Black holes, black strings and cosmological constant Yves Brihaye Université de Mons-Hainaut Mons, BELGIUM 418-th WE-Heraeus Seminar, Bremen, 25-29 August 2008
Outline of the talk Introduction. The models and the Ansätze Rotating Black Holes in higher dimensions. Einstein-Maxwell. Domain of existence. Black Strings in higher dimensions.einstein. Effect of Maxwell and/or of rotation Stability of AdS black strings Effect of Gauss-Bonnet
In collaboration with T. Delsate (Univ. Mons, Belgium) E. Radu (Univ. Maynooth, Ireland + Tours, France) Ch. Stelea (Perimeter Institute, Canada). T. D. Tchrakian (Univ. Maynooth, Ireland) Related work with A. Chakrabarti (E.P., Paris), B. Hartmann (Jacobs U., Bremen), J. Kunz (Univ. Oldenburg).
I. INTRODUCTION There are several reasons which lead us to study General Relativity and field theory in more than four dimensions:. Solution to the hierarchy problem.. String theory requires more than four dimensions. Brane worlds: Our universe is a 3-dimensional brane embedded into a higher dimensional bulk. Spectrum of solutions of General Relativity in higher dimensions is much bigger than in 4 dimensions
. Black hole solutions with different horizon topopogies occur: Problem of classification and stability. There are several reasons to include a cosmological constant Λ in the equations. Astronomical observations suggest the presence of a positive Λ.. (Anti-)de-Sitter/Conformal Field theory correspondence ((A)dS/CFT). Physical aspects: Dark Energy.
. Mathematical interest for understanding the pattern of solutions of The Einstein equations with cosmological constant.
II. Rotating Black Holes in higher dimensions The model * Einstein-Maxwell Lagrangian with a cosmological constant Λ in a d dimensional space-time. I = 1 16πG d M dd x g(r 2Λ F μν F μν ) 1 8πG d M dd 1 x hk, *G d denotes the d-dimensional Newton constant. Units are chosen in such a way that G d appears as an overal factor. * Define a ((Anti-)de-Sitter) radius l according to (d 2)(d 1) Λ=± 2l 2 * Einstein-Maxwell equations follow from the variation of the action with respect to the metric and the electromagnetic fields.
The ansatz: Space-times with odd dimensions, d =2N+1 +p(r) ds 2 = b(r)dt 2 + dr2 N 1 f(r) + g(r) +h(r) N k=1 N k 1 k=1 l=0 k 1 l=0 i=1 i 1 j=0 cos 2 θ j dθi 2 cos 2 θ l sin 2 θ k (dϕ k w(r)dt) 2 cos 2 θ l sin 2 θ k dϕ 2 k N k 1 k=1 l=0 cos 2 θ l sin 2 θ k dϕ k * N + 1 Killing vectors ϕk, N angular momenta. * Most general Maxwell potential consistent with these symmetries A μ dx μ = V (r)dt + a ϕ (r) N k=1 k 1 l=0 cos 2 θ l sin 2 θ k dϕ k
For d = 5, the metric above takes the form ds 2 = b(r)dt 2 + dr2 f(r) + g(r)dθ2. + h(r)((sin θ) 2 (dϕ 1 w(r)dt) 2 +(cosθ) 2 (dϕ 2 w(r)dt) 2 ). + p(r)(sin θ) 2 (cos θ) 2 (dϕ 1 dϕ 2 ) 2 * Inserting this ansatz in the Einstein-Maxwell equations results in a system of seven non-linear, coupled differential equations provided p(r) = g(r) h(r). * b(r),f(r),g(r),h(r),w(r) are unknown. * one can be fixed: e.g. g = r 2
Explicit Solutions (i) The vacuum black holes are recovered for a vanishing gauge fields (V = a ϕ =0)and f(r) =1 ɛ r2 l 2 2MΞ 2Ma2 + rd 3 r d 1, h(r) =r2 (1 + 2Ma2 ), rd 1 w(r) = 2Ma r d 3 h(r), g(r) =r2, b(r) = r2 f(r) h(r), * M and a are two constants related to the solution s mass and angular momentum and Ξ = 1 + a 2 /l 2. * Generalize the Myers-Perry (MP) solutions for the case of non-vanishing Λ. * Here and in the following, Ω w(r h ) will denote the angular velocity at the event horizon.
