PHYSICS 131 hw-7-soln

# PHYSICS 131 hw-7-soln - HOMEWORK SET 7, PHYS. 131...

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HOMEWORK SET 7, PHYS. 131 Instructor: Anthony Zee Email: [email protected] TA: Kevin Moore Email: [email protected] 12.4. Consider the spacetime specifed by the line element ds 2 = - ± 1 - M r ² 2 dt 2 + ± 1 - M r ² - 2 dr 2 + r 2 ( 2 + sin 2 θdφ 2 ) except For r = M , the coordinate t is always timelike and the coordinate r is spacelike. (a) ±ind a transFormation to new coordinates ( v, r, θ, φ ) analagous to (12.1) that sets g rr = 0 and shows that the geometry is not singular at r = M . Let t = f ( r, v ) while r, θ, and φ are unchanged. We are attempting to fnd a coordinate trans- Formation where the metric has g ± rr = 0, or 0= g μν ∂x μ ∂r ∂x ν ∂r = - ± 1 - M r ² 2 ± ∂t ∂r ² 2 + ± 1 - M r ² - 2 which leaves ∂t ∂r = ± 1 - M r ² - 2 Integrating, and choosing v as the constant oF integration, gives t = r + M 1 - r/M +2 M log( r - M )+ v and ds 2 = - ± 1 - M r ² 2 dv 2 +2 drdv + r 2 2 + r 2 sin 2 θdφ 2 which is maniFestly regular at r = M . (b) Sketch a ( ± t, r ) diagram analagous to ±igure 12.1 showing the world lines oF ingoing and outgoing light rays and light cones.

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## This note was uploaded on 07/15/2008 for the course PHYS 131 taught by Professor Kevin during the Spring '08 term at UCSB.

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PHYSICS 131 hw-7-soln - HOMEWORK SET 7, PHYS. 131...

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