Phases of AdS Higher Dimensional Black Strings

1
Phases of AdS Higher Dimensional Black Strings

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

2
Plan

• 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

3
Motivation

• Suppose dgt4. Gravity propagates in ED.

4
Motivation

• Suppose dgt4. Gravity propagates in ED.
• Black Objects theoretical labs for gravity.
• NB No more uniqueness thm in ED.

5
Motivation

• Suppose dgt4. Gravity propagates in ED.
• Black Objects theoretical labs for gravity.
• NB No more uniqueness thm in ED.
• If ED are compact, possible black strings

6
Motivation

• Suppose dgt4. 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.

7
Motivation

• Suppose dgt4. 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 ?

8
Motivation

• Suppose dgt4. 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

9
Motivation

• Suppose dgt4. 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 ?

10
Asymptotically Flat space Black Strings and the
GL instability

• d-dim Black string solution to Einstein equation,
  RMN 0

(d-1) Tangherlini
r0
11
Asymptotically Flat space Black Strings and the
GL instability

• d-dim Black string solution to Einstein equation,
  RMN 0

(d-1) Tangherlini
1 Ricci flat direction
12
Asymptotically Flat space Black Strings and the
GL instability
Black Strings are unstable towards long
wavelength perturbations (R. Gregory, R.
Laflamme 1993)
13
Asymptotically Flat space Black Strings and the
GL instability
Black Strings are unstable towards long
wavelength perturbations (R. Gregory, R.
Laflamme 1993)
14
Asymptotically Flat space Black Strings and the
GL instability
Black Strings are unstable towards long
wavelength perturbations (R. Gregory, R.
Laflamme 1993)
15
Asymptotically Flat space Black Strings and the
GL instability
For a given mass, long black string are
unstable short black strings are stable
16
Asymptotically Flat space Black Strings and the
GL instability
For a given mass, long black string are
unstable short black strings are stable
17
Asymptotically Flat space Phase diagram
-gt Unstable black strings. What should they decay
to?
18
Asymptotically Flat space Phase diagram
-gt Unstable black strings. What should they decay
to ? Localised Black Hole ?
19
Asymptotically Flat space Phase diagram

• Answer NO !
• -gtTakes an infinite proper time at the horizon
  for such a transition
• (Horowitz and Maeda, 2001)

20
Asymptotically Flat space Phase diagram

• Answer NO !
• -gtTakes an infinite proper time at the horizon
  for such a transition
• (Horowitz and Maeda, 2001)
• -gt Suggests the existance of something else

Non uniform Black String (Gubser 2002 , Wiseman
2003)
21
Asymptotically Flat space Phase diagram
Thermodynamical quantities Mass M (time
translation), Entropy S (quarter horizon area),
Temperature TH (regular Euclidean
sections), Tension T (z translation)
22
Asymptotically Flat space Phase diagram
Thermodynamical quantities Mass M (time
translation), Entropy S (quarter horizon area),
Temperature TH (regular Euclidean
sections), Tension T (z translation)
Dimensionless quantities n T /ML m
GdM/Ld-3
23
Asymptotically Flat space Phase diagram
Thermodynamical quantities Mass M (time
translation), Entropy S (quarter horizon area),
Temperature TH (regular Euclidean
sections), Tension T (z translation)
Dimensionless quantities n T /ML m
GdM/Ld-3
Harmark, Niarchos and Obers
24
Asymptotically Flat space Phase diagram
Thermodynamical quantities Mass M (time
translation), Entropy S (quarter horizon area),
Temperature TH (regular Euclidean
sections), Tension T (z translation)
Dimensionless quantities n T /ML m
GdM/Ld-3
Merger Point
Harmark, Niarchos and Obers
25
Asymptotically Flat space Phase diagram
Thermodynamical quantities Mass M (time
translation), Entropy S (quarter horizon area),
Temperature TH (regular Euclidean
sections), Tension T (z translation)
Dimensionless quantities n T /ML m
GdM/Ld-3
Topological phase transition
difficult to study
Merger Point
Harmark, Niarchos and Obers
26
And in AdS ???

• Do all these phenomenae have
• an AdS counterpart ?