(ii) The (Anti-)de-Sitter-Reissner-Nordstrom black holes are recovered in the limit w(r) =a ϕ (r) =0: f(r) = b(r) =1 ɛ r2 l 2 2M r d 3 + q 2 h(r) = g(r) =r 2, V(r) = 2(d 2)(d 3)r 2(d 3), q (d 3)r d 3 where M and q are related to the mass and electric charge of the solution. (iii) Explicit charged-rotating black holes are also know when a Chern-Simons term is added Klemm, Sabra (2000) Cvetic, Lu, Pope (2004)
Charged, rotating black holes with Λ=0: J. Kunz, F. Navarro-Lerida, J. Viebahn (2006) J. Kunz, F. Navarro-Lerida, A.K. Petersen (2005) Charged, rotating black holes with Λ < 0: J. Kunz, F. Navarro-Lerida, E. Radu (2007) These solutions are constructed numerically using isotropic coordinate.
Charged Rotating black holes for Λ > 0. * For Λ > 0, we expect a cosmological horizon to appear : horizons at r h and r c. * f(r h )=0, f(r c )=0,b(r h )=0, b(r c )=0. * Equations have two singular points. * Numerical Strategy (Y.B. and T. Delsate (2007)) :. Use a Schwarzchild coordinate g(r) =r 2..Fixr h, r c by hand and add an equation dλ/dr =0.. Implement the boundary conditions at r h, r c and solve Eqs. for r [r h,r c ], determining Λ..SolveEqs.forr [r c, ] with initial data at r = r c.
The profile of the metric and Maxwell functions for r c = 3,r h =1for a h =0.5 andω=0.62
* In the gauge g = r 2 the equations look like : Λ =0, f =...,,b =..., h =..., w =..., V =..., a ϕ =... * Invariance : b, w, V can be arbitrarily redefined b μ 2 b, w μw, V μv + C * The boundary conditions for r [r h,r c ] f(r h )=0, b(r h )=0, b (r h )=1, Γ h (r h )=0 w(r h )=w h, V(r h )=0, a ϕ (r h)=a h, Γ A (r h )=0 fixinginpassingthescaleofb f(r c )=0, b(r c )=0, Γ h (r c )=0, Γ A (r c )=0 * The parameters w h, a h are fixed by hand and control the angular and magnetic moments.
* The conditions Γ = 0 are necessary for regular solutions at the horizon. e.g. Γ A (r) 4a ϕ b h + r 4 f (hw V + a ϕhww a ϕb )(r) (1) * After the integration, the functions w, b, V have to be renormalized such that space-time is asymptotically desitter : b(r) Λr 2 +1+O(1/r 2 ) for r * Consequence: impossible to study solutions for fixed Q * The domain of existence of solutions in the plane a h, Ω h. (i) For fixed a h and w h varying black holes exist only on a finite interval of the horizon angular velocity Ω. (ii) In the critical limits, solutions converge to extremal black holes, i.e. with f(r h )=0,f (r h )=0.
Metric parameters and l 2 as functions of Ω for r c =3,r h = 1 and for several values of a h. Rem 1 : The value 1/l 2 is nearly constant. Rem 2 : Pattern is similar for Λ < 0, but looks different for y h fixed, with y isotropic coordinate.
Limiting Extremal solutions can be constructed numerically by implementing directly the appropriate BC. Extremal rotating black hole with Ω = 0.2 andλ< 0
Some Physical quantities: The asymptotic form of the solutions to the Einstein equations : b(r) = ɛ r2 l 2 +1+ f(r) = ɛ r2 l 2 +1+ α r d 3 + O(1/r2d 6 ), (2) β r d 3 + O(1/rd 1 ), (3) h(r) =r 2 (1 + ɛ l2 (β α) r d 1 + O(1/r 2d 4 )), w(r) = Ĵ r d 1 + O(1/r2d 4 ). (4) The mass-energy E and the angular momentum J E = V d 2 16πG d (β (d 1)α), J = V d 2 8πG d Ĵ. (5) Cons. charges, obtained with Counterterm Formalism. Balasubramanian and Kraus: 1999, and many authors.
Mass and Angular momentum of the Λ > 0 Black holes for r c =3,r h = 1 as function of the angular velocity Ω Rem: Conserved quantities can be defined at the two horizons and Smarr formulas relating them have been obtained.
