Spin, Charge, and Topology in low dimensions
BIRS, 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
(a) S 1  R3 or (b) R 2  S 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



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
(2+1) static axisymmetric spacetime
ds   dt  dl   d
2
2
t
Wick’s rotation
2
2
2
t  i
ds   d  dl   d
2
2
2
2
2
2
2
Black hole case:
  0,   0, R  S
No black hole case:
  0,   0, S  R
2
2
2
2
2
1
1
2
sub
critical
super
Two phases of BBH: sub- and super-critical
Euclidean topology
# dim: bulk 4, brane 3
Sub-critical:
Super-critical:
S R
1
2
R S
2
1
A transition between sub- and super-critical
phases changes the Euclidean topology of BBH
Our goal is to study these transitions
Merger transitions [Kol,’05]
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:
dS 2  g dx dx  FdT 2  F 1dr 2  r 2d 2
F
r0
 1
r
d 2  d 2  sin 2  d 2
X


a
   0,...,3
 a  0,..., 2
bulk coordinates
coordinates on the brane
Dirac-Nambu-Goto action
S   d  det ab ,
3
 ab  g X a X 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
   (r )
Coordinates on the brane
  (T  r )
a
Induced metric
ds   FdT  [ F  r (ddr ) ]dr  r sin  d 
2
2
1
S  2 T  dr L ,
2
2
2
2
2
L  r sin  1  Fr 2 (ddr ) 2
2
Brane equations
d  dL  dL
0


dr  ddr  d
d
d
 d 
 d 
 B3 
 B0  0
  B2 
  B1
2
dr
dr
 dr 
 dr 
2
3
2
cot 
3 1 dF
B0  
 B1  

2
Fr
r F dr
r dF 

B2   cot   B3  r  2 F 

2
dr


Far distance solutions
Consider a solution which approaches


 q(r )
2
2
d q 3 dq 1

 2 q0
2
dr
r dr r
p  p ln r
q
r
 p, p '
- asymptotic data
   2
Near critical branes
Zoomed vicinity
of the horizon
Brane near horizon
Proper distance
Z 
r
r0
dr
F
r  r0   Z  2, F   Z
2

2
2
is the surface gravity
Metric near the horizon
dS 2   2 Z 2 dT 2  dZ 2  dR2  R2 d 2
Brane surface:
F ( Z  R)  0
Parametric form: Z  Z ( ) 
R  R ( )
Induced metric
ds 2   2 Z 2 dT 2  [(dZ d  )2  (dRd  )2 ] d  2  R 2d 2
Reduced action:
W   d Z R
R  Z
S  2 T W
(dZ d  )  (dRd  )
2
symmetry
2
Brane equations near the horizon
2


ZRR  (RR  Z )(1  R )  0 ( for R  R(Z ))
This equation is invariant under rescaling
R(Z )  kR(Z )  Z  kZ
RZZ  (ZZ  R)(1  Z )  0 ( for Z  Z ( R))
2
This equation is invariant under rescaling
Z ( R)  kZ ( R)  R  kR
Boundary conditions
BC follow from finiteness of the curvature
It is sufficient to consider a scalar curvature
2

2 R R  6ZRR  2Z
R
Z 2 R 2 (1  R2)
2
dR
R Z 0  R0 
dZ
dZ
Z R  0  Z 0 
dR
2
0
Z 0
0
R 0


Z2
R  R0 
…
4 R0
R2
Z  Z0 
…
4Z 0
Critical solutions as attractors
RZ
Critical solution:
New variables:
x  R ,
y  Z RR  ds  dZ  ( yZ )
1
First order autonomous system
dx
2
 x(1  y )(1  x )
ds
dy
 y[1  2 y  x 2 (2  y )]
ds
Node (0,0)
Saddle (0,1/ 2)
Focus
( 1,1)
Phase portrait
n  1,
focus (1,1)
Near-critical solutions
R  Z   (Z )
Z    2Z    2  0
2
1
  Z     (1  i 7)
2

RZ Z
1 2
(CZ )    7 / 2
i
Scaling properties
C (kR0 )  k
Dual relations:
3/ 2 i 7 / 2
C ( R0 )
Z  R   ( R)
R   2R  2  0
2
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  R0  r0 sin0  { p, p '}
For critical solution
2(r  r0 )
{ p* , p* '}   
 r02
p  ( p  p )  ( p  p)
2
2
Near critical solutions
R0
 C ( R0 )  { p, p '}
Critical brane:
R0  0  C  0  { p* , p }
,
*
Under rescaling the critical brane does not move
C  ( R0 R0 )3 2i
7 /2
C,
(p)2 R03 [1  2  A  cos(2 ln R0  B)]

 | A | 1/ 2
3
2
(p)
R 0 [1  2  A  cos( B)]
Scaling and self-similarity
ln R0
 ln(p)  f (ln(p))  Q,
f ( z ) is a periodic function with the period
3

,
7
2
 
3
For both super- and sub-critical branes

Curvature at R=0 for sub-critical branes
D=3
D=4
ln()
D=6
 ln(p)
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]
Moving branes
Flachi and Tanaka, PRL 95, 161302 (2005) [ (3+1) brane in 5d]
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
Higher-dimensional generalization
BBH modeling of low (and higher) dimensional
black holes
Universality, scaling and discrete (continiuos)
self-similarity of BBH phase transitions
Singularity resolution in the field-theory analogue
of the topology change transition
BBHs and BH merger transitions