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The Phase Transition between Caged Black Holes and Black Strings

This study delves into the phase transition physics of caged black holes and black strings, exploring the energy release, order of transition, and free energy systems in higher dimensions and string theory.

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The Phase Transition between Caged Black Holes and Black Strings

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  1. The Phase Transition between Caged Black Holes and Black Strings Barak Kol Hebrew University - Jerusalem Bremen, Aug 2008 • Based on hep-th’s (BK) • 0411240 Phys. Rep. • 0206220 original • 0608001, 0609001 “optimal gauge” • Outline • Physical summary • Recent elements of the phase diagram

  2. Why study GR in higher dimensions? A parameter of GR String theory Large extra dimensions The novel feature – non-uniqueness of black objects The distinction between “no-hair” and “uniqueness” Uniqueness is special to 4d Phase transition physics: energy release, order of transition and free energy Systems The ring, the susy ring Black hole black string Einstein-YM Physical summary • Background • Set up • Processes • Phase diagram

  3. Generalizations Background BK-Sorkin Extra matter content: vectors, scalars… Charged Kudoh-Miyamoto Rotation Kleihaus-Kunz-Radu Liquid analogy Dias-Cardoso Brane-worlds Tanaka, Emparan, Fitzpatrick-Randall-Wiseman The system Theory: pure gravity, Background Coordinates z L r Set-up and formulation

  4. Black string Black hole Single dim’less parameter. e.g. or The phases Horizon topology Uniform/ Non

  5. Processes – personal view • String decay

  6. Processes – personal view • String decay

  7. Black hole decay

  8. Initial conditions for evaporation Critical string 1st Continuous transition: developing non-uniformity Continuously reaching the merger point 2nd Continuous transition to a black hole • Smooth transition (2nd order or higher)

  9. μ μ merger X merger GL GL X b/β b/β Main results - The phase diagram Prediction: BH and String merge, the end state is a BH Sorkin Un Str Un Str Non-Un Str Non-Un Str BH BH No stable non-uniform phase (for D<14)! 2nd order

  10. string merger GL BH GL’ Theory and “experiment” Kudoh-Wiseman Sep 2004 BK, hep-th/0206220

  11. μ merger X GL Caged BH b/β (Recent) elements of the phase diagram • Gregory-Laflamme instability (perturbative) • Caged black holes (perturbative) • Merger (qualitative-topological) • Numeric (non-perturbative)

  12. Gregory-Laflamme InstabilityThe uniform string

  13. Arbitrary d (D=d+1)Sorkin The instability • GL (1993) (D is d, r0=2) • Negative modes for Euclidean black holes • Gross-Perry-Yaffe (1982) (dim’less)

  14. Components of A system of nF=5 eq’s Gauge choice? nG=2 Possible to eliminate the gauge! BK “Gauge Invariant Perturbation Theory” – possible due to 1 non-homogeneous dim The gauge shoots twice: eliminates one field and makes another non-dynamic nD=nF-2 nG Here nD=5-2*2=1, hence Master field Master equation BK-Sorkin

  15. d as a parameter (like dim. Reg.) d=4 is numeric, but there are limits d→∞, λGL~d + pert. expansion d→3, λGL →0 Interpolation (Pade) Asymptotics for Good to 2% Asnin-Gorbonos-Hadar-BK-Levi-Miyamoto

  16. Order of transition Gubser • The zero mode can be followed to yield an emerging non-uniform branch • First or second order? Landau - Ginzburg theory of phase transitions

  17. Harmark, Gorbonos-BK Caged black holes – A dialogue of multipoles • Two zones • Matched asymptotic expansion • Asymptotically – zeroth order is Schw, first order is Newtonian • Inthe near zone – zeroth order is flat compactified space (origin removed), and we developed the form of the first correction in terms of the hypergeometric func

  18. eccentricity BH “makes space” for itself “BH Archimedes effect” Some results • Dialogue figure The perturbation ladder • Effective Field Theory approach, Chu-Goldberger-Rothstein, an additional order • CLEFT improvement, BK-Smolkin • more in next talk

  19. Morse theory is the topological theory of extrema of functions You may be familiar with the way Morse theory measures global properties of manifolds (Homology), but here we need properties of extrema that are invariant under deformations of the function. Solutions are extrema of the action Simplest formulation “Phase conservation law” n+1 n More (and most) general Merger - Morse theory BK So expect non-uniform St phase to connect with BH

  20. Numeric solutions Wiseman • Relaxation (see also Ricci-flow) • 2d -> Cauchy-Riemann identity for constraints Used to set b.c. Results for non-uniform strings ->

  21. Wiseman-Kudoh Sorkin BK Piran Results for the black hole Geometry Embedding diagrams Eccentricity x is the small dimensionless parameter

  22. BH thermodynamics Mass, tension Area, beta Correction to area - temperature relation

  23. Phase diagram End state Processes Critical dimensions Topology change of total Euclidean manifold BK Elements of the diagram Control/order parameters on axes Harmark-Obers, BK-Sorkin-Piran Limits first The GL uniform string Caged BHs Qualitative form of the diagram Merger Full numerical solutions Conclusions

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