Dynamics of Kerr Black Holes from CFT

Tom Hartman IAS

IPMU Workshop on Black Holes February 2011

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Applied Holography The Fermi Sea Conclusion • With AdS/CFT, string theory provides a detailed microscopic theory of certain black holes, assuming - Supersymmetry Extra dimensions Strominger, Vafa ’96 - Maldacena ’97 - Charge - Anti de Sitter space (AdS) - String theory 3d Thermal 4d Black Gauge Hole Theory Physics

• Can general lessons be extracted from string theory and applied to more realistic black holes, including 4d Kerr?

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Classical and Quantum Absorption & Superradiance The Fermi Sea Motivation Conclusion • I. Classical gravity - Membrane paradigm: modeling realistic black hole behavior as a thermal system/conductor. Damour, Thorne, Price, MacDonald, etc 1980s - We will “enhance” the membrane paradigm by showing that the thermal system living on the membrane is often a CFT. - Example: classical wave absorption. • II. Semiclassical gravity / QFT in curved space - Quantum fields propagating on a semiclassical black hole background = thermal QFT. - CFT often governs the low-energy dynamics. - Example: Fermions + ergosphere = non-Fermi liquid.

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Classical and Quantum Absorption & Superradiance The Fermi Sea Motivation Conclusion

• III. Effective theory of quantum gravity - Macroscopic black hole thermodynamics is universal: Area S = Entropy = 4GN T dS = dM ΩdJ H − etc. - String theory explanation is not universal; depends on many details and special cases. - There should be a microscopic, universal approach: the statistical mechanics of the membrane. - CFT provides a framework that is more universal, but still limited. • Example: Statistical derivation of the entropy of all extremal black holes.

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Outline The Fermi Sea Conclusion

I. Overview of the Kerr/CFT Correspondence Guica, TH, Song, Strominger 0809.4266 Hartman, Nishioka, Murata, Strominger 0811.4393 II. Application 1: Absorption & Superradiance Bredberg, TH, Song, Strominger 0907.3477 TH, Song, Strominger 0908.3909

III. Application 2: Fermi seas, ergospheres, and TH, Song, Strominger 0912.4265 collective phenomena TH, Hartnoll 1003.1918 • Superconductivity in curved space

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Caveat on “Realism” The Fermi Sea Conclusion

• We will consider two types of black holes: • 4d Kerr - Most results require large spin - Basic idea holds in an astrophysical setting, but quantum effects are observationally irrelevant • 4d Reissner-Nordstrom-AdS - Electric charge not realistic - However, ergosphere has similarities to Kerr

Friday, February 25, 2011 Part I: Overview of the Kerr/CFT Correspondence

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance The Kerr Spacetime The Fermi Sea Conclusion

near-horizon ergosphere flat space region

length log T ∼− H T 0 as J M 2 H → →

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Near Horizon Limit The Fermi Sea Conclusion Bardeen, Wagoner ’71 • Near-horizon Extreme Kerr Bardeen, Horowitz ’99 dr2 ds2 =2JΩ2 r2dt2 + + dθ2 + Λ2(dφ + rdt)2 − r2 ￿ ￿

Ω2, Λ2 = functions of θ

4 Isometries • r =0 U(1) rotating φ SL(2,R) acting on AdS r = 2 ∞

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Holography The Fermi Sea Conclusion

Guica, TH, Strominger ’08 • The Kerr/CFT Correspondence Quantum gravity on near-horizon extremal Kerr is equivalent to a 2d conformal field theory living on the boundary of the near horizon region.

Classical gravity on near-horizon has conformal symmetry.

near- horizon ergosphere flat space

horizon 2d CFT

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Asymptotic Conformal Absorption & Superradiance The Fermi Sea Symmetry Conclusion

• Infinite number of asymptotic conservation laws and conserved charges Qn methods of: Brown, Henneaux ’86 • Generalizations of the conserved angular momentum Q0 = J

• Obey the 2d conformal (Virasoro) algebra under Dirac brackets i Q ,Q =(m n)Q + J(m3 m)δ { m n} − m+n − m+n,0

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Symmetry Enhancement The Fermi Sea Conclusion

Extremal Kerr: 2 symmetries

Near horizon limit: -4 exact symmetries -infinite asymptotic symmetries

• Symmetry fixes density of states in CFT and can be used to derive the entropy-area law: stat-mech of extreme black holes. Area S = = S CFT 4 BH • Chiral: Full 2d CFTs have two copies of this algebra. cf. next talk

Friday, February 25, 2011 Part II: Absorption and Superradiance

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Absorption The Fermi Sea Conclusion • Goal: compute the absorption cross section of a scalar field against Kerr in two ways: 1. General relativity Starobinsky, Churilov / Teukolsky, Press / Chandrasekhar 2. 2d conformal field theory

