (k = 1, k = 0)

22 Behaviour on the AdS boundary Asymptotically Anti-de Sitter space-time Metric functions f (r 1) = k + r 2 L 2 eff + f 2 r 2 + O(r 4 ) a(r 1) = 1 + a 4 r 4 + O(r 6 ) f 2, a 4 constants (have to be determined numerically)

23 Properties of black holes specific heat in canonical ensemble: Energy E C = T H ( S/ T H ) C > 0 black hole stable C < 0 black hole unstable 16πGE = 1 αl ( V 2 3f 2 8 a ) 4 3 L 2 eff V 3 : volume of the 3-dimensional space Free energy F = E T H S µq with Hawking temperature T H and entropy S T H = f (r h )a(r h ) S, = r 3 ( h 1 6α ) 4π V 3 4G rh 2

24 Black holes without scalar hair (Boulware & Deser, 1982; Cai, 2003) For m 2 > mbf,2 2 ψ(r) 0 and φ(r) = Q r 2 h Q r 2 f (r) = k + r 2 2α ( 1 ) 1 4α L 2 + 4αM r 4 4αγQ2 r 6, a(r) 1 M: mass parameter, Q: charge (density) event horizon at f (r h ) = 0

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26 Uncharged black holes for k = 1 Uncharged black holes Q = 0; uncharged scalar field e = 0 for k = 1 extremal solution with f (r h ) = 0, f (r) r=rh = 0 exists extremal solution has T H = 0, r (ex) h = L/ 2 close to extremality horizon topology is AdS 2 H 3 (Astefanesei, Banerjee & Dutta, 2008) hyperbolic Gauss-Bonnet black holes in d = 5 have AdS 2 radius R = L 2 /4 α (Y. Brihaye & B.H., PRD 84, 2011) asymptotic AdS 5 stable, near-horizon AdS 2 unstable for mbf,5 2 = 4 L 2 m 2 1 eff 4R 2 = m2 BF,2

27 Black holes with scalar hair, γ 0, α 0 (Y. Brihaye & B.H., PRD 84, 2011) black holes with scalar hair thermodynamically preferred

28 Black holes with scalar hair, γ 0, α 0 (Y. Brihaye & B.H., PRD 84, 2011) the larger α the lower T H at which instability appears

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30 Planar Anti-de Sitter (AdS) space-time For α = 0, φ 0, ψ 0 ds 2 = r 2 L 2 dt 2 + L2 r 2 dr 2 + r 2 L 2 ( dx 2 + dy 2 + dz 2) r Dual theory lives here Dilatation symmetry dual theory scale-invariant r Boundary of AdS (t, x, y, z) λ(t, x, y, z), r r λ r: scale in dual theory x,y,z r=0 Taken from arxiv: r large dual theory strongly coupled (UV regime)

31 Planar Schwarzschild-Anti-de Sitter (SAdS) For α = 0, φ 0, ψ 0 ( r ds 2 2 = L 2 r ) ( 4 h r r 2 L 2 dt L 2 r ) 4 1 h r 2 L 2 dr 2 + r 2 ( dx 2 L 2 + dy 2 + dz 2) r r Dual theory lives here Boundary of SAdS AdS Black hole with planar horizon at r = r h temperature T = r h /(πl 2 ) asymptotically AdS 4+1 x,y,z r=r h horizon Taken from arxiv:

32 Planar Schwarzschild-Anti-de Sitter (SAdS) For α = 0, φ 0, ψ 0 ( r ds 2 2 = L 2 r ) ( 4 h r r 2 L 2 dt L 2 r ) 4 1 h r 2 L 2 dr 2 + r 2 ( dx 2 L 2 + dy 2 + dz 2) r Dual theory lives here new (dimensionless) scale r h /L scale-invariance broken at r < r Boundary of SAdS AdS BUT: scale-invariant dual theory at finite temperature (r ) x,y,z r=r h horizon all T 0 equivalent Taken from arxiv:

33 Planar Reissner-Nordström-Anti-de Sitter (RNAdS) For α = 0, ψ 0 ds 2 = f (r)a 2 (r)dt f (r) dr 2 + r 2 L 2 ( dx 2 + dy 2 + dz 2) with f (r) = r 2 L 2 2M r 2 + γq2 r 4, a(r) 1 planar (outer) horizon at r = r h, asymptotically AdS 4+1 U(1) gauge field in AdS A M dx M = A t (r)dt = Q r 2 h ( 1 r h 2 ) ( r 2 = µ 1 r h 2 ) r 2

34 Planar Reissner-Nordström-Anti-de Sitter (RNAdS) temperature T = 3r h /(πl 2 ) γµ 2 /(2πr h ) can be varied continuously (in particular T = 0 possible) r : scale-invariant dual theory at finite T global U(1) with µ chemical potential In principle: fluid/superfluid interpretation since U(1) not dynamical Mostly: conductor/superconductor interpretation (photons neglected in many condensed matter processes) only (dimensionless) scale: T /µ RNAdS is gravity dual of a conductor/fluid

