A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS

“Models of Gravity in Higher Dimensions”, Bremen, Aug. 25-29, 2008

Based on

Christensen, V.F., Larsen, Phys.Rev. D58, 085005 (1998)

V.F., Larsen, Christensen, Phys.Rev. D59, 125008 (1999)

V.F. Phys.Rev. D74, 044006 (2006)

V.F. and D.Gorbonos, hep-th/ 0808.3024 (2008)

BH critical merger solutions

2 2 2 2 2 2 2132 cos ( 4) DDds d d dt D d

B.Kol, 2005; V.Asnin, B.Kol, M.Smolkin, 2006

9D9D

2

` '

1... ... ?

CompleteEinstein Local theoryHilbert Non local

R R R R

`Golden Dream of Quantum Gravity’

Consideration of merger transitions, Choptuik critical collapse, and other topology change transitions might require using the knowledge of quantum gravity.

Topology change transitions

Change of the spacetime topology

Euclidean topology change

An example

A thermal bath at finite temperature: ST after the Wick’s rotation is the Euclidean manifolds

1 3S R

No black hole

Euclidean black hole

2 22 22dr

F dF

r dds 01 /F r r

22R S 2 2( )DSR

A static test brane interacting with a black hole

Toy model

If the brane crosses the event horizon of the bulk black hole the induced geometry has horizon

By slowly moving the brane one can “create” and “annihilate” the brane black hole (BBH)

In these processes, changing the (Euclidean) topology, a curvature singularity is formed

More fundamental field-theoretical description of a “realistic” brane “resolves” singularities

Approximations

In our consideration we assume that the brane is:

(i) Test (no gravitational back reaction)

(ii) Infinitely thin

(iii) Quasi-static

(iv) With and without stiffness

brane at fixed time

brane world-sheet

The world-sheet of a static brane is formed by Killing trajectories passing throw at a fixed-time brane surface

A brane in the bulk BH spacetime

black hole brane

event horizon

A restriction of the bulk Killing vector to the brane gives the Killing vector for the induced geometry. Thus if the brane crosses the event horizon its internal geometry is the geometry of (2+1)-dimensional black hole.

The temperature of the bulk BH and of the brane BH is the same.

0 .

.

,

, .

a ya y a

y

y

Let X bea positionof astaticunperturbedbrane

Consider braneperturbations X Decompose

X e n wheree areunit vectors

tothebrane andn areunit normal vectors

isasetof scalar fields propagatingal

tangent

.

ongthebrane

anddescribingthebraneexcitations

The brane BH emits Hawking radiation of -quanta.

2 2 2 2 2 2tds dt dl d

(2+1) static axisymmetric spacetime

Black hole case:2 2 2 10, 0, R S

Wick’s rotation t i2 2 2 2 2 2ds d dl d

2 2 1 20, 0, S R No black hole case:

Induced geometry on the brane

Two phases of BBH: sub- and super-critical

sub

supercritical

Euclidean topology change

A transition between sub- and super-critical phases changes the Euclidean topology of BBH

An analogy with merger transitions [Kol,’05]

Our goal is to study these transitions

Bulk black hole metric

2 2 1 2 2 2dS g dx dx FdT F dr r d

22 2 2sind d d 01 r

rF

No scale parameter – Second order phase transition

bulk coordinates

0,...,3X

0,..., 2a a coordinates on the brane

Dirac-Nambu-Goto action

3 det ,abS d ab a bg X X

We assume that the brane is static and spherically symmetric, so that its worldsheet geometry possesses the group of the symmetry O(2).

( )r

( )a T r

Brane equation

Coordinates on the brane

2 2 1 2 2 2 2 2 2[ ( ) ] sinds FdT F r d dr dr r d

Induced metric

2 ,S T drL 2 2sin 1 ( )L r Fr d dr

Main steps

1. Brane equations2. Asymptotic form of a solution at infinity3. Asymptotic data4. Asymptotic form of a solution near the horizon5. Scaling properties6. Critical solution as attractor7. Perturbation analysis of near critical solutions8. The brane BH size vs `distance’ of the asymptotic data from the critical one9. Choptuik behavior

Far distance solutions

Consider a solution which approaches 2

( )2

q r

lnp p rq

r

, 'p p - asymptotic data

Near critical branes

Zoomed vicinity of the horizon

is the surface gravity

Metric near the horizon

2 2 2 2 2 2 2 2dS Z dT dZ dR R d

Brane near horizon

2( )(1 ) 0 ( ( ))ZRR RR Z for R R ZR

This equation is invariant under rescaling

( ) ( )R Z kR Z Z kZ

Duality transformation

duality transformationmapsa

to a ( ) ,

( ) .

