Exploring Rotating NS5-Brane Penrose Process & Ergoregion in Kerr Black Holes

1. Tachyon Induced Penrose Process around non-BPSRotating NS5-Brane T. Eguchi, Y. Nakayama, SJR, Y. Sugawara hep-th/0611mmm, hep-th/0611nnn

2. NS-5! Spin up and you’ll disappear!I-Robot

3. Rotating NS5-brane background 2-dim. BH ⊂ Black NS5-brane background  Exactly solvable toy model for Schwarzshild BH • “Rotating” NS5-brane background (obtainable from TsT transformam of NS5-branes)  Exactly solvable model for Kerr BH Questions • bulk vs. localized tachyon? Stability? • Ergoregion? If yes, Penrose Process? • Hawking radiation? • Black hole entropy? • (Dual) gauge theory interpretation?

4. Kerr geometry and Ergosphere • Kerr black hole (a: angular momentum per mass M) • Event horizon: • Killing horizon: Outer horizon Inner horizon Ergo region Inside the ergosphere, gtt>0  particle cannot remain stationary.

5. Penrose process • Kerr black hole metric • Killing vector is not time-like inside the ergo sphere. • Energy momentum conservation • Usually E1 , E2 > 0, but inside the ergo sphere, E1 can be negative! (since generates space-like momentum) By suitably prescribing “precisely timed breakup” of incident particles, one can extract energy from Kerr BH (Penrose process)

6. Penrose Process and Superradiation • At the same time, angular momentum is extracted • Second law of BH thermodynamics • Energy (mass) loss is compensated by angular momentum loss • Superradiant scattering • With particular energy and momenta, wave scattering off Kerr geometry shows superradiation: • This also occurs for horizonless rotating nonsingular geometry(Cardoso-Dias-Hovdebo-Myers)  origin of Hawking radiation in Mathur’s picture?

7. Rotating Extremal NS5-brane Backgrounds We will studytwo different classes for rotating NS5-branes (Singular) NS5 Nonextremal NS5 Singularity Yes No Horizon No Yes Ergo region Yes Yes Nonsingular CFT Yes Yes Spontaneous Yes No Penrose Process

8. Construction of singular NS5-brane 1~2. Begin with (cigar geometry) × (time) [manifestly nonsingular] 3. Perform TsT transform  deformed cigar background As , linear dilaton cylinder with radius 4. Combine it with SU(2)/U(1)

9. 5. Take T-dual along φ2:

10. Properties • limit: NS5-branes on a ring • ergoregion but noBH horizon Therefore, Nonzero ω Rotating NS5-brane

11. Ergoregion Rotating NS5-branes solution contain ergosphere • Take small ω limit. • In ρ,ω 0 limit, the metric looks like • Rotating around with the speed

12. Singularity and FZZ resolution • As in the non-rotating limit, the metric has singularity at • Just corresponding to the source of NS5-branes. • Trumpet-like singularity, so FZZ duality should work. Trumpet  N=2 Liouville Singularity is removed by localized tachyon condensation (tachyon censorship)!

13. Tachyon Censorship • Indeed, If one expands the U(1) radius in S3, • The would-be marginal (N=2 Liouville) interaction becomes massive: This is the FZZ dual description of rotating NS5-brane solution (see alsoItzaki-Kutasov-Seiberg).

14. Exact CFT for Rotating NS5-brane Not only the SUGRA solution, but also exact worldsheet CFT is obtainable by applyingTsT transformation. • Since T-duality does not change partition function, deformed cigar is equivalent to

15. The partition function of deformed cigar (formally) = cigar with level × free non-compact boson • Combineit with fermions and SU(2)/U(1) part:  partition function for rotating NS5-branes

16. Rotating Nonextremal NS5-brane Background • Begin with Lorentzian 2d BH × U(1) • Boost as • Combine it with SU(2)/U(1) and T-dualize back:

17. Properties • ergosphere exists at • blackhole horizonexists at • Obtainable from general Kerr solution in M-theory by reduction Nonzero ω Nonextremal NS5-branes Rotating solution

18. Penrose process Rotating NS5-branes solution is FZZ dual to linear dilaton with U(1) (radius R) and time theory together with superpotential. • Conformality condition • The system can be viewed as an example of generalized FZZ duality  explicit correlation functionsare known. • This underlying tachyon condensation will become a source for (spontaneous) Penrose process

19. 2-Point Correlator • By Lorentz boost, we obtain 2pt function for rotating extremal NS5-branes: • Nothing special:no superradiant scattering.

20. 3-Point Correlator • By Lorentz boost, we also obtain 3pt function for rotating extremal NS5-branes:

21. After Lorentz boost: • In every term, is conserved • Nonconservation of c“energy” is not conserved! • Part of energy is absorbed into rotating NS5-branes (Penrose process) • Localized tachyon condensation (FZZ duality) breaks time translation invariance  source of energy nonconservation • Energy extraction is quantized. • In N-pointcorrelator upto N-2 units of energy is violated

22. Physics • Inside ergoregion, exists singularity • Singularity covered by: -- localized tachyon for extremal NS5-brane -- BH horizon for nonextremal NS5-brane • Penrose process is entirely stringy • localized tachyon indispensible for Penrose process • Even after tachyon condensation, rotating NS5-brane has instability. • ENRS Conjecture: rotating extremal NS5  non-rotating nonextremal NS5

23. Conclusions • We constructedexactly solvable CFTs for rotating NS5-branes(extremal and nonextremal). • Exact to all order in α’: essential feature for understanding BH/string transition(NRS) • Duality between tachyon condensation and geometry plays a crucial role (Penrose process) • Singularity is removed by tachyon condensation (tachyon censorship)

24. Issues in progress • Singular rotating NS5-branes v.s. Non-extremal rotating NS5-branes – further relations • Effect of localized tachyon condensation in geometry involvingLorentzian 2D black hole • Construction of horizonless nonsingular supergravity solution (in contact with Mathur’s) • Black hole/String transition (across k=1) • Gauge theory interpretation? What is rotating NS5-branes? Penrose process? Ergo region?