“models of gravity in higher dimensions”, bremen, aug. 25-29, 2008
TRANSCRIPT
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“Models of Gravity in Higher Dimensions”, Bremen, Aug. 25-29, 2008
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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. Phys.Rev. D74, 044006 (2006)
V.F. and D.Gorbonos, hep-th/ 0808.3024 (2008)
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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
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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.
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Topology change transitions
Change of the spacetime topology
Euclidean topology change
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An example
A thermal bath at finite temperature: ST after the Wick’s rotation is the Euclidean manifolds
1 3S R
No black hole
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Euclidean black hole
2 22 22dr
F dF
r dds 01 /F r r
22R S 2 2( )DSR
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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
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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
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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
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A brane in the bulk BH spacetime
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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.
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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.
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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
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Two phases of BBH: sub- and super-critical
sub
supercritical
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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
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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
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No scale parameter – Second order phase transition
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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).
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( )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
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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
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Far distance solutions
Consider a solution which approaches 2
( )2
q r
lnp p rq
r
, 'p p - asymptotic data
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Near critical branes
Zoomed vicinity of the horizon
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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
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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.
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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)
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Phase portrait
1, (1,1)n focus
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Near-critical solutions
1 2 ( ) 7 / 2 iR Z Z CZ
Scaling properties
3/ 2 7 / 20 0( ) ( )iC kR k C R
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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
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0gr Z R Near (Rindler)zone (scalingtransformationsare valid)
gr Z
Asymptoticregion {p,p’}
Global structure of near-critical solution
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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
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Phase portraits
2, ( 2,2)n focus
4, (2,4)n focus
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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
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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
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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)
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Exist scale parameter – First order phase transition
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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
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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
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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
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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)
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as a function of for n=2. The dashed line is the same function for DNG branes (without stiffness terms).
0Z 0Z
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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
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n=2, C=1
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R''(0) as a function of R_0 (supercritical) for n=2 and B=1
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THICK BRANE INTERACTING WITH BLACK HOLE
Morisawa et. al. , PRD 62, 084022 (2000); PRD 67, 025017 (2003)
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Flachi and Tanaka, PRL 95, 161302 (2005) [ (3+1) brane in 5d]
Moving brines
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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.