T. Delsate University of Mons-Hainaut Talk included in the Rencontres de Moriond 2009 – La Thuile (Italy).

 Motivations  Asymptotically flat space:  Black Strings and the GL instability  Phase diagram of higher dimensional black strings  Asymptotically AdS space:  Uniform black string in AdS  Still a GL instability ?  Perturbative non uniform black string in AdS: a first step in the phase diagram  Non-perturbative analysis : hint for localised black holes  Conclusion

 Suppose d>4. Gravity propagates in ED.

 Black Objects = theoretical labs for gravity. NB : No more uniqueness thm in ED.

 Suppose d>4. Gravity propagates in ED.  Black Objects = theoretical labs for gravity. NB : No more uniqueness thm in ED.  If ED are compact, possible black strings

 Suppose d>4. Gravity propagates in ED.  Black Objects = theoretical labs for gravity. NB : No more uniqueness thm in ED.  If ED are compact, possible black strings  Better understanding of Black String phases.  Relatively well understood in asymptotically flat space.

 Suppose d>4. Gravity propagates in ED.  Black Objects = theoretical labs for gravity. NB : No more uniqueness thm in ED.  If ED are compact, possible black strings  Better understanding of Black String phases.  Relatively well understood in asymptotically flat space.  Why AdS ?

 Suppose d>4. Gravity propagates in ED.  Black Objects = theoretical labs for gravity. NB : No more uniqueness thm in ED.  If ED are compact, possible black strings  Better understanding of Black String phases.  Relatively well understood in asymptotically flat space.  Why AdS ?  AdS / CFT duality

 Suppose d>4. Gravity propagates in ED.  Black Objects = theoretical labs for gravity. NB : No more uniqueness thm in ED.  If ED are compact, possible black strings  Better understanding of Black String phases.  Relatively well understood in asymptotically flat space.  Why AdS ?  AdS / CFT duality  Why not ?

 d-dim Black string solution to Einstein equation, R MN = 0: r z L r0r0 (d-1) Tangherlini

 d-dim Black string solution to Einstein equation, R MN = 0: r z L (d-1) Tangherlini+ 1 Ricci flat direction

Black Strings are unstable towards long wavelength perturbations (R. Gregory, R. Laflamme – 1993)

Black Strings are unstable towards long wavelength perturbations (R. Gregory, R. Laflamme – 1993)  k

Black Strings are unstable towards long wavelength perturbations (R. Gregory, R. Laflamme – 1993)  k kckc UnstableStable

For a given mass, long black string are unstableshort black strings are stable  k kckc UnstableStable

For a given mass, long black string are unstableshort black strings are stable  k kckc UnstableStable

-> Unstable black strings. What should they decay to? r z L

-> Unstable black strings. What should they decay to ? Localised Black Hole ? r z LL ?

 Answer : NO ! ->Takes an infinite proper time at the horizon for such a transition… (Horowitz and Maeda, 2001)

 Answer : NO ! ->Takes an infinite proper time at the horizon for such a transition… (Horowitz and Maeda, 2001) -> Suggests the existance of something else L Non uniform Black String (Gubser 2002, Wiseman 2003)

Thermodynamical quantities : Mass M (time translation), Entropy S (quarter horizon area), Temperature T H (regular Euclidean sections), Tension T (z translation)

Thermodynamical quantities : Mass M (time translation), Entropy S (quarter horizon area), Temperature T H (regular Euclidean sections), Tension T (z translation) Dimensionless quantities : n = T /ML  = G d M/L d-3

Thermodynamical quantities : Mass M (time translation), Entropy S (quarter horizon area), Temperature T H (regular Euclidean sections), Tension T (z translation) Dimensionless quantities : n = T /ML  = G d M/L d-3 Harmark, Niarchos and Obers n  Uniform Black String Non Uniform BS Localised BH

Thermodynamical quantities : Mass M (time translation), Entropy S (quarter horizon area), Temperature T H (regular Euclidean sections), Tension T (z translation) Dimensionless quantities : n = T /ML  = G d M/L d-3 n  Uniform Black String Non Uniform BS Localised BH cc Merger Point Harmark, Niarchos and Obers

Thermodynamical quantities : Mass M (time translation), Entropy S (quarter horizon area), Temperature T H (regular Euclidean sections), Tension T (z translation) Dimensionless quantities : n = T /ML  = G d M/L d-3 cc Topological phase transition = difficult to study n  Uniform Black String Non Uniform BS Localised BH Harmark, Niarchos and Obers Merger Point

Do all these phenomenae have an AdS counterpart ?

 Uniform black string solution in AdS (Mann, Radu, Stelea – 2006)

 Uniform black string solution in AdS (Mann, Radu, Stelea – 2006)

 Uniform black string solution in AdS (Mann, Radu, Stelea – 2006)  f 1 =f 1 (r 0,l,d), a 0, b 1 arbitrary constants (fixed by asymptotically AdS requirement)  l² being the AdS radius  No Closed form solution -> numerics

 Thermodynamics :

 T H, S as usual

 Thermodynamics :  T H, S as usual  Mass, Tension : Counter term procedure (Balasubramanian, Kraus 1999) ▪ Involves integration over z from 0 to L.

