Hawking radiation as tunneling from squashed Kaluza-Klein BH Ken Matsuno and Koichiro Umetsu (Osaka city university) (Kyoto sangyo university) Phys. Rev. D 83, (2011) 1

I. Introduction String theory, brane world models: Studies of higher-dimensional black hole solutions Construction of more and more general solutions (charges, horizon topologies, asymptotic structures,...) Physical properties of solutions (geodesics, stabilities, thermodynamics, Hawking radiation,...) Applying to our universe (mini-BHs, geodetic precession, gravitational lensing,...) We consider Hawking radiation from 5D squashed Kaluza-Klein BH on the basis of tunneling mechanism 2

Derivation of Hawking radiation on the basis of tunneling mechanism (Srinivasan & Padmanabhan) Forming of a particle-antiparticle pair close to horizon inside black hole 3 ingoing mode: trapped inside horizon outgoing mode: escape from BH by tunneling effect ⇔ radiation from black hole Black hole outgoing mode ingoing mode PP Horizon Our universe

Derivation of Hawking temperature (Parikh & Wilczek) WKB tunneling probability amplitude of outgoing particle escapes from black hole for classically forbidden trajectory 4 ( ω : energy of particle, κ : surface gravity on black hole horizon ) Comparing Γ with Boltzmann factor in thermal equilibrium state at temperature T : Using this method, we can evaluate backreaction of radiation : Hawking temperature

II. 5D charged static Kaluza-Klein black hole with squashed S 3 horizons 5 M : mass of black hole, Q : electric charge : parameter regions

Horizon and asymptotic structures Squashed S 3 horizons in the form of Hopf bundle 6 Surface gravity on outer horizon of black hole At infinity (ρ → ∞) A twisted S 1 fiber bundle over 4D Minkowski spacetime ( r ∞ : size of compactified extra dimension at infinity ) : outer horizon : inner horizon

III. Dimensional reduction near BH horizon 5D complex scalar field Φ in squashed Kaluza-Klein BH background 7 Partial wave decomposition of Φ: S lmλ : spin-weighted spherical function : orthonormal condition

8 Defining tortoise coordinate: Near black hole horizon region (ρ ≃ ρ + ): F(ρ + )=0 ⇒ we can ignore S int term and retain dominant kinetic term

2D effective field near black hole horizon By using technique of dimensional reduction near BH horizon, 5D squashed Kaluza-Klein BH metric effectively becomes 2D metric 9 [ 2D Einstein-Maxwell-dilaton system ]

IV. Hawking radiation as tunneling process  2D effective metric as background metric near black hole horizon 10 Tunneling mechanism: quantum effect near black hole horizon ⇒ derivation of Hawking radiation as tunneling process by using technique of dimensional reduction is justified To describe across-horizon phenomena, defining Painlevé-like coordinate:  outgoing radial null geodesic

We can ignore mass of matter field near black hole horizon by discussion of dimensional reduction ⇒ derivation of Hawking radiation by using null geodesic is justified Restricting ourselves to uncharged radiations from BH and taking into account backreaction of radiation ( ω : energy of particle escapes from BH by tunneling mechanism ) 11 ω M-ωM-ω M Before radiationAfter radiation

WKB Tunneling probability amplitude for classically forbidden trajectory: Γ ≃ exp(-2ImS) Imaginary part of action for outgoing positive energy particle from ρ in to ρ out 12

Hawking temperature and backreaction WKB tunneling probability amplitude 13 Considering effect in 2 nd order of ω 1 st order in ω : Hawking temperature of squashed Kaluza-Klein BH 2 nd order in ω : correction by backreaction of radiation

ρ 0 =0 limit WKB tunneling probability amplitude 14 2D effective metric near black hole horizon 1 st order in ω : Hawking temperature of 4D Reissner-Nordström BH 2 nd order in ω : correction by backreaction of radiation ( coincides with that by Parikh & Wilczek )

Summary We discuss Hawking radiation from 5D charged static Kaluza-Klein black hole with squashed horizons on the basis of tunneling mechanism We use 2D effective metric obtained by dimensional reduction near black hole horizon as background metric We obtain desired result of Hawking temperature effect of backreaction associated with radiation 15