Spin, Charge, and Topology in low dimensions BIRS,

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 wicks rotation the Euclidean manifolds have the topology (a) S 1 R 3

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 2 2 t 2 2 2

2 2 2 2 ds dt dl d Wicks rotation 2 t i 2

2 ds d dl d Black hole case: No black hole case: 2 2 2 1 2

2 1 2 0, 0, R S 0, 0, S R sub critical super

Two phases of BBH: sub- and super-critical Euclidean topology # dim: bulk 4, brane 3 Sub-critical: Super-critical: 1 S R 2 2 R S 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 2 d 2 F 1 r0 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 detab ,

3 ab g X X a 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

a (T r ) Induced metric 2 2 1 2 2 2

2 2 ds FdT [ F r (ddr ) ]dr r sin d S 2 T dr L , L r sin 1 Fr 2 (ddr ) 2 2 Brane equations d dL dL 0 dr ddr d

2 3 2 d d d d B3 B0 0 B2 B1 2 dr dr dr

dr 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 pln r q r

p, p ' - asymptotic data 2 Near critical branes Zoomed vicinity of the horizon Brane near horizon Proper distance r Z

r0 2 dr F 2 r r0 Z 2, F Z 2 is the surface gravity Metric near the horizon

dS 2 2 Z 2 dT 2 dZ 2 dR 2 R 2 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 (dRd ) 2 ] d 2 R 2 d 2 Reduced action: W d Z R R

Z S 2 T W (dZ d ) 2 (dR d ) 2 symmetry Brane equations near the horizon ZRR ( RR Z )(1 R2) 0 ( for R R( Z )) This equation is invariant under rescaling R( Z ) kR ( Z ) Z kZ 2 RZZ ( ZZ R )(1 Z ) 0 ( for Z Z ( R ))

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 R 2 R2 6ZRR 2Z 2 R Z 2 R 2 (1 R2) R Z 0 R0 Z R 0 Z 0 dR dZ

dZ dR 0 Z 0 0 R 0 Z2 R R0 4 R0

R2 Z Z 0 4Z 0 Critical solutions as attractors R Z Critical solution: New variables: x R, 1 y Z RR ds dZ ( yZ ) First order autonomous system

dx 2 x(1 y )(1 x ) ds dy 2 y[1 2 y x (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 ) 2 Z 2 Z 2 0 1 Z ( 1 i 7) 2 R Z Z

1 2 i (CZ ) 7 / 2 Scaling properties C (kR0 ) k Dual relations: 3/ 2 i 7 / 2 C ( R0 ) Z R ( R )

R 2 R 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

2 p ( p p ) ( p p) 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

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