 Thermodynamics :  T H, S as usual  Mass, Tension : Counter term procedure (Balasubramanian, Kraus 1999) ▪ Involves integration over z from 0 to L. ▪ NB : No obvious background for background substraction methods

 2 phases :

 2 phases :  Small black string (r 0 /l <<1): ▪ Essentially same feature as flat case (thermodynamically unstable)

 2 phases :  Small black string (r 0 /l <<1): ▪ Essentially same feature as flat case (thermodynamically unstable)  Big black string : ▪ Becomes thermodynamically stable (« AdS acts like a confining box ») NB : This phenomena occurs for AdS black Holes (Hawking, Page 1983)

 Non-uniform ansatz

 Non-uniform ansatz

 Non-uniform ansatz  Xi’s = Fourier modes  k c =2  /L fixes the length of the black string.

 Order  : Stability. (=static perturbation,  = 0) (Brihaye, Delsate and Radu – 2007)

 Order  : Stability. (=static perturbation,  = 0) (Brihaye, Delsate and Radu – 2007)  Equations of Motion : Eigen value problem for k c ².  Also numerical (background is numerical…)

 Order  : Stability. (=static perturbation,  = 0) (Brihaye, Delsate and Radu – 2007)  Equations of Motion : Eigen value problem for k c ².  Also numerical (background is numerical…)  k c ² > 0 = Exists GL instability  k c ² < 0 = Dynamically stable. NB : AdS radius provides a lengthscale -> µ 2 = L/l = 1/(l k c )

 Results :

 Results :  Small AdS black string dynamically unstable

 Results :  Small AdS black string dynamically unstable  Big AdS black string dynamically stable

 Results :  Small AdS black string dynamically unstable  Big AdS black string dynamically stable THE DYNAMICAL AND THERMODYNAMICAL INSTABILITIES MATCH ! (=Gubser-Mitra conjecture, 2001)

 Order  ² : (Delsate – 2008)

 Corrections on thermodynamical quantities

 Order  ² : (Delsate – 2008)  Corrections on thermodynamical quantities Recall the integration over z from 0 to L=2  /k c :  Order  : linear in cos(k c z)

 Order  ² : (Delsate – 2008)  Corrections on thermodynamical quantities Recall the integration over z from 0 to L=2  /k c :  Order  : linear in cos(k c z)  Order  ² : ▪Linear in X 0, X 2 cos(2k c z) -> X 2 terms vanish ▪Terms of the form X 1 ²

 Corrections on thermodynamical quantities at fixed length n 

n  µ 1 (1) µ 1 (3) µ 1 (2) n UBS (µ 1 (i) ) n pNUBS (µ 1 (i) ) n UBS (µ)/L n 

 Corrections on thermodynamical quantities at fixed length n  µ 1 (1) µ 1 (3) µ 1 (2) n UBS (µ 1 (i) ) n pNUBS (µ 1 (i) ) n UBS (µ)/L n  Effect of the new Length scale !!

 Corrections on thermodynamical quantities at fixed length n  µ 1 (1) µ 1 (3) µ 1 (2) n UBS (µ 1 (i) ) n pNUBS (µ 1 (i) ) n UBS (µ)/L n  ?

 Corrections on thermodynamical quantities at fixed length (S UBS /L independant of L) S/L  µ 1 (1) µ 1 (3) µ 1 (2) S/L UBS (µ 1 (i) ) S/L pNUBS (µ 1 (i) ) S UBS (T H )/L

 Corrections on thermodynamical quantities at fixed length (S UBS /L independant of L) S/L  µ 1 (1) µ 1 (3) µ 1 (2) µ 1 (0)  S/  T H >0 : new thermodynamically stable phases (T. Delsate – 12/2008)

 Corrections on thermodynamical quantities at fixed length (S UBS /L independant of L) S/L  µ 1 (1) µ 1 (3) µ 1 (2) µ 1 (0)  S/  T H >0 : new thermodynamically stable phases L large, r 0 small : « Long small NUBS » (T. Delsate – 12/2008)

 First results in non perturbative approach confirms the perturbative results

 Regime of strong deformation suggests the existance of localised black holes

 First results in non perturbative approach confirms the perturbative results  Regime of strong deformation suggests the existence of localised black holes  Only partial results, still under investigation

Prediction from perturbative analysis are confirmed within the numerical accuracy d0d0

Embedding of the horizon in euclidean space z r

Embedding of the horizon in euclidean space Preiodicity in z direction -> Localised black hole phase ? z r

Embedding of the horizon in euclidean space Preiodicity in z direction -> Localised black hole phase ? NB : d 0 controls the deformation z r

 Small black Strings in AdS are unstable - Large black strings are not,  neither thermo nor dynamically (Gubser Mitra)  Small AdS BS follow the same pattern as Flat case  New thermo stable pNUBS  To be confirmed with non perturbative analysis  Hint for the localised AdS black hole phase  First approximation for AdS black rings ?  (thin black rings ; works for  =0).  Connection with boundary CFT in AdS/CFT context?  New backgrounds for the dual theory (S d-3 xS 1 ).  Dual of the new stable long-small black strings?