35 Planar Anti-de Sitter soliton double Wick rotation (t iη, z it) of SAdS metric ds 2 = r 2 ( r L 2 dt L 2 r ) 4 1 ( h r r 2 L 2 dr L 2 r ) 4 h r 2 L 2 dη 2 + r 2 L 2 ( dx 2 + dy 2) to avoid conical singularity η periodic with period τ η = πl2 r 0 where r 0 > 0 space-time unchanged for A M dx M = µdt can be considered at any temperature T

36 Planar Anti-de Sitter soliton only (dimensionless) scale T /µ energy spectrum discrete and strictly positive (mass gap) unstable to decay to planar SAdS for large T (Surya, Schleich & Witt, 2001) AdS soliton is gravity dual of an insulator

37 Onset of superconductivity RNAdS black hole forms scalar hair for T < T c, µ 0 (Gubser, 2008) conductor/superconductor phase transition fluid/superfluid phase transition AdS soliton forms scalar hair for µ > µ c (even at T = 0) (Nishioka, Ryu, Takayanagi, 2010) insulator/superconductor phase transition insulator/superfluid phase transition

38 Including Gauss-Bonnet corrections (Brihaye & B. Hartmann, Phys. Rev. D 81, 2010) Gauss-Bonnet coupling 0 α L 2 /4 =16 G/e 2 = condensation gets harder for α > 0, but not suppressed

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40 Spherically symmetric AdS solitons For k = 1 solitons regular on r [0 : [ exist (Basu, Mukherjee & Shieh, 2009; Dias, Figueras, Minwalla, Mitra, Monteiro & Santos, 2011 ) Boundary conditions at r = 0 f (0) = 1, φ (0) = 0, ψ (0) = 0, a (0) = 0

41 Charged solitons with scalar hair exist only in limited domain of Q-e 2 -plane at e = e c (Q, α): numerical results suggest a(0) 0 for α 0 range of allowed Q and e values enlarged solitons no solitons

42 Charged black hole with scalar hair, k = 1 (Brihaye & B. Hartmann, PRD 85 (2012) ) small large For small α: solution exists down to r h = 0 soliton? For large α: solution has a(r h ) 0 for r h r (cr) h > 0 extremal black hole?

43 Charged black hole with scalar hair, k = 1 (Brihaye & B. Hartmann, PRD 85 (2012) ) For α = 0: Black hole tend to soliton solutions in the limit r h 0

44 Charged black hole with scalar hair, k = 1 (Brihaye & B. Hartmann, PRD 85 (2012) ) α 0: Gauss-Bonnet solitons with scalar hair exist black holes with scalar hair do not tend to corresponding solitons for r h 0

45 Charged black hole with scalar hair, k = 1 (Brihaye & B. Hartmann, PRD 85 (2012) There exist no extremal Gauss-Bonnet black holes with scalar hair.

46 Charged black hole with scalar hair, k = 1 Proof: assume near-horizon geometry to be AdS 2 S 3 : ( ds 2 = v 1 ρ 2 dτ ) ρ 2 dρ2 +v 2 (dψ 2 + sin 2 ψ (dθ 2 + sin 2 θdϕ 2)) v 1, v 2 : positive constants Combination of equations of motion yields ( ρ 2 0 = 16πG ψ 2 + e2 φ 2 ψ 2 ) v 1 ρ 2 v 1 This leads to: ψ = 0 and φ 2 ψ 2 = 0 in near horizon geometry φ 2 = 0 ruled out ψ 0 in near horizon geometry q.e.d.

47 The planar limit (α = 0: Gentle, Pangamani & Withers, (2011)) (α 0: Brihaye & B. Hartmann, PRD 85 (2012) ) apply rescaling r λr, t λ 1 t, M λ 4 M, Q λ 3 Q to Boulware-Deser-Cai solution with ( ( f (r) = r 2 1 r )) 1 1 4α 2α L 2 + 4αM r 4 4αγQ2 r 6 For λ : λ 2 dω 2 3 dx 2 + dy 2 + dz 2 For Q the k = 1 solution becomes k = 0 solution, i.e. becomes comparable in size to AdS radius L electromagnetic repulsion balanced by gravitational attraction present in AdS

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49 Conclusions Two mechanisms can make AdS black holes unstable to scalar condensation uncharged scalar field: black holes with AdS 2 factor in near-horizon geometry (near-extremal black holes) Example in this talk: uncharged, static black holes with hyperbolic horizon (k = 1) charged scalar field: coupling to gauge field lowers effective mass of scalar field Examples in this talk: charged, static black holes with flat or spherical horizon (k = 0, k = 1) Black holes with scalar hair thermodynamically preferred

50 Outlook Instabilities of other black holes and solitons in AdS d charged and rotating Einstein black holes: Y. Brihaye & B.H., JHEP 1203 (2012) 050 charged and rotating Gauss-Bonnet black holes: Y. Brihaye, B.H. & S. Tojiev, Phys. Rev. D 87(2013) charged boson stars: Y. Brihaye, B.H. & S. Tojiev, CQG 30 (2013) solitons in conformal theories: Y. Brihaye, B.H. & S. Tojiev, Phys. Rev. D 88 (2013) Implications on CFT side? Implications for stability of String Theory/Supergravity black holes?

51 Still sceptical about Holography? Street Art in Germany Muito orbigada pela sua atenção!