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