A Toy Model for Topology Change Transitions

1. A Toy Model for Topology Change Transitions Valeri P. Frolov University of Alberta Spin, Charge, and Topology in low dimensions BIRS, Banff, July 29 - August 3, 2006

2. 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)

3. A thermal bath at finite temperature with (a) and without (b) black hole. After the wick’s rotation the Euclidean manifolds have the topology Topology change transitions Change of the spacetime topology Euclidean topology change An example

4. More fundamental field-theoretical description of a “realistic” brane “resolves” singularities 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

5. 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

6. A brane in the bulk BH spacetime

7. 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

8. Black hole case: No black hole case: (2+1) static axisymmetric spacetime Wick’s rotation

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

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

11. 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:

12. 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).

13. Brane equation Coordinates on the brane Induced metric

14. Brane equations

15. - asymptotic data Far distance solutions Consider a solution which approaches

16. Near critical branes Zoomed vicinity of the horizon

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

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

19. Brane equations near the horizon This equation is invariant under rescaling This equation is invariant under rescaling

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

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

22. Phase portrait

23. Near-critical solutions

24. Scaling properties Dual relations:

25. A solution is singled out by the value of We study super-critical solutions close to the critical one. Consideration of sub-critical solutions is similar. For critical solution

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

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

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

29. 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]

30. Moving branes Flachi and Tanaka, PRL 95, 161302 (2005) [ (3+1) brane in 5d]

32. THICK BRANE INTERACTING WITH BLACK HOLE Morisawa et. al. , PRD 62, 084022 (2000)

33. 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.

34. 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