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

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Contents

• Motivation
• Surface Gravity
• Wormhole Temperature
• Hamilton-Jacobi Equation
• Summary

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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

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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)

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Wormhole temperature

• By Killing vector definition (Hong & Kim, 2006)
• Negative temperature

with exotic matter

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Trapping Horizon

• A sphere of radius r=(A/4π)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

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Surface gravity at trapping horizon (Hayward, 1998)

• Kodama vector: preferred flow of time
• Normal to sphere of symmetry
• Surface gravity
• Spherically symmetric metric

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Wormhole Case

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

Hamilton-Jacobi equation (1/3)

• With redefinition of v =t*+r* as

dt =C1/2dt*, dr = eφCdr*, C =1-2m/r

• Advanced Eddington-Finkelstein form (φ =Ψ)
• WKB approximation of the tunneling probability Г

is the imaginary part of action I

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Hamilton-Jacobi equation (2/3)

• In thermal form
• Action
• Energy and momentum
• Hamilton-Jacobi equation
• r
• Solution for outgoing mode
• I has a pole

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Hamilton-Jacobi equation (3/3)

• For the case of (2m=8πr 3τ )
• The imaginary part of the action
• By comparing with the thermal form

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Summary

• Various definitions of the surface gravity
• Wormhole’s Hawking temperature
• Checking by Hamilton-Jacobi tunneling method, we

can consider the Hawking radiation

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