27
Asymptotically AdS space Uniform black String

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

28
Asymptotically AdS space Uniform black String

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

29
Asymptotically AdS space Uniform black String

• Uniform black string solution in AdS
• (Mann, Radu, Stelea 2006)
• f1f1(r0,l,d), a0, b1 arbitrary constants (fixed
  by asymptotically AdS requirement)
• l² being the AdS radius
• No Closed form solution -gt numerics

30
Asymptotically AdS space Uniform black String
Thermodynamics

• Thermodynamics

31
Asymptotically AdS space Uniform black String
Thermodynamics

• Thermodynamics
• TH, S as usual

32
Asymptotically AdS space Uniform black String
Thermodynamics

• Thermodynamics
• TH, S as usual
• Mass, Tension Counter term procedure
• (Balasubramanian, Kraus 1999)
• Involves integration over z from 0 to L.

33
Asymptotically AdS space Uniform black String
Thermodynamics

• Thermodynamics
• TH, 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

34
Asymptotically AdS space Uniform black String
Thermodynamics

• 2 phases

35
Asymptotically AdS space Uniform black String
Thermodynamics

• 2 phases
• Small black string (r0/l ltlt1)
• Essentially same feature as flat case
  (thermodynamically unstable)

36
Asymptotically AdS space Uniform black String
Thermodynamics

• 2 phases
• Small black string (r0/l ltlt1)
• 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)

37
Asymptotically AdS space Still a GL instability
?

• Non-uniform ansatz

38
Asymptotically AdS space Still a GL instability
?

• Non-uniform ansatz

39
Asymptotically AdS space Still a GL instability
?

• Non-uniform ansatz
• Xis Fourier modes
• kc 2p/L fixes the length of the black string.

40
Asymptotically AdS space Still a GL instability
?

• Order e Stability. (static perturbation, W
  0) (Brihaye, Delsate and Radu 2007)

41
Asymptotically AdS space Still a GL instability
?

• Order e Stability. (static perturbation, W
  0) (Brihaye, Delsate and Radu 2007)
• Equations of Motion Eigen value problem for
  kc².
• Also numerical (background is numerical)

42
Asymptotically AdS space Still a GL instability
?

• Order e Stability. (static perturbation, W
  0) (Brihaye, Delsate and Radu 2007)
• Equations of Motion Eigen value problem for
  kc².
• Also numerical (background is numerical)
• kc² gt 0 Exists GL instability
• kc² lt 0 Dynamically stable.
• NB AdS radius provides a lengthscale -gt µ2
  L/l 1/(lkc)

43
Asymptotically AdS space Still a GL instability
?

• Results

44
Asymptotically AdS space Still a GL instability
?

• Results
• Small AdS black string dynamically unstable

45
Asymptotically AdS space Still a GL instability
?

• Results
• Small AdS black string dynamically unstable
• Big AdS black string dynamically stable

46
Asymptotically AdS space Still a GL instability
?

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

47
Perturbative NUBS in AdSA first step in the
phase diagram

• Order e² (Delsate 2008)

48
Perturbative NUBS in AdSA first step in the
phase diagram

• Order e² (Delsate 2008)
• Corrections on thermodynamical quantities

49
Perturbative NUBS in AdSA first step in the
phase diagram

• Order e² (Delsate 2008)
• Corrections on thermodynamical quantities
• Recall the integration over z from 0 to L2p/kc
• Order e linear in cos(kcz)

50
Perturbative NUBS in AdSA first step in the
phase diagram

• Order e² (Delsate 2008)
• Corrections on thermodynamical quantities
• Recall the integration over z from 0 to L2p/kc
• Order e linear in cos(kcz)
• Order e²
• Linear in X0, X2 cos(2kcz) -gt X2 terms vanish
• Terms of the form X1²

51
Perturbative NUBS in AdSA first step in the
phase diagram

• Corrections on thermodynamical quantities at
  fixed length

52
Perturbative NUBS in AdSA first step in the
phase diagram

• Corrections on thermodynamical quantities at
  fixed length

53
Perturbative NUBS in AdSA first step in the
phase diagram

• Corrections on thermodynamical quantities at
  fixed length

Effect of the new Length scale !!
54
Perturbative NUBS in AdSA first step in the
phase diagram

• Corrections on thermodynamical quantities at
  fixed length

?
55
Perturbative NUBS in AdSA first step in the
phase diagram

• Corrections on thermodynamical quantities at
  fixed length (SUBS/L independant of L)

