1. Higher Dimensional Black Holes Tsvi Piran Evgeny Sorkin & Barak Kol The Hebrew University, Jerusalem Israel E. Sorkin & TP, Phys.Rev.Lett. 90 (2003) 171301 B. Kol, E. Sorkin & TP, Phys.Rev. D69 (2004) 064031 E. Sorkin, B. Kol & TP, Phys.Rev. D69 (2004) 064032 E. Sorkin, Phys.Rev.Lett. (2004) in press

2. Kaluza-Klein, String Theory and other theories suggest that Space- time may have more than 4 dimensions. The simplest example is: 4d space-time R 3,1 Higher dimensions R 3,1 x S 1 - one additional dimension z r

3. d – the number of dimensions is the only free parameter in classical GR. It is interesting, therefore, from a pure theoretical point of view to explore the behavior of the theory when we vary this parameter (Kol, 04). 2d trivial 3d almost trivial 4d difficult 5d… you ain’t seen anything yet

4. What is the structure of Higher Dimensional Black Holes? Black String 4d space-time z Asymptotically flat 4d space- time x S 1 Horizon topology S 2 x S 1 z r

5. Or a black hole? 4d space-time z Asymptotically flat 4d space- time x S 1 Locally: Horizon topology S 3 z r

6. Black String Black Hole

7. A single dimensionless parameter Black hole L Black String 

8. What Happens in the inverse direction - a shrinking String? r z L Uniform B-Str:Non-uniform B-Str - ? Black hole Gregory & Laflamme (GL) : the string is unstable below a certain mass. Compare the entropies (in 5d): S bh ~  3/2 vs. S bs ~  2 Expect a phase tranistion  ?

9. Horowitz-Maeda ( ‘01 ): Horizon doesn’t pinch off! The end-state of the GL instability is a stable non-uniform string. ? Perturbation analysis around the GL point [ Gubser 5d ‘01 ] the non- uniform branch emerging from the GL point cannot be the end- state of this instability.  non-uniform >  uniform S non-uniform > S uniform Dynamical: no signs of stabilization (5d) [CLOPPV ‘03]. This branch non-perturbatively (6d) [ Wiseman ‘02 ]

10. Dynamical Instaibility of a Black String Choptuik, Lehner, Olabarrieta, Petryk, Pretorius & Villegas 2003

11. Non-Uniform Black String Solutions (Wiseman, 02)

12. Objectives: Explore the structure of higher dimensional spherical black holes. Establish that a higher dimensional black hole solution exists in the first place. Establish the maximal black hole mass. Explore the nature of the transition between the black hole and the black string solutions

13. The Static Black hole Solutions

15. Equations of Motion

16. Poor man’s Gravity – The Initial Value Problem (Sorkin & TP 03) Consider a moment of time symmetry: Higher Dim Black holes exist? There is a Bh-Bstr transition

17. Proper distance Log(  -  c )

18. Apparent horizons: From a simulation of bh merger Seidel & Brügmann A similar behavior is seen in 4D

19.  

20. The Anticipated phase diagram The Anticipated phase diagram [ Kol ‘02 ] order param  merger point ? x GL Uniform Bstr We need a “good” order parameter allowing to put all the phases on the same diagram. [Scalar charge] Non-uniform Bstr Gubser 5d,’01 Wiseman 6d,’02 Black hole Sorkin & TP 03

21. Asymptotic behavior At r  : The metric become z-independent as: [ ~exp(-r/L) ] Newtonian limit Sorkin,Kol,TP ‘03 Harmark & Obers ‘03 [Townsend & Zamaklar ’01,Traschen,’03]

22. Asymptotic Charges: The mass: m   T 00 d d-1 x The Tension:   -  T zz d d-1 x /L   Ori 04

23. The asymptotic coefficient b determines the length along the z-direction. b=k d [(d-3)m-  L] The mass opens up the extra dimension, while tension counteracts. BH Bstr Archimedes for BHs For a uniform Bstr: both effects cancel: But not for a BH:

24. Together with We get (gas ) Smarr’s formula (Integrated First Law) This formula associate quantities on the horizon (T and S) with asymptotic quantities at infinity (a). It will provide a strong test of the numerics.

25. Numerical Solution Sorkin Kol & TP

26. Numerical Convergence I:

27. Numerical Convergence II:

28. Numerical Test I (constraints):

29. Numerical Test II (The BH Area and Smarr’s formula):

31. Ellipticity “Archimedes” Analytic expressions Gorbonos & Kol 04 eccentricity distance

32. A Possible Bh Bstr Transition?

33. Anticipated phase diagram Anticipated phase diagram [ Kol ‘02 ] order param  merger point ? x GL Uniform Bstr Non-uniform Bstr Gubser 5d,’01 Wiseman 6d,’02 d=10 a critical dimension Black hole Kudoh & Wiseman ‘03 ES,Kol,Piran ‘03

34. Scalar charge GL x merger point ? The phase diagram: BHs Uniform Bstr Non-uniform Bstr Gubser: First order uniform-nonuniform black strings phase transition. explosions, cosmic censorship? Universal? Vary d ! E. Sorkin 2004 Motivations: (1) Kol’s critical dimension for the BH-BStr merger (d=10) (2) Problems in numerics above d=10 For d*>13 a sudden change in the order of the phase transition. It becomes smooth Scalar charge BHs Perturbative Non- uniform Bstr Uniform Bstr ?

35. The deviations of the calculated points from the linear fit are less than 2.1% with  = 0.686

36. corrected BH: Harmark ‘03 Kol&Gorbonos for a given mass the entropy of a caged BH is larger Entropies:

37. The curves intersect at d~13. This suggests that for d>13 A BH is entropically preferable over the string only for  c. A hint for a “missing link” that interpolates between the phases.

38. A comparison between a uniform and a non-uniform String: Trends in mass and entropy For d>13 the non-uniform string is less massive and has a higher entropy than the uniform one. A smooth decay becomes possible.

39. Interpretations & implications Above d*=13 the unstable GL string can decay into a non-uniform state continuously. BHs Perturbative Non- uniform Bstr Uniform Bstr ? b filling the blob a smooth merger discontinuous a new phase, disconnected

40. Outlook Continue the non-uniform phase to the non- linear regime. Check time-evolution: does the evolution stabilize? Is the smooth p.t. general: more extra dimensions, other topologies. Would the critical dimension become smaller than d=10?

41. Summary We have demonstrated the existence of BH solutions. Indication for a BH-Bstr transition. The global phase diagram depends on the dimensionality of space-time

42. Open Questions Non uniquness of higher dim black holes Topology change at BH-Bstr transition Cosmic censorship at BH-Bstr transition Thunderbolt (release of L=c 5 /G) Numerical-Gravitostatics Stability of rotating black strings