Spin, Charge, and Topology in low dimensionsBIRS, Banff, July 29 - August 3, 2006

Based on

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

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

V.F. gr-qc/0604114 (2006)

Topology change transitions

Change of the spacetime topology

Euclidean topology change

An example

A thermal bath at finite temperature with (a) and without (b) black hole. After the wick’s rotation the Euclidean manifolds have the topology

1 3 2 2( ) ( )a S R or b R S

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

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.

2 2 2 2 2 2tds dt dl d

(2+1) static axisymmetric spacetime

Wick’s rotation t i

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

2 2 2 2 2 2ds d dl d

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

Two phases of BBH: sub- and super-critical

sub

supercritical

Euclidean topology

Sub-critical: 1 2S R

# dim: bulk 4, brane 3

Super-critical: 2 1R S

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

Merger transitions [Kol,’05]

Our goal is to study these transitions

Let us consider a static test brane interacting with a bulk static spherically symmetrical black hole. For briefness, we shall refer to such a system (a brane and a black hole) as to the BBH-system.

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

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

Brane equations

0d dL dL

dr d dr d

3 22

3 2 1 020

d d d dB B B B

dr dr dr dr

0 12

cot 3 1 dFB B

F r r F dr

2 3cot 22

r dFB B r F

dr

Far distance solutions

Consider a solution which approaches 2

( )2

q r

2

2 2

3 10

d q dqq

dr r dr r

lnp p rq

r

, 'p p - asymptotic data

Near critical branes

Zoomed vicinity of the horizon

Proper distance0

r

r

drZ

F

2 2 20 2,r r Z F Z

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

Brane surface: ( ) 0F Z R

Parametric form: ( ) ( )Z Z R R

Induced metric

2 2 2 2 2[( ) ( ) ]dZ d dR d d R d 2 2 2 2ds Z dT

Reduced action: 2S TW 2 2( ) ( )W d ZR dZ d dR d

symmetryR Z

Brane equations near the horizon

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

2( )(1 ) 0 ( ( ))RZZ ZZ R Z for Z Z R

This equation is invariant under rescaling

This equation is invariant under rescaling

( ) ( )R Z kR Z Z kZ

( ) ( )Z R kZ R R kR

Boundary conditions

BC follow from finiteness of the curvature

It is sufficient to consider a scalar curvature2 22

2 2 2

2 6 2

(1 )

R ZRR ZRRZ R R

0 00

0RR

dZZ Z

dR

2

004

RZ Z …

Z

0 00

0ZZ

dRR R

dZ

2

004

ZR R …

R

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

( )R Z Z

2 2 2 0Z Z

1( 1 7)2

Z i

1 2 ( ) 7 / 2iR Z Z CZ

Scaling properties

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

Dual relations: ( )Z R R

2 2 2 0R R

We study super-critical solutions close to the critical one. Consideration of sub-critical solutions is similar.

A solution is singled out by the value of 0

0 0 0 0sin { , '}R r p p

0* * 2

0

2( ){ , '}

r rp p

r

For critical solution

22 ( )( ) pp p p p

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

3 2 7 / 20 0( ) ,iC R R C

320 0

320

[1 2 cos(2 ln )]( )| | 1/ 2

( ) [1 2 cos( )]

R A R BpA

p A BR

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 branes

Curvature at R=0 for sub-critical branes

ln( )

ln( )p

D=6

D=3

D=4

Choptuik critical collapse

Choptuik (’93) has found scaling phenomena in gravitational collapse

A one parameter family of initial data for a spherically symmetric field coupled to gravity

The critical solution is periodic self similar

A graph of ln(M) vs. ln(p-p*) is the sum of a linear function and a periodic function

For sub-critical collapse the same is true for a graph of ln(Max-curvature) [Garfinkle & Duncan, ’98]

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

Moving branes

THICK BRANE INTERACTING WITH BLACK HOLE

Morisawa et. al. , PRD 62, 084022 (2000)

Emergent gravity is an idea in quantum gravity that spacetime background emerges as a mean field approximation of underlying microscopic degrees of freedom, similar to the fluid mechanics approximation of Bose-Einstein condensate. This idea was originally proposed by Sakharov in 1967, also known as induced gravity.

Summary and discussions

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

BBH modeling of low (and higher) dimensional black holes

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

BBHs and BH merger transitions

Higher-dimensional generalization