56
Perturbative NUBS in AdSA first step in the
phase diagram

• Corrections on thermodynamical quantities at
  fixed length (SUBS/L independant of L)

dS/dTH gt0 new thermodynamically stable phases
(T. Delsate 12/2008)
57
Perturbative NUBS in AdSA first step in the
phase diagram

• Corrections on thermodynamical quantities at
  fixed length (SUBS/L independant of L)

dS/dTH gt0 new thermodynamically stable phases
L large, r0 small  Long small NUBS 
(T. Delsate 12/2008)
58
Non perturbative analysis

• First results in non perturbative approach
  confirms the perturbative results

59
Non perturbative analysis

• First results in non perturbative approach
  confirms the perturbative results

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

60
Non perturbative analysis

• 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

61
Non perturbative analysis
d0
Prediction from perturbative analysis are
confirmed within the numerical accuracy
62
Non perturbative analysis
Embedding of the horizon in euclidean space
r
z
63
Non perturbative analysis
Embedding of the horizon in euclidean
space Preiodicity in z direction -gt Localised
black hole phase ?
r
z
64
Non perturbative analysis
Embedding of the horizon in euclidean
space Preiodicity in z direction -gt Localised
black hole phase ?
r
z
NB d0 controls the deformation
65
Stability of AdS Higher Dimensional BSConclusion
- Outlook

• 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 L0).
• Connection with boundary CFT in AdS/CFT context?
• New backgrounds for the dual theory (Sd-3xS1).
• Dual of the new stable long-small black strings?

66
Stability of AdS Higher Dimensional BSConclusion
- Outlook

• 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 L0).
• Connection with boundary CFT in AdS/CFT context?
• New backgrounds for the dual theory (Sd-3xS1).
• Dual of the new stable long-small black strings?

67
Stability of AdS Higher Dimensional BSConclusion
- Outlook

• 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 L0).
• Connection with boundary CFT in AdS/CFT context?
• New backgrounds for the dual theory (Sd-3xS1).
• Dual of the new stable long-small black strings?

68
Stability of AdS Higher Dimensional BSConclusion
- Outlook

• 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 L0).
• Connection with boundary CFT in AdS/CFT context?
• New backgrounds for the dual theory (Sd-3xS1).
• Dual of the new stable long-small black strings?

69
Stability of AdS Higher Dimensional BSConclusion
- Outlook

• 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 L0).
• Connection with boundary CFT in AdS/CFT context?
• New backgrounds for the dual theory (Sd-3xS1).
• Dual of the new stable long-small black strings?

70
Stability of AdS Higher Dimensional BSConclusion
- Outlook

• 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 L0).
• Connection with boundary CFT in AdS/CFT context?
• New backgrounds for the dual theory (Sd-3xS1).
• Dual of the new stable long-small black strings?

71
Stability of AdS Higher Dimensional BSConclusion
- Outlook

• 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 L0).
• Connection with boundary CFT in AdS/CFT context?
• New backgrounds for the dual theory (Sd-3xS1).
• Dual of the new stable long-small black strings?

72
Stability of AdS Higher Dimensional BSConclusion
- Outlook

• 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 L0).
• Connection with boundary CFT in AdS/CFT context?
• New backgrounds for the dual theory (Sd-3xS1).
• Dual of the new stable long-small black strings?

73
Stability of AdS Higher Dimensional BSConclusion
- Outlook

• 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 L0).
• Connection with boundary CFT in AdS/CFT context?
• New backgrounds for the dual theory (Sd-3xS1).
• Dual of the new stable long-small black strings?

74
Stability of AdS Higher Dimensional BSConclusion
- Outlook

• 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 L0).
• Connection with boundary CFT in AdS/CFT context?
• New backgrounds for the dual theory (Sd-3xS1).
• Dual of the new stable long-small black strings?

75
Stability of AdS Higher Dimensional BSConclusion
- Outlook

• 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 L0).
• Connection with boundary CFT in AdS/CFT context?
• New backgrounds for the dual theory (Sd-3xS1).
• Dual of the new stable long-small black strings?

76
Phases of AdS Higher Dimensional BSThank you
Thank you for your attention!
77
Stability of AdS Higher Dimensional BSReferences

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• R. Mann, E. Radu and C. Stelea, Black string
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• S.W. Hawking, D.N. Page, Commun. Math. Phys. 87
  (1983) 577.
• B. Kol, The Phase Transition between Caged Black
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