JHEP 06 (2004) 053, hep-th/ The Hebrew University July Freie Universität Berlin Class.Quant.Grav. 22 (2005) , hep-th/ Talk at:

The goal and related works Description of the method matched asymptotic expansion in –Example: monopole match Results and implications for the phase diagram Summary

- solution for a small BH Analytical Harmark 03’, (Harmark and Obers 02’) Karasik et al. 04’ Chu, Goldberger and Rothstein 06’ 5d Interpolating coordinates EFT formalism 2 nd order Only monopole match Sorkin, Kol and Piran 03’ 5d Kudoh and Wiseman 03’, 04’ 5d, 6d Numerical M / M crit n / n bs

Coordinates for small caged BH r z cylindrical coordinates spherical coordinates + two dimensionful parameters: one dimensionless parameter:

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

The near zone The asymptotic zone The overlap region

Asymptotic zone - post-Newtonian expansion Near zone – Black hole static perturbations (4d - Regge Wheeler 57’) The solution is determined up to solutions of the homogeneous equation Einstein’s equations The leading terms in the radial part Weak field

BH (6d) monopole quadrupoleHexadecapole Asymptotic zone Near zone order in A dialogue of multipoles

Asymptotic zone Nearzone OverlapRegion angular terms

The area of a unit Near zone Asymptotic zone

Eccentricity The “Archimedes” effect The BH “repels” the space of the compact dimension

6d Numerical results: Kudoh and Wiseman 03’, 04’ nn bs MM crit GL BH US NUS 1 st order M / M crit n / n bs US 1 st order GL M / M crit n / n bs BH NUS

GL BH US NUS 1 st order 2 nd order n / n bs M / M crit 6d Numerical results: Kudoh and Wiseman 03’, 04’

GL BH US NUS 1 st order 2 nd order n / n bs M / M crit 6d Numerical results: Kudoh and Wiseman 03’, 04’

6d Numerical results: Kudoh and Wiseman 03’, 04’ GL BH US NUS 1 st order 2 nd order n / n bs M / M crit Inflection point ??

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. 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.