“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

D

 9

D

 9

ds

2 

d

 2 

D

1  2  2

d

 2  cos 2 

dt

2  (

D

 4)

d

 2

D

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

`Golden Dream of Quantum Gravity’

R Einstein



Hilbert



R

2  ...

Local



R

1

R

 ...

 ?

`

Complete

'

theory

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 No black hole

S

1 

R

3

Euclidean black hole

ds

2 

F d

 2 

dr

2

F

 2

r d

 2

F r r

0 /

R

2 

S

2 (

R

2 

S D

 2 )

Toy model

A static test brane interacting with a black hole 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

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.

black hole   brane event horizon

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

Let X be a position of a static unperturbed brane

0 .

 .

Decompose



X

  

a e a

  

y y a tangent

.

y



y is a set of scalar fields propagating al ong the brane and describing the brane excitations

.

Induced geometry on the brane (2+1) static axisymmetric spacetime

ds

2  

t

2

dt

2 

dl

2   2 2 Wick’s rotation

ds

2 

t

2 

i

 

dl

2  2 Black hole case:   2  0,   2  0,

R

2 

S

1 No black hole case:   2  0,   2  0,

S

1 

R

2

sub critical super Two phases of BBH: sub- and super-critical

Euclidean topology change

A transition between sub- and super critical phases changes the Euclidean topology of BBH Our goal is to study these transitions An analogy with merger transitions [Kol,’05]

Bulk black hole metric

dS

2 

g dx dx

  

FdT

2  2   2

F r

0

r d d

 2  sin 2 2

No scale parameter – Second order phase transition

X

    0,...,3  

a



a

 0,..., 2  bulk coordinates coordinates on the brane Dirac-Nambu-Goto action

S

 

d

3   det 

ab

, 

ab

 

g X X

a

 

b

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

Brane equation Coordinates on the brane 

a

) Induced metric

ds

2  

FdT

2  [

F

 1  2 (   2 ) ]

dr

2 

r

2 sin 2 

d

 2

S

 2   ,

L



r

sin  1  2 (   ) 2

Main steps

1. Brane equations 2. Asymptotic form of a solution at infinity 3. Asymptotic data 4. Asymptotic form of a solution near the horizon 5. Scaling properties 6. Critical solution as attractor 7. Perturbation analysis of near critical solutions 8. The brane BH size vs `distance’ of the asymptotic data from the critical one 9. Choptuik behavior



Far distance solutions

Consider a solution which approaches  2

q



p



p

 ln

r r

2   - asymptotic data

Near critical branes

Zoomed vicinity of the horizon

Brane near horizon

Metric near the horizon

dS

2    2 2

Z dT

2 

dZ

2 

dR

2  2

R d

 2  is the surface gravity

ZRR

  (

RR

 

Z

)(1 

R

 2 )  0 (

for R

 This equation is invariant under rescaling  

Z



kZ

Duality transformation

R



Z

duality transformation maps a

supercritical brane

to a

subcritical one : then Z

 .

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

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

Critical solutions as attractors Critical solution:

R



Z

New variables:

x



R

 ,

y

  1

Z RR

 

ds

 First order autonomous system

dx ds dy

 

ds x

(1 

y

)(1 

x

2 )

y



y



x

2 (2 

y

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

Phase portrait

n

 1,

focus

(1,1)

R

Near-critical solutions

Z



(

CZ i



)

  

7 / 2

Scaling properties ( 0 ) 

k

 3/ 2 

i

7 / 2 ( 0 )

R

0 Near critical solutions 

(

0

)

 Critical brane:

R

0  0 

C

 0  {

p p

* , * , } Under rescaling the critical brane does not move (

p



p

 ) 2  (

p

 

p

  ) 2

Global structure of near-critical solution

Asymptotic region {p,p’}

r g Z R

0

r g Z

Near (Rindler) zone (scaling transformations are valid)

Scaling and self-similarity

ln

R

0  ln(

f

(ln( 

p

)) 

Q

, is a periodic function with the period    3  7 ,   2 3 For both super- and sub-critical brines

Phase portraits

n

 2,

focus

( 2, 2)

n

 4,

focus

(2, 4)

Scaling and self-similarity ln

R

0  ln(

p

)

f

(ln( 

p

)) 

Q

, (

D

 6)  is a periodic function with the period   (

n

 2) 4  4

n



n

2 ,  

n

2  2 ,

n



D

- 2  ln

R

0  ln( 

p

) , (

D

 6)  

n n

2  4(

n

 1) 4

n

 4 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

 [1    2 

C K



K

 ] Polyakov 1985 Set “fundamental length”: C=1   

L K

      (

n

) extrinsic curvature

EOM: 4 th order ODE

L

   [1    2 

C K



K

 ]

Z Z

(0) '(0)  

Z

0 0

Z

''(0)  ?

Z

'''(0)  0

Axial symmetry Highest number of derivatives of the fields 1 2

Z Z max(

B

,

C

) R R

 1

n R

  Z Z

4 th order linear equation for



4 modes:

  

e R

 1 2 

n

  2 

n

 1

n

2 

R n

4 

e

2 

n

  1 

R

3 stable 1 unstable R R Tune the free parameter

Z

''(0)

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)

Z Z

0 same function for DNG branes (without stiffness terms).

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

Z

0

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)

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

Final remarks

DNG vs stiff branes: Second order vs first order phase transitions Dynamical picture: Asymmetry of BBH and BWH BH Merger transition: New examples of `cosmic censorship’ violation?

Spacetime singularities during phase transitions?

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