A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS

1. A TOY MODEL FOR TOPOLOGY CHANGE TRANSITIONS Valeri P. Frolov University of Alberta “Models of Gravity in Higher Dimensions”, Bremen, Aug. 25-29, 2008

2. 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) V.F. and D.Gorbonos, hep-th/ 0808.3024 (2008)

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

4. `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.

5. Topology change transitions Change of the spacetime topology Euclidean topology change

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

7. Euclidean black hole

8. 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

9. 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

10. 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

11. A brane in the bulk BH spacetime

12. 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

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

14. (2+1)- brane near (3+1)- bulk black hole

15. (2+1) static axisymmetric spacetime Wick’s rotation Black hole case: No black hole case: Induced geometry on the brane

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

17. Euclidean topology change A transition between sub- and super-critical phases changes the Euclidean topology of BBH Our goal is to study these transitions An analogy with merger transitions [Kol,’05]

18. Bulk black hole metric

19. DNG Branes (without Stiffness) No scale parameter – Second order phase transition

20. 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).

21. Brane equation Coordinates on the brane Induced metric

22. Main steps 1. Brane equations2. 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

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

24. Near critical branes Zoomed vicinity of the horizon

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

26. Duality transformation The critical solution is invariant under both scaling and dual transformations. Combining the scaling and duality transformations one can obtain any noncritical solution from any other one.

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

28. Phase portrait

29. Near-critical solutions Scaling properties

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

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

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

33. D-dim brane near N-dim bulk black hole

34. Phase portraits

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

36. 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

37. Beyond the adopted approximations • Thickness effects • Interaction of a moving brane with a BH • Irreversability • Role of the brane tension • Curvature corrections (V.F. and D.Gorbonos, • under preparation)

38. Branes with Stiffness Exist scale parameter – First order phase transition

39. Adding Stiffness extrinsic curvature Polyakov 1985 Set “fundamental length”: C=1 Energy density

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

41. Linear perutbations to the attractor 4th order linear equation for Z R 4 modes: 3 stable Tune the free parameter 1 unstable

42. 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)

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

44. 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

45. n=2, C=1

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

47. Additional Subjects

48. THICK BRANE INTERACTING WITH BLACK HOLE Morisawa et. al. , PRD 62, 084022 (2000); PRD 67, 025017 (2003)

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