JHEP 06 (2004) 053, hep-th/0406002

1. Matched Asymptotic Expansion for Caged Black Holes Dan Gorbonos Barak Kol Talk at: MG11 The Hebrew University Freie Universität Berlin July 27 2006 JHEP 06 (2004) 053, hep-th/0406002 Class.Quant.Grav. 22 (2005) 3935-3960, hep-th/0505009

2. Matched Asymptotic Expansion for Caged Black Holes Outline • The goal and related works • Description of the methodmatched asymptotic expansion in • Example: monopole match • Results and implications for the phase diagram • Summary

3. Analytical Numerical 5d • Sorkin, Kol and Piran 03’ • Harmark 03’, (Harmark • and Obers 02’) Interpolating coordinates 5d, 6d • Kudoh and Wiseman 03’, 04’ 5d • Karasik et al. 04’ M / Mcrit 3 2.5 2nd order • Chu, Goldberger • and Rothstein 06’ EFT formalism 2 1.5 Only monopole match 1 0.5 n /nbs 0.1 0.2 0.3 0.4 0.6 0.8 1 Goal: - solution for a small BH Related works:

4. Description of the method Coordinates for small caged BH z cylindrical coordinates r spherical coordinates + two dimensionful parameters: one dimensionless parameter:

5. b.c. Description of the method The matching idea Perturbative expansion of Einstein’s equations • A small parameter • Input: - An exact solution - Boundary conditions Two zones: Near horizon Asymptotic Fixed parameter Small parameter The exact solution Large Overlap Region Schwarzschild-Tangherlini Minkowsky + periodic

6. Description of the method The two zones The near zone The overlap region The asymptotic zone

7. Description of the method What is actually matched ? • Asymptotic zone - post-Newtonian expansion • Near zone – Black hole static perturbations Einstein’s equations (4d -Regge Wheeler 57’) The solution is determined up to solutions of the homogeneous equation Multipole moments Weak field The leading terms in the radial part

8. monopole Hexadecapole quadrupole A dialogue of multipoles Description of the method Matching patterns (6d) Newtonian PN Asymptotic zone 2 3 1 4 5 6 7 8 order in 0 Near zone BH

9. PN angular terms Description of the method Example: monopole match Asymptotic zone Newtonian Overlap Region Near zone

10. The area of a unit Matching results Near zone Asymptotic zone Monopole matching

11. Matching results Effects of higher multipoles • Eccentricity • The “Archimedes” effect The BH “repels” the space of the compact dimension

12. M / Mcrit M / Mcrit M M crit 1st order 1st order 3 3 US US 2.5 2.5 2 BH 2 1.5 1.5 NUS GL GL 1 1 0.5 0.5 n / nbs n / nbs n n bs 0.1 0.2 0.3 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.6 0.8 1 Implications for the phase diagram 6d Numerical results: Kudoh and Wiseman 03’, 04’ BH NUS

13. 3 2.5 2 1.5 1 0.5 0.1 0.2 0.3 0.4 0.6 0.8 1 Implications for the phase diagram 6d Numerical results: Kudoh and Wiseman 03’, 04’ M / Mcrit 1st order 2nd order US BH NUS GL n / nbs

14. 3 2.5 2 1.5 1 0.5 0.1 0.2 0.3 0.4 0.6 0.8 1 Implications for the phase diagram 6d Numerical results: Kudoh and Wiseman 03’, 04’ M / Mcrit 1st order 2nd order US BH NUS GL n / nbs

15. 3 2.5 2 1.5 1 0.5 0.1 0.2 0.3 0.4 0.6 0.8 1 Implications for the phase diagram 6d Numerical results: Kudoh and Wiseman 03’, 04’ M / Mcrit 1st order 2nd order US BH NUS GL n / nbs Inflection point ??

16. Summary • Matched asymptotic expansion was used to obtain an approximate • analytical solution for a small BH in . The method can be • carried in principle to an arbitrarily high order in the small parameter. • The method yields approximations to the whole metric providing not • only the thermodynamic quantities but also BH eccentricity and • "the BH Archimedes effect" • A comparison with the numerical simulation in 6d shows an excellent • agreement in the first order approximation when the second order • indicates that there should be an inflection point which is not seen • in the simulations so far.