Non-uniform black strings/branes:non-linear effects of gauge charge

Umpei MIYAMOTOWaseda U.

“Einstein's Gravity in Higher Dims” 18-22 Feb ’07 @Hebrew Univ of Jerusalem

U. Miyamoto Non-Uniform Charged BS 2

PlanPlan

1.1. Gregory-Laflamme instabilityGregory-Laflamme instability

– BasicsBasics– Correlation of stabilitiesCorrelation of stabilities– Possible final state & Critical Dims.Possible final state & Critical Dims.

2.2. Static perturbations of charged stringsStatic perturbations of charged strings

– ““Proof” of GM conjectureProof” of GM conjecture– (New) stable phase of non-uniform string(New) stable phase of non-uniform string

3.3. SummarySummaryReviews:•hep-th/0411240, Kol•hep-th/0701022, Harmark-Niarchos-Obers

1. Introduction

U. Miyamoto Non-Uniform Charged BS 4

GL instability GL instability (vacuum)(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 k0

[Gregory-Laflamme, ‘93]

U. Miyamoto Non-Uniform Charged BS 5

Correlation of stabilitiesCorrelation of stabilities

[G-L ’93, Gubser-Mitra,'00,'01Reall ’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]

U. Miyamoto Non-Uniform Charged BS 6

Possible final state Possible final state (vacuum)(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]

U. Miyamoto Non-Uniform Charged BS 7

Critical Dims. Critical Dims. (vacuum)(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]

U. Miyamoto Non-Uniform Charged BS 8

Motivation:Motivation:why non-uniform charged BS?why non-uniform charged BS?

Black strings/branes have charges (gauge/dilaton/aBlack strings/branes have charges (gauge/dilaton/angular momenta).ngular 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: Smeared black p-branes (boost + U-duality) [Harmark-Obsers ’02..,Kudoh-UM ‘05]– cf: Thin black ring (boosted BS) [Hobdebo-Myers ‘06]cf: Thin black ring (boosted BS) [Hobdebo-Myers ‘06]

– Phase structurePhase 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?

2. Static perturbationsof charged BS

・ JHEP 12(2006)048・ Works in progress

collaborations with

Hideaki KUDOH

U. Miyamoto Non-Uniform Charged BS 10

Setup: action & background (magnetic BS)Setup: action & background (magnetic BS)[Gibbons,Horowitz,Townsend ’95]

U. Miyamoto Non-Uniform Charged BS 11

Perturbation schemePerturbation scheme

Expansion of X = (a,b,c)

[Gubser '02]

z

U. Miyamoto Non-Uniform Charged BS 12

Forbiddenby GMC

NS

Linear Linear O(ε):O(ε):

realization of GMC & “critical phenomena”realization of GMC & “critical phenomena”

“2nd-order phase transition”

U. Miyamoto Non-Uniform Charged BS 13

The “optimal” gauge & “proof” of GMCThe “optimal” gauge & “proof” of GMC

Master equation

[Kol ’06,’06]

**

Non-existence of GL mode for

U. Miyamoto Non-Uniform Charged BS 14

Higher order: non-linear backreactionsHigher order: non-linear backreactions

D=6

D=14

D=10

NUBS is favored !!(2nd-order transition)

Entropy comparison btw NUBS & UBSin microcanonical ensemble (same (M, Q)) cf: D-dep.

U. Miyamoto Non-Uniform Charged BS 15

Other ensemblesOther ensembles

D=6 D=14

Canonical ensemble(same T & Q)

Grandcanonical ensemble(same T & ΦH)

3. Summary

U. Miyamoto Non-Uniform Charged BS 17

SummarySummary

• Final fate of GL instability is open.Final fate of GL instability is open.– It will depend on D (critical dim. in vacuum).It will depend on D (critical dim. in vacuum).– How about string/brane with charges?How about string/brane with charges?

• Static perturbations of magnetic strings (5<D<15).Static perturbations of magnetic strings (5<D<15).– Linear order:Linear order:

• Simplest(?) master equation & Simplest(?) master equation & no GL for Q>Qno GL for Q>QGMGM

• kkGLGL~|Q-Q~|Q-QGMGM||1/21/2 near Q=Q near Q=QGMGM

– Higher orders:Higher orders:• Critical charges appear: QCritical charges appear: QI,crI,cr , , QQII,crII,cr, , QQIII,crIII,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).

U. Miyamoto Non-Uniform Charged BS 18

Future prospects / To doFuture prospects / To do

• Understanding the critical charges, QUnderstanding the critical charges, Qn,crn,cr (n=I,II,III)(n=I,II,III)– Dilatonic branes with no GM point (NS5,D5..) (in progress) Dilatonic branes with no GM point (NS5,D5..) (in progress) – Fully non-linear deformation (in progress) Fully non-linear deformation (in progress) * * ****– Extension of Extension of Landau-Ginzburg argumentLandau-Ginzburg argument

• Dynamical (perturbative) stability of NUBSDynamical (perturbative) stability of NUBS• Dynamical evolutionDynamical evolution for Q for QI,crI,cr< Q < Q< Q < QII,crII,cr

– Charge (in 6D) would be easier than D=14.Charge (in 6D) would be easier than D=14.

2 control parameters

END