barak kol hebrew university - jerusalem bremen, aug 2008 outline physical summary recent elements of...
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Barak KolHebrew University - Jerusalem
Bremen, Aug 2008
Outline
• Physical summary
• Recent elements of the phase diagram
Based on hep-th’s (BK)
• 0411240 Phys. Rep.
• 0206220 original
• 0608001, 0609001 “optimal gauge”
Physical summary
Why study GR in higher dimensions?
• A parameter of GR• String theory• Large extra dimensions
The novel feature – non-uniqueness of black
objects• The distinction between “no-
hair” and “uniqueness”• Uniqueness is special to 4d• Phase transition physics:
energy release, order of transition and free energy
Systems• The ring, the susy ring• Black hole black string• Einstein-YM
•Background•Set up•Processes•Phase diagram
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-Randall-Wiseman
The system• Theory: pure gravity,
• Background
• Coordinates
Set-up and formulation
0
1,1 1, 1
5
d S d D
D
L z
rz z L
1,1
3 3
,
, , ,
d c
c
X d c D
X T K CY
The phases
• Black string • Black hole
2 1dS S 2DS Horizon topology
Uniform/ Non
Single dim’less parameter. e.g. or3D
GM
L
L
• String decay
Processes – personal view
• String decay
Processes – personal view
• Black hole decay
• Smooth transition (2nd order or higher)
Initial conditions for evaporationCritical string1st Continuous transition: developing non-uniformity
Continuously reaching the merger point
2nd Continuous transition to a black hole
Main results - The phase diagram
Prediction: BH and String merge, the end state is a BH
b/β
μ
5 13D
GL
X
merger
No stable non-uniform phase (for D<14)!
b/β
μ
14 D
GL merger
X
2nd order
UnStr
UnStr
BHBH
Non-UnStr
Non-UnStr
Sorkin
Theory and “experiment”
BH
merger
GL
string
GL’
Kudoh-Wiseman Sep 2004BK, hep-th/0206220
(Recent) elements of the phase diagram
• Gregory-Laflamme instability (perturbative)
• Caged black holes (perturbative)
• Merger (qualitative-topological)
• Numeric (non-perturbative)
b/β
μ
GL
X
merger
Caged BH
Gregory-Laflamme InstabilityThe uniform string
2 2 2
2 2
2 2 1 2 2 22
3
0
30
2
1
16
( 2)
Schw X
X
Schw d
d
d
d
ds ds ds
ds dz
ds f dt f d d
f
GM
d
•Negative modes for Euclidean black holes Gross-Perry-Yaffe (1982)
The instability
• GL (1993) (D is d, r0=2)
( , ) ( ) GL
Schw GPY GPY
ik zGL GPY
GL
L h h
h r z h r e
k
•Arbitrary d (D=d+1) Sorkin0: 2GL
rk
L (dim’less)
Master equation
• Components of• 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 non-homogeneous 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
h
BK-Sorkin
Asymptotics for
• d as a parameter (like dim. Reg.)
• d=4 is numeric, but there are limits
• d→∞, λGL~d
+ pert. expansion
• d→3, λGL →0
• Interpolation (Pade)
GL GL d
Asnin-Gorbonos-Hadar-BK-Levi-Miyamoto
1 1
1
1
ˆ1ˆ ˆˆ1 1
ˆ : 3
0.71515
c d cd d
c d
d d
c
Good to 2%
Order of transition• The zero mode can be followed to yield an emerging non-uniform branch• First or second order? Landau - Ginzburg theory of phase transitions
Gubser
Caged black holes – A dialogue of multipoles
• 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
Harmark, Gorbonos-BK
• Dialogue figure
The perturbation ladder
Some results
eccentricity
BH “makes space” for itself
“BH Archimedes effect”
•Effective Field Theory approach, Chu-Goldberger-Rothstein, an additional order
•CLEFT improvement, BK-Smolkin
more in next talk
Merger - Morse theory
• Morse theory is the topological theory of extrema of functions
• You may be familiar with the way Morse theory measures global properties of manifolds (Homology), but here we need properties of extrema that are invariant under deformations of the function.
• Solutions are extrema of the action
• Simplest formulation “Phase conservation law”
More (and most) general
nn+1
So expect non-uniform St phase to connect with BH
BK
Numeric solutions
• Relaxation (see also Ricci-flow)
• 2d -> Cauchy-Riemann identity for constraints
Used to set b.c.
Results for non-uniform strings ->
Wiseman
Results for the black holeWiseman-KudohSorkin BK Piran
Geometry
Embedding
diagrams
Eccentricity
x is the small dimensionless parameter
BH thermodynamics
Mass, tensionArea, 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, BK-Sorkin-Piran
Limits first
• The GL uniform string• Caged BHs
Qualitative form of the diagram
• Merger• Full numerical solutions