Higher Dimensional Black Holes

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. z r Kaluza-Klein, String Theory and other theories suggest that Space-time may have more than 4 dimensions.The simplest example is: R3,1 x S1 - one additional dimension Higher dimensions 4d space-time R3,1

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 almosttrivial 4d difficult 5d … you ain’t seen anything yet

4. z r What is the structure of Higher Dimensional Black Holes? Black String Asymptotically flat 4d space-time x S1 z 4d space-time Horizon topology S2 x S1

5. z r Or a black hole? Locally: Asymptotically flat 4d space-time x S1 z 4d space-time Horizon topology S3

6. Black String Black Hole

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

8. What Happens in the inverse direction - a shrinking String? m ? Non-uniform B-Str - ? Uniform B-Str: z L r Black hole Gregory & Laflamme (GL) : the string is unstable below a certain mass. Compare the entropies (in 5d): Sbh~m3/2 vs. Sbs ~ m2 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. mnon-uniform > muniform Snon-uniform > Suniform 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: There is a Bh-Bstr transition Higher Dim Black holes exist?

17. Log(m-mc) Proper distance

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

19. m m

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

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

22. t t Asymptotic Charges: The mass:m  T00dd-1x The Tension:t  -Tzzdd-1x/L Ori 04

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

24. Smarr’s formula (Integrated First Law) Together with (gas ) We get 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. eccentricity Ellipticity Analytic expressions Gorbonos & Kol 04 distance “Archimedes”

32. A Possible Bh Bstr Transition?

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

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

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

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

37. The curves intersect at d~13. This suggests that for d>13 A BH is entropically preferable over the string only for m<mc . 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 a smooth merger Uniform Bstr Perturbative Non-uniform Bstr filling the blob ? discontinuous BHs a new phase, disconnected Above d*=13 the unstable GL string can decay into a non-uniform state continuously. b

40. 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

41. 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=c5/G) • Numerical-Gravitostatics • Stability of rotating black strings