2. Barak Kol Hebrew University - Jerusalem Bremen, Aug 2008 Outline Physical summary Recent elements of the phase diagram Based on hep-th’s (BK) 0411240 Phys. Rep. 0206220 original 0608001, 0609001 “optimal gauge”

3. Physical summary Why study GR in higher dimensions? A parameter of GR String theory Large extra dimensions The novel feature – non-uniqueness of black objects The distinction between “no- hair” and “uniqueness” Uniqueness is special to 4d Phase transition physics: energy release, order of transition and free energy Systems The ring, the susy ring Black hole black string Einstein-YM Background Set up Processes Phase diagram

4. Generalizations Background BK-Sorkin Extra matter content: vectors, scalars… Charged Kudoh-Miyamoto Rotation Kleihaus-Kunz-Radu Liquid analogy Dias-Cardoso Brane-worlds Tanaka, Emparan, Fitzpatrick-Randall- Wiseman The system Theory: pure gravity, Background Coordinates Set-up and formulation L z r

5. The phases Black stringBlack hole Horizon topology Uniform/ Non Single dim’less parameter. e.g. or

6. String decay Processes – personal view

7. String decay Processes – personal view

8. Black hole decay

9. Smooth transition (2 nd order or higher ) Initial conditions for evaporationCritical string1st Continuous transition: developing non-uniformity Continuously reaching the merger point 2 nd Continuous transition to a black hole

10. Main results - The phase diagram Prediction: BH and String merge, the end state is a BH b/β μ GL X merger No stable non-uniform phase (for D<14)! b/β μ GL merger X 2 nd order Un Str Un Str BH Non-Un Str Non-Un Str Sorkin

11. Theory and “experiment” BH merger GL string GL’ Kudoh-Wiseman Sep 2004 BK, hep-th/0206220

12. (Recent) elements of the phase diagram Gregory-Laflamme instability (perturbative) Caged black holes (perturbative) Merger (qualitative-topological) Numeric (non-perturbative) b/β μ GL X merger Caged BH

13. Gregory-Laflamme Instability The uniform string

14. Negative modes for Euclidean black holes Gross-Perry-Yaffe (1982) The instability GL (1993) (D is d, r0=2) Arbitrary d (D=d+1) Sorkin (dim’less)

15. Master equation Components of A system of n F =5 eq’s Gauge choice? n G =2 Possible to eliminate the gauge! BK “Gauge Invariant Perturbation Theory” – possible due to 1 non- homogeneous dim The gauge shoots twice: eliminates one field and makes another non-dynamic n D =n F -2 n G Here n D =5-2*2=1, hence Master field BK- Sorkin

16. Asymptotics for d as a parameter (like dim. Reg.) d=4 is numeric, but there are limits d→∞, λ GL ~d + pert. expansion d→3, λ GL →0 Interpolation (Pade) Asnin-Gorbonos-Hadar-BK-Levi- Miyamoto Good to 2%

17. Order of transition The zero mode can be followed to yield an emerging non-uniform branch First or second order? Landau - Ginzburg theory of phase transitions Gubser

18. Caged black holes – A dialogue of multipoles Two zones Matched asymptotic expansion Asymptotically – zeroth order is Schw, first order is Newtonian In the near zone – zeroth order is flat compactified space (origin removed), and we developed the form of the first correction in terms of the hypergeometric func Harmark, Gorbonos-BK

19. Dialogue figure The perturbation ladder Some results eccentricity BH “makes space” for itself “BH Archimedes effect” Effective Field Theory approach, Chu-Goldberger-Rothstein, an additional order CLEFT improvement, BK-Smolkin more in next talk

20. Merger - Morse theory Morse theory is the topological theory of extrema of functions You may be familiar with the way Morse theory measures global properties of manifolds (Homology), but here we need properties of extrema that are invariant under deformations of the function. Solutions are extrema of the action Simplest formulation “Phase conservation law” More (and most) general n n+1 So expect So expect non-uniform St phase to connect with BH BK

21. Numeric solutions Relaxation (see also Ricci- flow) 2d -> Cauchy-Riemann identity for constraints Used to set b.c. Results for non-uniform strings -> Wiseman

22. Results for the black hole Wiseman-Kudoh Sorkin BK Piran Geometry Embedding diagrams Eccentricity x is the small dimensionless parameter

23. BH thermodynamics Mass, tension Area, beta Correction to area - temperature relation

24. Conclusions Phase diagram End state Processes Critical dimensions Topology change of total Euclidean manifold BK Elements of the diagram Control/order parameters on axes Harmark-Obers, BK- Sorkin-Piran Limits first The GL uniform string Caged BHs Qualitative form of the diagram Merger Full numerical solutions

25. The Issues Static The end state The phase diagram Time evolution Naked singularity? Cosmic censorship violation? Thunderbolt? Local geometry? Quantum gravity? Much progress!No progress!

