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Matched Asymptotic Expansion for Caged Black Holes

This talk at MG11 discusses using matched asymptotic expansion to approximate small black holes, covering method details, monopole matching example, phase diagram implications, analytical and numerical results in 5D and 6D caged black holes scenarios.

eugenia-klein
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Matched Asymptotic Expansion for Caged Black Holes

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  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.

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