Einstein Gravity in Higher Dimensions Jerusalem 18 22

“Einstein Gravity in Higher Dimensions”, Jerusalem, 18 -22 Feb. , 2007

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.

Based on Christensen, V. F. , Larsen, Phys. Rev. D 58, 085005 (1998) V. F. , Larsen, Christensen, Phys. Rev. D 59, 125008 (1999) V. F. Phys. Rev. D 74, 044006 (2006)

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

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

Static black holes in higher dimensions Tangherlini (1963) metric: N is the number of ST dimensions is the metric on a unit n-dim sphere

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 brane event horizon

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

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

Euclidean topology # dim: bulk 4, brane 3 Sub-critical: Super-critical: 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:

bulk 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

Brane equations

Far distance solutions Consider a solution which approaches - asymptotic data

Near critical branes Zoomed vicinity of the horizon

Brane near horizon Proper distance is the surface gravity Metric near the horizon

Brane surface: Parametric form: Induced metric Reduced action: symmetry

Brane equations near the horizon This equation is invariant under rescaling

Boundary conditions BC follow from finiteness of the curvature It is sufficient to consider a scalar curvature

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

Phase portrait

Near-critical solutions

Scaling properties Dual relations:

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 For critical solution

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

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

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 5 d]

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.

Euclidean topology # dim: bulk N, brane D, n=D-2 Sub-critical: Super-critical: A transition between sub- and super-critical phases changes the Euclidean topology of BBH Merger transition [Kol, ’ 05]

Phase portraits

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

Curvature at R=0 for sub-critical branes D=3 D=4 D=6

The plot ln(Rmax) vs. ln(p-p*) from Garfinkle & Duncan (’ 98) paper A similar plot for BBH system for D=4 after rescaling:

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

Final remarks Phase transitions, near critical behavior Asymmetry of BBH and BWH Dynamical picture Spacetime singularities during phase transitions? New examples of `cosmic censorship’ violation