26. Main results - continuation Critical dimensions D* merger =10 no tachyons for D>D* BK D* GL =“13.5” second order for D>D* Sorkin Topology change around the “pinch” solution – all are Ricci flat metrics

27. A critical dimension blue line: log(μ GL )/log(μ s )-1 Sorkin Indications μ s is the mass for equal areas, S BH (μ s )=S St (μ s ) BH approximated to be spherical Results Upper – lowest order change in mass Lower – same for entropy

28. The phase diagram The end state? For D≥14 there is a natural candidate – the non-uniform St. (For D≤13) GL argued for the BH (5d formulae) Horowitz-Maeda disagreed. Argued that pinching is impossible (at finite horizon time) due to increasing area of event horizon. Predicted a stable non-uniform string.

29. In order to know all possible end states it is enough to map out the static solutions – the phase diagram The order parameter and general identities Qualitative tools Morse theory The “merger” topology change BK hep-th/0206220

30. The order parameter and general identities Far away, for r>>L, metric becomes z- independent 2 asymptotic constants b is the “scalar charge” For St b=0, while for BH b and a are both proportional to m through Φ N Hence, b/m is a good order parameter: It is zero for St and finite for BH The BH makes space for itself –”Archimedes effect” St - tension counteracts BK, Sorkin & Piran Harmark & Obers

31. a and b can be traded for 2 physical quantities: m, τ These are defined through thermodynamics: For idea gas dE=T dS – P dV Here dm= T dS + τ dL τ is the tension “positive tension theorem” Relation with a,b A useful relation – Smarr formula or integrated first law Relates horizon quantities with asymptotic ones

32. Merger Topology change (conifold - like) The cone Deformed cone BK

33. A critical dimension The ansatz Quadratic part of action for ε - after change of variables EOM is like a Schrödinger problem with V=-c/r^2 is critical. Hence D*=10

34. Theory and experiment BH merger GL string GL’ Kudoh-Wiseman Sep 2004 BK, hep-th/0206220 Morse theory/ Poincare’s method -> merger ←

35. Obtaining solutions Numeric – relaxation Non uniform St, BH Analytic – Dialogue of multipoles

36. Harmark-Obers Coordinates Define R through the Newtonian potential Define a second coordinate v, to be orthogonal with a specific parameterization according to a “weighted Cauchy Riemann eq” The profit: one of the EOM is algebraic, leaving us with 2 functions

37. Time evolution Garfinkle-Lehner-Pretorius gr-qc/0412014 Choptuik et al gr-qc/0304085. Re-anlysis: Time evolution is consistent with pinching at infinite “horizon time” (affine parameter λ=e s ) and finite asymptotic time

38. Related work Gauge theory (Gross-Witten transition) Bubbles Harmonics Limits on intersections in the phase diagram Black holes in brane-worlds Harmark Obers Susskind(1997) Aharony Marsano Minwalla Wiseman Epilogue

39. JHEP hep-th/0206220 contains the central ideas: merger of phases with topology change, one of the critical dimensions, and the correct phase diagram for d<14. Interestingly, it was rejected for publication by both JHEP and PRD. A possible summary: Referee report on hep-th/0206220 ~“looks like a research proposal rather than a paper…’ ~“Should be massively rewritten before I look at it” On hep-th/0502033 (to be mentioned) “Merger has no relation with neither the BH – Str transition, nor with Choptuik scaling” “Long on formalism and buzzwords but short on physics” ←

40. Reprise Main results Static case is reasonably well understood Phases merge End state Critical dimensions Merger of different topologies Additional results Order parameter – scalar charge and tension Smarr formula Simulation via relaxation GL critical k Dialogue of multipoles – matched asymptotic expansion including “Archimedes effect”

41. Open issues and developments Issues Time evolution – spacetime diagram, naked singularity? a thunderbolt? Merger and Choptuik scaling (threshold for BH production) – hep- th/0502033 (BK) (rejected by JHEP):rejected by JHEP Self similarity for naked singularities. Log-period for “echoing” – a fractal Scaling exponents. Self-organized criticality

42. Jerusalem