“Models of Gravity in Higher Dimensions”, Bremen, Aug , 2008

Based on Christensen, V.F., Larsen, Phys.Rev. D58, (1998) V.F., Larsen, Christensen, Phys.Rev. D59, (1999) V.F. Phys.Rev. D74, (2006) V.F. and D.Gorbonos, hep-th/ (2008)

BH critical merger solutions B.Kol, 2005; V.Asnin, B.Kol, M.Smolkin, 2006

`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.

Topology change transitions Change of the spacetime topology Euclidean topology change

An example A thermal bath at finite temperature: ST after the Wick’s rotation is the Euclidean manifolds No black hole

Euclidean black hole

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

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

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

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.

The temperature of the bulk BH and of the brane BH is the same.

(2+1) static axisymmetric spacetime Black hole case: Wick’s rotation No black hole case: Induced geometry on the brane

Two phases of BBH: sub- and super-critical sub super critical

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

Bulk black hole metric

No scale parameter – Second order phase transition

bulk coordinates coordinates on the brane Dirac-Nambu-Goto action 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 Coordinates on the brane Induced metric

Main steps 1. Brane equations 2. Asymptotic form of a solution at infinity 3. Asymptotic data 4. Asymptotic form of a solution near the horizon 5. Scaling properties 6. Critical solution as attractor 7. Perturbation analysis of near critical solutions 8. The brane BH size vs `distance’ of the asymptotic data from the critical one 9. Choptuik behavior

Far distance solutions Consider a solution which approaches - asymptotic data

Near critical branes Zoomed vicinity of the horizon

is the surface gravity Metric near the horizon Brane near horizon This equation is invariant under rescaling

Duality transformation 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.

Critical solutions as attractors Critical solution: New variables: First order autonomous system Node Saddle Focus

Phase portrait

Near-critical solutions Scaling properties

Near critical solutions Critical brane: Under rescaling the critical brane does not move

Near (Rindler) zone (scaling transformations are valid) Asymptotic region {p,p’} Global structure of near-critical solution

Scaling and self-similarity is a periodic function with the period For both super- and sub-critical brines

Phase portraits

Scaling and self-similarity is a periodic function with the period For both super- and sub-critical brines

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

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)

Exist scale parameter – First order phase transition

extrinsic curvature Set “fundamental length”: C=1 Energy density Polyakov 1985

2 1 EOM: 4 th order ODE Axial symmetry Z R Highest number of derivatives of the fields

4 th order linear equation for 4 th order linear equation for 4 modes: 3 stable 1 unstable Tune the free parameter R Z

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)

as a function of for n=2. The dashed line is the same function for DNG branes (without stiffness terms).

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

n=2, C=1

R''(0) as a function of R_0 (supercritical) for n=2 and B=1

THICK BRANE INTERACTING WITH BLACK HOLE Morisawa et. al., PRD 62, (2000); PRD 67, (2003)

Flachi and Tanaka, PRL 95, (2005) [ (3+1) brane in 5d] Moving brines

Final remarks DNG 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.