• Scatter a scalar field, iωt+imφ Φ e− ∼ ω =frequency m = angular momentum

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Superradiance The Fermi Sea Conclusion • Penrose process extracts energy from rotating black hole • Wave version of the Penrose process: superradiance ω

CFT description. compare: talk by T. Hanada on high-E collisions talk by E. Barausse on the self-force

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Gravity computation I The Fermi Sea Conclusion • To compute absorption cross section, solve the scalar wave equation on the Kerr background ￿Φ =0 • Compare ingoing and outgoing fluxes to compute σabs • Separate contribution from near-horizon region: σ σnearσfar abs ∼ abs abs

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Gravity Computation II The Fermi Sea Conclusion • Final result: n n σnear = T 2β sinh L + R abs H 2T 2T ￿ L R ￿ 1 in 1 in Γ( + β L ) 2 Γ( + β + R ) 2 | 2 − 2πTL | | 2 2πTR |

β2 = numerical separation constant

nL = m (angular momentum of mode) n ω m Ω (freq. away from superradiant bound) R ∼ − H T T R ∼ H TL =1/2π (chemical potential for J)

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Dual Description The Fermi Sea Conclusion • Now, rederive this from the dual CFT

near- horizon ergosphere flat space

2d CFT Remove the black hole; horizon replace it with a membrane described by a CFT

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance CFT Computation The Fermi Sea Conclusion • Black hole language: scatter a field off the black hole, compute absorption (near-extremal, near-superradiant) • CFT language: perturb the CFT by some operator, compute transition rate H H + J CFT → CFT O • The transition rate is given by Fermi’s Golden Rule:

σnear = J 2 d2xeikx (0) (x) abs | | ￿O O ￿ ￿ • Recall infinite symmetries: correlator is fixed by conformal invariance!

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance CFT answer The Fermi Sea Conclusion • Result agrees with the gravity computation, n n σnear = T 2β sinh L + R abs H 2T 2T ￿ L R ￿ 1 in 1 in Γ( + β L ) 2 Γ( + β + R ) 2 | 2 − 2πTL | | 2 2πTR | • β , essentially the Carter constant, is a scaling dimension in the CFT • Comments: - CFT correlation function = gravity observable - purely classical on the gravity side, strongly interacting and quantum in CFT - near-superradiant, near-extremal; however, also works for non-extremal black holes under some extra assumptions Castro, Maloney, Strominger ’10

Friday, February 25, 2011 Part III: The Fermi Sea

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Why a Fermi Sea? The Fermi Sea Conclusion Charged: Liu, McGreevy, Vegh, 0903.2477 Rotating: TH, Song, Strominger 0912.4265 • Absorption of a massive particle has poles, Φout Φin →∞ • For a scalar, these are instabilities • For a fermion, these indicate a Fermi surface

Claim: Semiclassical fermion ground state has a filled Fermi Sea

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Black Hole Bomb The Fermi Sea Conclusion • The ergosphere makes Kerr unstable if the world has massive scalars. - σ abs < 0 for ω

Negative energy orbits horizon

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Why a Fermi Sea? The Fermi Sea Conclusion • Negative-energy scalars instability • Negative-energy fermions Fermi sea

Vef f

Fermi sea

Negative energy orbits horizon

• Caveat: This is the semiclassical (Hartle-Hawking) vacuum, but equilibration time 10 65 years ∼ Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Stability The Fermi Sea Conclusion • Physics near the Fermi surface: - Kerr Fermi sea: classically unstable - Reissner-Nordstrom Fermi sea: quantum instability toward Cooper pairing • BCS Superconductivity is a negative “mass-squared” for Cooper pairs: - Define Cooper pair Φ = ψCγ5ψ - Compute effective action≡ Φ G Φ V (Φ) m2 Φ2 eff ∼ eff G

- Poles in G at the Fermi surface dominate this integral

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Superconductivity The Fermi Sea Conclusion

G ψψ ψψ G • This G is related to the absorption rate considered earlier. Poles at the Fermi surface are controlled by the near-horizon region, and therefore by the dual CFT • Infrared divergence at low T Hawking drive effective mass-squared negative; this signals the formation of a charged condensate and BCS superconductivity ￿6

￿8

￿10 log TC ￿12

￿14

Cuts off when CFT ￿16

scaling dimension ￿18 ∆ < 1 qfermion

Friday, February 25, 2011 Introduction Kerr/CFT Correspondence Absorption & Superradiance Conclusions The Fermi Sea Conclusion

Conformal field theory near the event horizon controls certain black hole phenomena ω mΩ ∼ H • Application I: Absorption • Application II: Semiclassical and quantum fermions - Condensed matter in curved space

• Theoretical motivation: - “Statistical mechanics” of quantum gravity?

Friday, February 25, 2011