III Black string Model and metric Model : * Einstein equations with the Einstein-Hilbert action in d-dimensions and with a cosmological constant. * One of the spacelike dimensions of space-time, z x d 1 plays a special role. * Space-time is the warped product of a d 1-dimensional black hole metric with the extra-dimension z. * The corresponding horizon has the topology of S d 3 S 1. * Black Strings have a metric ds 2 bs = a(r)dz2 + ds 2 * ds 2 see above. * Metric does not depend on the coordinate z : uniform black strings. * Non-uniform black strings exist for sufficiently large values of the radius of the extra coordinate metric depends on r and z (knownforλ=0).
* In absence of the electromagnetic field and of rotation, the above model leads to a system of three differential equations: f =...,, a =..., b =..., = Q(f, a, b, b, Λ) w(r) =0, h(r) =r 2, g(r) =r 2 * The BC for black strings at the horizon r h : f(r h )=0, a(r h )=1, b(r h )=0, b (r h )=1 using the invariance of the equations under an arbitrary rescaling of a, b.
Solutions Λ < 0 *ForΛ< 0, solutions were construced by Mann, Radu and Stelea (2006). * Once suitably renormalized, the fields obey asymptotically f(r),a(r),b(r) Λr 2 +1+O(1/r 2 ), * Regular solutions on r [0, ] exists i.e. f(0) = 1,b(0) = 1. * In the limit r h 0, the black strings approach the regular solution on r ]0, [. * Rotating solutions with w(r) > 0, g(r) r 2 exist (Y.B., E.Radu, Ch. Stelea (2007)).
Profile of the metric functions for a rotating AdS black string for r h =1, Ω=1
* The AdS black Strings can be characterized by their mass M and tension T given by the asymptotic decay of the metric functions: M = ld 4 16π LV d 3[c z (d 2)c t ]+M c (d) a(r) = + c z ( l r )d 3 +..., b(r) = + c t ( l r )d 3 +... * Thermodynamical quantities can also be determined; they depend on the value of the metric at the horizon r h : the entropy S S = 1 4 rd 3 h LV d 3 a(r h ) and Hawking temperature T H T H = 1 4 b (r h ) r h ((d 4) + (d 1)r 2 h /l2 )
Local thermodynamical stability is related to the sign of the heat capacity S C = T H, for L fixed T H Solutions with C>0arestable(C<0 are unstable). Remark: * Asymptotically flat black strings with different r h are related by a rescaling of the radial coordinate, * AdS black strings with Λ fixed and r h varying form a family of intrinsically different solutions. * The solutions obtained by varying r h form two branches distinguished thermodynamically: solutions with small r h have C>0, solutions with large r h have C<0.
5 4 4S/(LV 1,2 ) 2 3 d=5 Λ=-1 2 1 k 2 1 2 0-1 1.2 1.35 1.5 1.65 1.8 1.95 T H Entropy as function of T H for the family of black strings corresponding to Λ = 1 andd =5 1
* Charged black String with A = V (r)dt were also considered. * The Maxwell equation can be solved directly : F tr = q r d 3 f(r) a(r)b(r), q = constant * Charged black strings exist for r h >r h,min > 0forq>0. * Charge or rotation change the thermodynamical stability. Addition of charge or rotation tends to stabilize the solution. * This property holds for fixed L, Q (electric charge) and for fixed L, J (angular momentum) : Grand canonical ensemble.
