Barak Kol
Hebrew University - Jerusalem
Bremen, Aug 2008
Based on hep-th’s (BK)
Outline
• Physical summary
• Recent elements of the phase
diagram
• 0411240 Phys. Rep.
• 0206220 original
• 0608001, 0609001
“optimal gauge”
Physical summary
•Background
Why study GR in higher
Systems
dimensions?
•Set up • The ring, the susy ring
• A parameter of GR
• Black hole black string
•Processes
• String theory
• Einstein-YM
•Phase
diagram
• Large extra dimensions
The novel feature –
non-uniqueness of black
objects
• The distinction between “nohair” and “uniqueness”
• Uniqueness is special to 4d
• Phase transition physics:
energy release, order of
transition and free energy
Set-up and formulation
The system
• Theory: pure gravity,
d 1,1
0
 S , d 1  D
1
D5
• Coordinates
z
L
z
zL
 X c, d  c  D
X  T c , K3 , CY3 ,
• Background
d 1,1
Generalizations
• Background BK-Sorkin
• Extra matter content:
vectors, scalars…
Charged Kudoh-Miyamoto
• Rotation Kleihaus-Kunz-Radu
• Liquid analogy Dias-Cardoso
• Brane-worlds Tanaka,
Emparan, Fitzpatrick-RandallWiseman
r
The phases
• Black string
S
d 2
S
• Black hole
1
Horizon topology
S
D2
Uniform/ Non
Single dim’less parameter. e.g.
GM
  D 3 or
L
 

L
Processes – personal view
• String decay
Processes – personal view
• String decay
• Black hole decay
• Smooth transition (2nd order or higher)
Continuously
Initial
2nd1st
Continuous
conditions
Continuous
Critical
reaching
for
string
transition
transition:
evaporation
the merger
to a
developing
black
point
non-uniformity
hole
Main results - The phase diagram
Prediction: BH and String merge,
the end state is a BH
14  D
5  D  13
μ
Un
Str
μ
Un
Str
merger
Non-Un
Str
X
GL
Sorkin
GL
Non-Un
Str
BH
b/β
No stable non-uniform phase (for D<14)!
merger
X
BH
b/β
2nd order
Theory and “experiment”

string
merger
GL
BH
GL’

BK, hep-th/0206220
Kudoh-Wiseman Sep 2004
(Recent) elements
of the phase diagram
•
•
•
•
Gregory-Laflamme instability (perturbative)
Caged black holes (perturbative)
Merger (qualitative-topological) μ
Numeric (non-perturbative)
merger
X
GL
Caged
BH
b/β
Gregory-Laflamme Instability
The uniform string
ds 2  dsSchw 2  ds X 2
ds X 2  dz 2
dsSchw 2   f dt 2  f 1 d  2   2 d  d  2 2
d 3
 0 
f  1  
  
16  G M
d 3
0 
(d  2)  d  2
The instability
• GL (1993)
(D is d, r0=2)
•Negative modes for
Euclidean black holes
Gross-Perry-Yaffe (1982)
LSchw hGPY   hGPY
hGL (r , z )  hGPY (r ) eikGL z
kGL  
r0
kGL : 2
L
(dim’less)
•Arbitrary d
(D=d+1)
Sorkin
Master equation
• Components of h
• A system of nF=5 eq’s
• Gauge choice? nG=2
• Possible to eliminate
the gauge!
BK
• “Gauge Invariant
Perturbation Theory”
– possible due to 1 nonhomogeneous dim
The gauge shoots
twice: eliminates one
field and makes another
non-dynamic
nD=nF-2 nG
• Here nD=5-2*2=1,
hence Master field
BKSorkin
Asymptotics for GL  GL  d 
• d as a parameter (like
dim. Reg.)
• d=4 is numeric, but
there are limits
• d→∞, λGL~d
+ pert. expansion
• d→3, λGL →0
Asnin-Gorbonos-Hadar-BK-LeviMiyamoto
• Interpolation (Pade)
ˆc
1

c
d


1
1
  dˆ   dˆ
1  c1  dˆ  1
dˆ : d  3
c1  0.71515
Good to 2%
Order of transition
Gubser
• The zero mode can be followed to yield an emerging
non-uniform branch
• First or second order?
Landau - Ginzburg theory of phase transitions
Caged black holes –
A dialogue of multipoles
Harmark,
Gorbonos-BK
• Two zones
• Matched asymptotic
expansion
• Asymptotically – zeroth order is Schw, first order is
Newtonian
•In the near zone – zeroth order is flat compactified
space (origin removed), and we developed the form of
the first correction in terms of the hypergeometric func
• Dialogue figure
Some results
eccentricity
The perturbation ladder
•Effective Field Theory approach,
Chu-Goldberger-Rothstein, an
additional order
•CLEFT improvement, BK-Smolkin
more in next talk
BH “makes space” for itself
“BH Archimedes effect”
Merger - Morse theory
BK
• Morse theory is the
• Simplest formulation
topological theory of
“Phase conservation law”
extrema of functions
• You may be familiar with
the way Morse theory
measures global
properties of manifolds
(Homology), but here we n+1
n
need properties of
extrema that are invariant
under deformations of the
function.
• Solutions are extrema of
More (and most) general
the action
So expect non-uniform St
phase to connect with BH
Numeric solutions
• Relaxation (see also Ricciflow)
• 2d -> Cauchy-Riemann
identity for constraints
Used to set b.c.
Results for non-uniform
strings ->
Wiseman
Results for the black hole
Geometry
Embedding
diagrams
Eccentricity
x is the small dimensionless parameter
Wiseman-Kudoh
Sorkin BK Piran
BH thermodynamics
Mass, tension
Area, beta
Correction to
area - temperature relation
Conclusions
•
•
•
•
•
Phase diagram
End state
Processes
Critical dimensions
Topology change of total
Euclidean manifold BK
Elements of the diagram
• Control/order parameters
on axes Harmark-Obers, BKSorkin-Piran
Limits first
• The GL uniform string
• Caged BHs
Qualitative form of the diagram
• Merger
• Full numerical solutions