 Small black Strings in AdS are unstable - Large black strings are not,  neither thermo nor dynamically (Gubser Mitra)  Small AdS BS follow the same pattern as Flat case  New thermo stable pNUBS  To be confirmed with non perturbative analysis  Hint for the localised AdS black hole phase  First approximation for AdS black rings ?  (thin black rings ; works for  =0).  Connection with boundary CFT in AdS/CFT context?  New backgrounds for the dual theory (S d-3 xS 1 ).  Dual of the new stable long-small black strings?

 Small black Strings in AdS are unstable - Large black strings are not,  neither thermo nor dynamically (Gubser Mitra)  Small AdS BS follow the same pattern as Flat case  New thermo stable pNUBS  To be confirmed with non perturbative analysis  Hint for the localised AdS black hole phase  First approximation for AdS black rings ?  (thin black rings ; works for  =0).  Connection with boundary CFT in AdS/CFT context?  New backgrounds for the dual theory (S d-3 xS 1 ).  Dual of the new stable long-small black strings?

 Small black Strings in AdS are unstable - Large black strings are not,  neither thermo nor dynamically (Gubser Mitra)  Small AdS BS follow the same pattern as Flat case  New thermo stable pNUBS  To be confirmed with non perturbative analysis  Hint for the localised AdS black hole phase  First approximation for AdS black rings ?  (thin black rings ; works for  =0).  Connection with boundary CFT in AdS/CFT context?  New backgrounds for the dual theory (S d-3 xS 1 ).  Dual of the new stable long-small black strings?

 Small black Strings in AdS are unstable - Large black strings are not,  neither thermo nor dynamically (Gubser Mitra)  Small AdS BS follow the same pattern as Flat case  New thermo stable pNUBS  To be confirmed with non perturbative analysis  Hint for the localised AdS black hole phase  First approximation for AdS black rings ?  (thin black rings ; works for  =0).  Connection with boundary CFT in AdS/CFT context?  New backgrounds for the dual theory (S d-3 xS 1 ).  Dual of the new stable long-small black strings?

 Small black Strings in AdS are unstable - Large black strings are not,  neither thermo nor dynamically (Gubser Mitra)  Small AdS BS follow the same pattern as Flat case  New thermo stable pNUBS  To be confirmed with non perturbative analysis  Hint for the localised AdS black hole phase  First approximation for AdS black rings ?  (thin black rings ; works for  =0).  Connection with boundary CFT in AdS/CFT context?  New backgrounds for the dual theory (S d-3 xS 1 ).  Dual of the new stable long-small black strings?

 Small black Strings in AdS are unstable - Large black strings are not,  neither thermo nor dynamically (Gubser Mitra)  Small AdS BS follow the same pattern as Flat case  New thermo stable pNUBS  To be confirmed with non perturbative analysis  Hint for the localised AdS black hole phase  First approximation for AdS black rings ?  (thin black rings ; works for  =0).  Connection with boundary CFT in AdS/CFT context?  New backgrounds for the dual theory (S d-3 xS 1 ).  Dual of the new stable long-small black strings?

 Small black Strings in AdS are unstable - Large black strings are not,  neither thermo nor dynamically (Gubser Mitra)  Small AdS BS follow the same pattern as Flat case  New thermo stable pNUBS  To be confirmed with non perturbative analysis  Hint for the localised AdS black hole phase  First approximation for AdS black rings ?  (thin black rings ; works for  =0).  Connection with boundary CFT in AdS/CFT context?  New backgrounds for the dual theory (S d-3 xS 1 ).  Dual of the new stable long-small black strings?

 Small black Strings in AdS are unstable - Large black strings are not,  neither thermo nor dynamically (Gubser Mitra)  Small AdS BS follow the same pattern as Flat case  New thermo stable pNUBS  To be confirmed with non perturbative analysis  Hint for the localised AdS black hole phase  First approximation for AdS black rings ?  (thin black rings ; works for  =0).  Connection with boundary CFT in AdS/CFT context?  New backgrounds for the dual theory (S d-3 xS 1 ).  Dual of the new stable long-small black strings?

 Small black Strings in AdS are unstable - Large black strings are not,  neither thermo nor dynamically (Gubser Mitra)  Small AdS BS follow the same pattern as Flat case  New thermo stable pNUBS  To be confirmed with non perturbative analysis  Hint for the localised AdS black hole phase  First approximation for AdS black rings ?  (thin black rings ; works for  =0).  Connection with boundary CFT in AdS/CFT context?  New backgrounds for the dual theory (S d-3 xS 1 ).  Dual of the new stable long-small black strings?

 Small black Strings in AdS are unstable - Large black strings are not,  neither thermo nor dynamically (Gubser Mitra)  Small AdS BS follow the same pattern as Flat case  New thermo stable pNUBS  To be confirmed with non perturbative analysis  Hint for the localised AdS black hole phase  First approximation for AdS black rings ?  (thin black rings ; works for  =0).  Connection with boundary CFT in AdS/CFT context?  New backgrounds for the dual theory (S d-3 xS 1 ).  Dual of the new stable long-small black strings?

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