0.8 Q=5 d=5 k=1 0.6 0.4 S 0.2 0 Q=1 Q=0.2 Q=0.001 0.3 0.4 0.5 0.6 Entropy as function of T H for families of black strings corresponding to Λ = 1 andd = 5 with fixed electric charge T H
1 0.8 0.6 J=0.25 J=0.0025 J=0 S 0.4 0.2 0 0.4 0.45 0.5 0.55 0.6 0.65 0.7 T H Entropy as function of T H for families of black strings corresponding to Λ = 1 and d = 5 and fixed angular momentum
Solutions with Λ > 0 (Y. B. and T. Delsate 2007) * No solutions that have both a regular horizon at r = r h and are asymptotically de Sitter f(r),a(r),b(r) Λr 2 +1+O(1/r 2 ), * Instead, the solutions with a regular horizon at r = r h evolve into a configuration such that α = 2(d 3) a(r) r α, b(r) r α, f(r) r γ 2(d 2)(d 3),γ =2(d 2)+ 2(d 2)(d 3) * These are solutions of the asymptotic equations which are different from ds space-time. * Singular at r = * Regular solutions at the origin exist, with the same asymptotics * Imposing a regular cosmological horizon at r = r c leads to an essential singularity inside
Solution with r h =0.5, Ω = 0.5, d =5andΛ> 0
III Stability of AdS black strings * Let the extra-dimension z be periodic : z [0,L] * Asymptotically flat black strings present a long wavelenghth instability (R. Gregory and R. Laflamme, 1993). * What about AdS black strings...? * Does the thermodynamical instability lead to an instability of the Gregory-Laflamme (GL) type? * Is the Gubser-Mitra conjecture fulfilled? * Gubser-Mitra conjecture (2001) : For black brane solution to be free of dynamical instabilities it is necessary and sufficient for it to be locally thermodynamically stable
Compare to an older problem in the Electroweak model. * Yang-Mills-Higgs equations admit a solution: the Klinkhamer- Manton sphaleron (1984). * Linearized equations for a time-dependent fluctuation about the sphaleron Φ(t, r) =Φ KM (r)+e ωt η(r) Hη = ωη * Eigenvalues ω(m H ) can be determined as functions of M H, the mass of the Higgs field. * For values M H,c,wehaveω(M H,c )=0. *There are zero modes and the number of negative eigenvalues depends on M H. * Consequence : New solutions exist for M H M H,c bifurcating from the KM sphaleron: the bisphaleron. Y. B. and J. Kunz (1988); L. Yaffe (1988)
Analogies Yang-Mills Higgs equations Einstein equations Higgs Field Mass M H Length of co-dimension L KM-Sphaleron Uniform Black String (Schwarzschild) BI-Sphaleron Non Uniform Black String Morse theorem + Catastrophe theory Gubser-Mitra conjecture
* To construct an instability of the GL-type, we consider a deformation of the metric: ds 2 = b(r)e 2A(r,z) dt 2 + e 2B(r,z) ( dr 2 f(r) + a(r)dz2 ) +r 2 e 2F (r,z) dω 2 d 3 assuming no time dependance, i.e. Ω = 0, with X(r, z) =ɛx 1 (r)cos(kz)+ɛ 2 (X 0 (r)+x 2 (r)cos(2kz)) + O(ɛ 2 ) where X is A, B, F, k =2π/L. * L is the length of coordinate z and ɛ is an infinitesimal parameter. * Extracting the linear terms in ɛ from Einstein equations leads to a system of linear differential equations in A 1 (r),b 1 (r),f 1 (r)
Properties of these equations: The potentials : the functions a(r),b(r),f(r) are known only numerically. The perturbations should vanish asymptotically. The function B 1 (r) can be eliminated from the system. Regularity at r = r h leads to two conditions of the form. Γ(A, C, A,C )(r = r h )=0
Special values of k 2 have to be determined such that boundary conditions are fulfilled up to a global factor Eigenvalue problem. A rescaling of the radial coordinate can be used to set either r h or Λ to a canonical value, e.g. Λ = 1. Solutions with k 2 > 0 are unstable, k 2 < 0arestable.
2 1.5 1 k 2 0.5 d=5 d=8 0 d=6-0.5 κ=1 Λ=-1-1 0.5 1 1.5 2 2.5 r h The value of k 2 as a function of r h for d = 5, 6, 8 and Λ= 1 Y.B., T. Delsate and E. Radu, 2008.
5 4 4S/(LV 1,2 ) 2 3 d=5 Λ=-1 2 1 k 2 1 2 0 1-1 1.2 1.35 1.5 1.65 1.8 1.95 T H The entropy S and the eigenvalue k 2 as function of T H for d = 5. The figure shows that the GM conjecture is obeyed.
25 20 1 d=8 κ=1 Λ=-1 15 S/(5LV 1,5 ) 10 2 5 20k 2 2 0 1 1.74 1.75 1.76 1.77 1.78 1.79 1.8 T H The entropy S and the eigenvalue k 2 as function of T H for d = 8. The figure shows that the GM conjecture is obeyed.
IV. Einstein-Gauss-Bonnet (EGB) AdS black strings. Motivations: * Gauss-Bonnet interaction is the first curvature correction to General Relativity from the low energy effective action of string theory * Quartic in the metric, but second order in derivatives. The EGB action: I = 1 16πG M dd x g R is the Ricci scalar, ( R 2Λ + α ) 4 L GB, L GB = R 2 4R μν R μν + R μνστ R μνστ, is the GB term and Λ = (d 2)(d 1)/2l 2.