R Z

If R F Z isasolution

thenZ F R isalsoasolution

supercritical

brane subcritical one :

Combining the scaling and duality transformations one can obtain any noncritical solution from any other one.

The critical solution is invariant under both scaling and dual transformations.

Critical solutions as attractors

Critical solution: R Z

New variables:1, ( )x R y Z RR ds dZ yZ

First order autonomous system

2(1 )(1 )dx

x y xds

2[1 2 (2 )]dy

y y x yds

Node (0,0) Saddle (0,1/ 2) Focus ( 1,1)

Phase portrait

1, (1,1)n focus

Near-critical solutions

1 2 ( ) 7 / 2 iR Z Z CZ

Scaling properties

3/ 2 7 / 20 0( ) ( )iC kR k C R

Near critical solutions

0 0( ) { , '}R C R p p

,0 * *0 0 { , }R C p p

Critical brane:

Under rescaling the critical brane does not move

22 ( )( ) pp p p p

0gr Z R Near (Rindler)zone (scalingtransformationsare valid)

gr Z

Asymptoticregion {p,p’}

Global structure of near-critical solution

Scaling and self-similarity

0ln ln( ) (ln( )) ,R p f p Q

2

3

( )f z is a periodic function with the period 3

,7

For both super- and sub-critical brines

Phase portraits

2, ( 2,2)n focus

4, (2,4)n focus

Scaling and self-similarity

0ln ln( ) (ln( )) , ( 6)R p f p Q D

2, - 2

2n D

n

( )f z is a periodic function with the period 2

( 2),

4 4

n

n n

0ln ln( ) , ( 6)R p D 22 4 4

4( 1)

n n n

n

For both super- and sub-critical brines

BBH modeling of low (and higher) dimensional black holes

Universality, scaling and discrete (continuous) self-similarity of BBH phase transitions

Singularity resolution in the field-theory analogue of the topology change transition

BBHs and BH merger transitions

Beyond the adopted approximations

(i) Thickness effects

(ii) Interaction of a moving brane with a BH

(iii) Irreversability

(iv) Role of the brane tension

(v) Curvature corrections (V.F. and D.Gorbonos,

under preparation)

Exist scale parameter – First order phase transition

L extrinsic curvatureextrinsic curvature

( )K n

2[1 ]B K CK K

Set “fundamental length”: C=1Set “fundamental length”: C=1

Energy density Energy density L , 0B C

Polyakov 1985Polyakov 1985

L 2[1 ]B K CK K

21

EOM: 4EOM: 4thth order ODE order ODE

R

Z

max( , )B C

0(0)

'(0) 0

''(0) ?

'''(0) 0

Z Z

Z

Z

Z

Axial symmetry

Z

R

Highest number of

derivatives of the fields

R

Z

1Z R R R

n 1

Z R R Rn

44thth order linear equation for order linear equation for R

4 modes:4 modes: 21

4 42

n n nR R

1

2

nR

B CR e

1

2

nR

B CR e

3 stable

1 unstableTune the

free parameter''(0)Z

R

Z

RESULTS

`Symmetric’ case: n=1, B=0 (C=1). A plot for super-critical phase is identical to this one. When B>0 symmetry is preserved (at least in num. results)

as a function of for n=2. The dashed line is the same function for DNG branes (without stiffness terms).

0Z 0Z

The energy density integrated for < R <5 as a function of Z_0 comparing two branches in the segment (1 < Z_0 < 1.25). Note that the minimal energy is obtained at the point which corresponds approximately to 0Z

n=2, C=1

R''(0) as a function of R_0 (supercritical) for n=2 and B=1

THICK BRANE INTERACTING WITH BLACK HOLE

Morisawa et. al. , PRD 62, 084022 (2000); PRD 67, 025017 (2003)

Flachi and Tanaka, PRL 95, 161302 (2005) [ (3+1) brane in 5d]

Moving brines

Final remarksDNG vs stiff branes: Second order vs first order

phase transitions

Spacetime singularities during phase transitions?

BH Merger transition: New examples of `cosmic censorship’ violation?

Dynamical picture: Asymmetry of BBH and BWH

`Resolution of singularities’ in the `fundamental field’ description.