Hawking Radiation around the Wormhole Sung-Won Kim (Ewha Womans - PowerPoint PPT Presentation

Hawking Radiation around the Wormhole Sung-Won Kim (Ewha Womans University) In collaboration with S. Hayward APS2012 - Yukawa 1

Contents • Motivation • Surface Gravity • Wormhole Temperature • Hamilton-Jacobi Equation • Summary APS2012 - Yukawa 2

Motivation • Recent active researches on dynamical horizon • Morris-Thorne wormhole throat is a double trapping horizon • Potential form around the throat is similar to that of event horizon • Temperature of wormhole issue is revisited APS2012 - Yukawa 3

Surface Gravity • Several Definitions – Killing vector definition (Wald) – Acceleration (Abreu & Visser) – 2D Expansion (Jacobson & Parentani) – Minimality condition (Hayward) • Recent review (Nielson & Yoon; Pielahn, Kunstatter, & Nielson) APS2012 - Yukawa 4

Wormhole temperature • By Killing vector definition (Hong & Kim, 2006) • Negative temperature with exotic matter APS2012 - Yukawa 5

Trapping Horizon • A sphere of radius r =( A /4 π ) 1/2 is – Untrapped for spatial g -1 ( dr ) – Marginal for null g -1 ( dr ) – Trapped for temporal g -1 ( dr ) • Trapping horizon: A hypersurface foliated by marginal spheres APS2012 - Yukawa 6

Surface gravity at trapping horizon (Hayward, 1998) • Kodama vector: preferred flow of time • Normal to sphere of symmetry • Surface gravity • Spherically symmetric metric APS2012 - Yukawa 7

Wormhole Case • Morris-Thorne wormhole: • Regge-Wheeler tortoise coordinate form • Surface gravity • By Einstein’s equation (2 m ≠8 πr 3 τ ) • Flare-out condition APS2012 - Yukawa 8

Hamilton-Jacobi equation (1/3) • With redefinition of v = t * + r * as dt = C 1/2 dt * , dr = e φ Cdr * , C =1-2 m / r • Advanced Eddington-Finkelstein form ( φ =Ψ) • WKB approximation of the tunneling probability Г is the imaginary part of action I APS2012 - Yukawa 9

Hamilton-Jacobi equation (2/3) • In thermal form • Action • Energy and momentum • Hamilton-Jacobi equation or • Solution for outgoing mode • I has a pole APS2012 - Yukawa 10

Hamilton-Jacobi equation (3/3) • For the case of (2 m =8 πr 3 τ ) • The imaginary part of the action • By comparing with the thermal form APS2012 - Yukawa 11

Summary • Various definitions of the surface gravity • Wormhole’s Hawking temperature • Checking by Hamilton-Jacobi tunneling method, we can consider the Hawking radiation APS2012 - Yukawa 12