“Einstein Gravity in Higher Dimensions”, Jerusalem, 18-22 Feb., 2007

The merger transitions are in many aspects similar to the topology change transitions in the classical and quantum gravity. One can expect that during both types of transitions the spacetime curvature can infinitely grow. It means that the classical theory of gravity is not sufficient for their description and a more fundamental theory (such as the string theory) is required. It might be helpful to have a toy model for the merger and topology changing transitions, which is based on the physics which is well understood. In this talk we discuss such a toy model.

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)

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

Static black holes in higher dimensions

Tangherlini (1963) metric:

2 2 1 2 2 22NdS g dx dx FdT F dr r d

2 2 2 21 1 1sin ii i id d d

301 ( )NF r r

N is the number of ST dimensions

2nd is the metric on a unit n-dim sphere

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

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

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

1 2( )DS R 2 2( )DR S

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 curvature

2 22

2 2 2

6 22 '(1 )

ZRR ZR RZ R 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

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.

Euclidean topology

Sub-critical: 1 1DS R

# dim: bulk N, brane D, n=D-2

Super-critical: 2 2DR S

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

Merger transition [Kol,’05]

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, - 22n 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 branes

Curvature at R=0 for sub-critical branes

ln( )

ln( )p

D=6

D=3

D=4

The plot ln(Rmax) vs. ln(p-p*) from Garfinkle & Duncan (’98) paper

A similar plot for BBH system for D=4 after rescaling:

ln( ) 2 ln( ),

ln( ) (1/ 3) ln( )

p p

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

Final remarks

Phase transitions, near critical behavior

Spacetime singularities during phase transitions?

New examples of `cosmic censorship’ violation

Asymmetry of BBH and BWH

Dynamical picture