Download - A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS
![Page 1: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/1.jpg)
“Models of Gravity in Higher Dimensions”, Bremen, Aug. 25-29, 2008
![Page 2: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/2.jpg)
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)
![Page 3: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/3.jpg)
BH critical merger solutions
2 2 2 2 2 2 2132 cos ( 4) DDds d d dt D d
B.Kol, 2005; V.Asnin, B.Kol, M.Smolkin, 2006
9D9D
![Page 4: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/4.jpg)
2
` '
1... ... ?
CompleteEinstein Local theoryHilbert Non local
R R R R
`Golden Dream of Quantum Gravity’
Consideration of merger transitions, Choptuik critical collapse, and other topology change transitions might require using the knowledge of quantum gravity.
![Page 5: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/5.jpg)
Topology change transitions
Change of the spacetime topology
Euclidean topology change
![Page 6: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/6.jpg)
An example
A thermal bath at finite temperature: ST after the Wick’s rotation is the Euclidean manifolds
1 3S R
No black hole
![Page 7: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/7.jpg)
Euclidean black hole
2 22 22dr
F dF
r dds 01 /F r r
22R S 2 2( )DSR
![Page 8: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/8.jpg)
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
![Page 9: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/9.jpg)
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
![Page 10: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/10.jpg)
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
![Page 11: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/11.jpg)
A brane in the bulk BH spacetime
![Page 12: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/12.jpg)
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.
![Page 13: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/13.jpg)
The temperature of the bulk BH and of the brane BH is the same.
0 .
.
,
, .
a ya y a
y
y
Let X bea positionof astaticunperturbedbrane
Consider braneperturbations X Decompose
X e n wheree areunit vectors
tothebrane andn areunit normal vectors
isasetof scalar fields propagatingal
tangent
.
ongthebrane
anddescribingthebraneexcitations
The brane BH emits Hawking radiation of -quanta.
![Page 14: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/14.jpg)
![Page 15: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/15.jpg)
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
![Page 16: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/16.jpg)
Two phases of BBH: sub- and super-critical
sub
supercritical
![Page 17: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/17.jpg)
Euclidean topology change
A transition between sub- and super-critical phases changes the Euclidean topology of BBH
An analogy with merger transitions [Kol,’05]
Our goal is to study these transitions
![Page 18: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/18.jpg)
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
![Page 19: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/19.jpg)
No scale parameter – Second order phase transition
![Page 20: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/20.jpg)
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).
![Page 21: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/21.jpg)
( )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
![Page 22: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/22.jpg)
Main steps
1. Brane equations2. Asymptotic form of a solution at infinity3. Asymptotic data4. Asymptotic form of a solution near the horizon5. Scaling properties6. Critical solution as attractor7. Perturbation analysis of near critical solutions8. The brane BH size vs `distance’ of the asymptotic data from the critical one9. Choptuik behavior
![Page 23: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/23.jpg)
Far distance solutions
Consider a solution which approaches 2
( )2
q r
lnp p rq
r
, 'p p - asymptotic data
![Page 24: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/24.jpg)
Near critical branes
Zoomed vicinity of the horizon
![Page 25: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/25.jpg)
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
2( )(1 ) 0 ( ( ))ZRR RR Z for R R ZR
This equation is invariant under rescaling
( ) ( )R Z kR Z Z kZ
![Page 26: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/26.jpg)
Duality transformation
duality transformationmapsa
to a ( ) ,
( ) .
R Z
If R F Z isasolution
thenZ F R isalsoasolution
supercritical
brane subcritical one :
Combining the scaling and duality transformations one can obtain any noncritical solution from any other one.
The critical solution is invariant under both scaling and dual transformations.
![Page 27: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/27.jpg)
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)
![Page 28: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/28.jpg)
Phase portrait
1, (1,1)n focus
![Page 29: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/29.jpg)
Near-critical solutions
1 2 ( ) 7 / 2 iR Z Z CZ
Scaling properties
3/ 2 7 / 20 0( ) ( )iC kR k C R
![Page 30: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/30.jpg)
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
22 ( )( ) pp p p p
![Page 31: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/31.jpg)
0gr Z R Near (Rindler)zone (scalingtransformationsare valid)
gr Z
Asymptoticregion {p,p’}
Global structure of near-critical solution
![Page 32: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/32.jpg)
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 brines
![Page 33: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/33.jpg)
![Page 34: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/34.jpg)
Phase portraits
2, ( 2,2)n focus
4, (2,4)n focus
![Page 35: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/35.jpg)
Scaling and self-similarity
0ln ln( ) (ln( )) , ( 6)R p f p Q D
2, - 2
2n 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 brines
![Page 36: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/36.jpg)
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
![Page 37: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/37.jpg)
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)
![Page 38: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/38.jpg)
Exist scale parameter – First order phase transition
![Page 39: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/39.jpg)
L extrinsic curvatureextrinsic curvature
( )K n
2[1 ]B K CK K
Set “fundamental length”: C=1Set “fundamental length”: C=1
Energy density Energy density L , 0B C
Polyakov 1985Polyakov 1985
![Page 40: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/40.jpg)
L 2[1 ]B K CK K
21
EOM: 4EOM: 4thth order ODE order ODE
R
Z
max( , )B C
0(0)
'(0) 0
''(0) ?
'''(0) 0
Z Z
Z
Z
Z
Axial symmetry
Z
R
Highest number of
derivatives of the fields
![Page 41: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/41.jpg)
R
Z
1Z R R R
n 1
Z R R Rn
44thth order linear equation for order linear equation for R
4 modes:4 modes: 21
4 42
n n nR R
1
2
nR
B CR e
1
2
nR
B CR e
3 stable
1 unstableTune the
free parameter''(0)Z
R
Z
![Page 42: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/42.jpg)
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)
![Page 43: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/43.jpg)
as a function of for n=2. The dashed line is the same function for DNG branes (without stiffness terms).
0Z 0Z
![Page 44: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/44.jpg)
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 0Z
![Page 45: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/45.jpg)
n=2, C=1
![Page 46: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/46.jpg)
R''(0) as a function of R_0 (supercritical) for n=2 and B=1
![Page 47: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/47.jpg)
![Page 48: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/48.jpg)
THICK BRANE INTERACTING WITH BLACK HOLE
Morisawa et. al. , PRD 62, 084022 (2000); PRD 67, 025017 (2003)
![Page 49: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/49.jpg)
Flachi and Tanaka, PRL 95, 161302 (2005) [ (3+1) brane in 5d]
Moving brines
![Page 50: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/50.jpg)
![Page 51: A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS](https://reader036.vdocuments.us/reader036/viewer/2022062807/568150f3550346895dbf10c4/html5/thumbnails/51.jpg)
Final remarksDNG vs stiff branes: Second order vs first order
phase transitions
Spacetime singularities during phase transitions?
BH Merger transition: New examples of `cosmic censorship’ violation?
Dynamical picture: Asymmetry of BBH and BWH
`Resolution of singularities’ in the `fundamental field’ description.