1. Non-uniform black strings/branes: non-linear effects of gauge charge Umpei MIYAMOTO Waseda U. “Einstein's Gravity in Higher Dims” 18-22 Feb ’07 @Hebrew Univ of Jerusalem

2. U. MiyamotoNon-Uniform Charged BS2 Plan 1.Gregory-Laflamme instability –Basics –Correlation of stabilities –Possible final state & Critical Dims. 2.Static perturbations of charged strings –“ Proof ” of GM conjecture –(New) stable phase of non-uniform string 3.Summary Reviews: hep-th/0411240, Kol hep-th/0701022, Harmark-Niarchos-Obers

3. 1. 1.Introduction

4. U. MiyamotoNon-Uniform Charged BS4 GL instability (vacuum) Fluid analogue: Jeans instability Rayleigh-Plateau instability [Cardoso-Dias '06] – –Dispersion relation – –Large D-dependence – –s-wave is only unstable mode – –Critical Dim. D*~10 [Gregory-Laflamme, ‘93] r z GL critical mode k 0 [Gregory-Laflamme, ‘93]

5. U. MiyamotoNon-Uniform Charged BS5 Correlation of stabilities [G-L ’93, Gubser-Mitra,'00,'01 Reall ’01, Ross-Wiseman ‘05] Gubser-Mitra (correlated stability) conjecture: For black objects with a non-compact translational symmetry, Dynamically stability  Locally thermodynamical stability Possible violation & refinement of the conjecture (unstable test scalar field) [Freiss-Gubser-Mitra ’05, Kol] Charging up [Gregory-Laflamme, ‘93]

6. U. MiyamotoNon-Uniform Charged BS6 Possible final state (vacuum) [Gubser 02, Harmark ’04,Kol-Sorkin-Piran ‘04 Gorbonos-Kol ‘04, Wiseman’03, Kudoh-Wiseman '05, Sorkin 06] Time evolution of unstable BS [Choptuik-Lehner et al, ‘03] D=5 D=6 [Horowitz-Maeda ’01, Choptuik-Lehner et al, ‘03] [Kudoh-Wiseman '05]

7. U. MiyamotoNon-Uniform Charged BS7 Critical Dims. (vacuum) 13.5 12.5 Entropy comparison (fixed M) Free-energy comparison (fixed T) [Sorkin ’04, Kudoh-UM ‘05] Landau-Ginzburg theory  D*=12 [Kol-Sorkin '06]  D > 13 : NUBS is favored!! (second-order transition)  D > 12 : NUBS is favored!! (second-order transition) Rayleigh-Plateau  D*=10 [Cardoso-Dias '06]

8. U. MiyamotoNon-Uniform Charged BS8 Motivation: why non-uniform charged BS? Black strings/branes have charges (gauge/dilaton/angular momenta). Final fate of GL instability for charged strings/branesFinal fate of GL instability for charged strings/branes –cf: Smeared black p-branes (boost + U-duality) [Harmark-Obsers ’ 02..,Kudoh-UM ‘ 05] –cf: Thin black ring (boosted BS) [Hobdebo-Myers ‘ 06] –Phase structure Do charges stabilize or destabilize non-uniform phase?Do charges stabilize or destabilize non-uniform phase? Can critical Dims. be reduced/disappear?Can critical Dims. be reduced/disappear?

9. 2. Static perturbations of charged BS ・ JHEP 12(2006)048 ・ Works in progress collaborations with Hideaki KUDOH

10. U. MiyamotoNon-Uniform Charged BS10 Setup: action & background (magnetic BS) [Gibbons,Horowitz,Townsend ’95]

11. U. MiyamotoNon-Uniform Charged BS11 Perturbation scheme Expansion of X = (a,b,c) [Gubser '02] z

12. U. MiyamotoNon-Uniform Charged BS12 Forbidden by GMC NS Linear O(ε): realization of GMC & “ critical phenomena ” “2nd-order phase transition”

13. U. MiyamotoNon-Uniform Charged BS13 The “ optimal ” gauge & “ proof ” of GMC Master equation [Kol ’06,’06] *** Non-existence of GL mode for

14. U. MiyamotoNon-Uniform Charged BS14 Higher order: non-linear backreactions D=6 D=14 D=10 NUBS is favored !! (2nd-order transition) Entropy comparison btw NUBS & UBS in microcanonical ensemble (same (M, Q)) cf: D-dep.

15. U. MiyamotoNon-Uniform Charged BS15 Other ensembles D=6 D=14 Canonical ensemble (same T & Q) Grandcanonical ensemble (same T & Φ H )

16. 3. Summary

17. U. MiyamotoNon-Uniform Charged BS17 Summary Final fate of GL instability is open.Final fate of GL instability is open. –It will depend on D (critical dim. in vacuum). –How about string/brane with charges? Static perturbations of magnetic strings (5<D<15).Static perturbations of magnetic strings (5<D<15). –Linear order: Simplest(?) master equation & no GL for Q>Q GMSimplest(?) master equation & no GL for Q>Q GM k GL ~|Q-Q GM | 1/2 near Q=Q GMk GL ~|Q-Q GM | 1/2 near Q=Q GM –Higher orders: Critical charges appear: Q I,cr, Q II,cr, Q III,crCritical charges appear: Q I,cr, Q II,cr, Q III,cr Entropically favored non-uniform string in any D.Entropically favored non-uniform string in any D. Charge controls the stability also in non-linear regime.Charge controls the stability also in non-linear regime. The final fate will depend on Q/M (extremality).The final fate will depend on Q/M (extremality).

18. U. MiyamotoNon-Uniform Charged BS18 Future prospects / To do Understanding the critical charges, Q n,cr (n=I,II,III)Understanding the critical charges, Q n,cr (n=I,II,III) –Dilatonic branes with no GM point (NS5,D5..) (in progress) –Fully non-linear deformation (in progress) ** *** ********** –Extension of Landau-Ginzburg argument Dynamical (perturbative) stability of NUBSDynamical (perturbative) stability of NUBS Dynamical evolution for Q I,cr < Q < Q II,crDynamical evolution for Q I,cr < Q < Q II,cr –Charge (in 6D) would be easier than D=14. 2 control parameters

19. END