The Einstein-Gauss-Bonnet (EGB) equations: Variation of the action with respect to the metric results in the EGB equations: where R μν 1 2 Rg μν +Λg μν + α 4 H μν =0, H μν =2(R μσκτ R σκτ ν 2R μρνσ R ρσ 2R μσ R σ ν +RR μν) 1 2 L GBg μν.
Effective AdS radius: It is useful to define an effective Anti-de-Sitter radius by means of 1+U l c = l 2, with U = 1 α(d 3)(d 4) l 2, * the counterterms take a simpler form (see next) * appears naturally in the black string asymptotics. * Leads naturally to the existence of an upper bound for the Gauss-Bonnet coefficient, α α max = l 2 /(d 3)(d 4), which holds for all asymptotically AdS solutions.
Counterterm Formalism * The various solutions of the Lagrangians studied here do not have a finite action because of the non-compact character of space-times with Λ = 0 or Λ < 0. * In order to have a finite action, one technique consists in adding suitable counterterms to the action. (Balasubramanian, Kraus, 1999) * These counterterms have to fulfill several requirements: (i) depend on curvature invariants associated with the geometry at the boundary of space-time (ii) do not affect the equations (iii) are also infinite, in order to cancel the divergences. * Lead to a boundary stress tensor Ta b allowing to define conserved quantities like mass,... (Ghezelbash-Mann, 2002). * Known for Einstein-Hilbert action. NOT Inthecaseof Einstein-Gauss-Bonnet.
For d < 8 and even, the appropriate counterterms read (Y.B. and E. Radu, 2008) Ict 0 = 1 8πG M dd 1 x γ { ( d 2 )( 2+U l c 3 ) l cθ (d 4) (2 U)R 2(d 3) l3 c Θ (d 6) 2(d 3) 2 (d 5) [ ( U R ab R ab d 1 ) 4(d 2) R2 d 3 ]} 2(d 4) (U 1)L GB, where * γ in the induced metric of the boundary of space-time. * R, R ab and L GB are the curvature, the Ricci tensor and the Gauss-Bonnet term associated with γ. * Θ(x) is the step-function with Θ (x) =1providedx 0, andzerootherwise.
Rem1: Appears as a truncated series of powers of R abcd and l c Rem2: The known counterterms of d-dimension Einstein gravity are recovered for α 0(U 1). Rem3 : For odd values the expression is more involved.
Black string solutions * Black string solutions of the Einstein-Gauss-Bonnet equations are obtained as smooth deformations of the Einstein black strings. * Around the horizon the profile deviates significantly even for infinitesimal values of α. * The Smarr relation available for α = 0 is still obeyed M + T L = T H S * The curve S(T H ) demonstrates drastic effects of the Gauss-Bonnet interaction on the thermodynamical properties.
Effect of the Gauss-Bonnet term on the solution. 2 1.5 a(α)/a(α=0)+1 α=0.00005 α=0.001 d=8 r h =0.01 k=1 1 f(α)/f(α=0) 0.5 0-0.5-1 -b(α)/b(α=0) 1.5 f(r) 1 a(r) 0.5 0 d=8 α=0-0.5 -b(r) -1-1.5 0 0.1 0.2 0.3 0.4 0.5 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 r Comparison of profiles of E-BS and EGB-BS for small values of α and d =8
Effect of the Gauss-Bonnet term on the thermodynamical properties 0.08 0.07 d=6, k=1 0.06 α=0.1 0.05 α=0 S 0.04 0.03 0.02 0.01 α=0.05 α=0.01 0 0 0.2 0.4 0.6 0.8 1 The entropy S as a function of T H for different values of α and d =6 T H
Domain of existence recall: For α = 0, regular solutions exist for r [0, ] and BS for r [r h, ]. What about EGB black string? i.e. if α>0forr h 0? * The numerical results strongly suggest that the EGB black strings have no regular limit. *Inparticular,f (r h )andb (r h )tendtozeroforr h 0. *Ford = 6 the EGB black strings converge to a singular configuration for r h 0.
The parameters at the horizon as functions of r h for different values of α and d =6
Conclusions * We have studied several extensions of black holes and black strings solutions in the presence of a cosmological constant. * Up to our knowledge, the extensions that we have discussed do not allow explicit solutions of the equations. * We therefore used numerical methods to solve the equations. * We hope these results contribute to a more general understanding of the classification of solutions of Einstein equations in d>4 * The problems we presented in this talk lead to differential equations. The numerical construction of non uniform AdS black strings and the study their properties are under investigation. It involves partial differential equations.