Finsler and Lagrange Geometries in Einstein and String Gravity
“Einstein Gravity in Higher Dimensions”, Jerusalem, 18-22 Feb., 2007.
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Transcript of “Einstein Gravity in Higher Dimensions”, Jerusalem, 18-22 Feb., 2007.
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“Einstein Gravity in Higher Dimensions”, Jerusalem, 18-22 Feb., 2007
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The merger transitions are in many aspects similar to the topology change transitions in the classical and quantum gravity. One can expect that during both types of transitions the spacetime curvature can infinitely grow. It means that the classical theory of gravity is not sufficient for their description and a more fundamental theory (such as the string theory) is required. It might be helpful to have a toy model for the merger and topology changing transitions, which is based on the physics which is well understood. In this talk we discuss such a toy model.
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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)
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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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Static black holes in higher dimensions
Tangherlini (1963) metric:
2 2 1 2 2 22NdS g dx dx FdT F dr r d
2 2 2 21 1 1sin ii i id d d
301 ( )NF r r
N is the number of ST dimensions
2nd is the metric on a unit n-dim sphere
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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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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
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
1 2( )DS R 2 2( )DR S
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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
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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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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
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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
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Near critical branes
Zoomed vicinity of the horizon
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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
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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
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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
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Boundary conditions
BC follow from finiteness of the curvature
It is sufficient to consider a scalar curvature
2 22
2 2 2
6 22 '(1 )
ZRR ZR RZ R R
R
0 00
0RR
dZZ Z
dR
2
004
RZ Z …
Z
0 00
0ZZ
dRR R
dZ
2
004
ZR R …
R
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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
( )R Z Z
2 2 2 0Z Z
1( 1 7)2
Z i
1 2 ( ) 7 / 2iR Z Z CZ
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Scaling properties
3/ 2 7 / 20 0( ) ( )iC kR k C R
Dual relations: ( )Z R R
2 2 2 0R R
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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
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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
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
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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 branes
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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]
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Flachi and Tanaka, PRL 95, 161302 (2005) [ (3+1) brane in 5d]
Moving branes
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THICK BRANE INTERACTING WITH BLACK HOLE
Morisawa et. al. , PRD 62, 084022 (2000)
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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.
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Euclidean topology
Sub-critical: 1 1DS R
# dim: bulk N, brane D, n=D-2
Super-critical: 2 2DR S
A transition between sub- and super-critical phases changes the Euclidean topology of BBH
Merger transition [Kol,’05]
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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, - 22n 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 branes
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Curvature at R=0 for sub-critical branes
ln( )
ln( )p
D=6
D=3
D=4
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The plot ln(Rmax) vs. ln(p-p*) from Garfinkle & Duncan (’98) paper
A similar plot for BBH system for D=4 after rescaling:
ln( ) 2 ln( ),
ln( ) (1/ 3) ln( )
p p
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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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Final remarks
Phase transitions, near critical behavior
Spacetime singularities during phase transitions?
New examples of `cosmic censorship’ violation
Asymmetry of BBH and BWH
Dynamical picture