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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
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:
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
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)
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]
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