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Unformatted text preview: Physics 523, Introduction to Relativity Homework 5 Due Thursday, 17 th November 2011 Hans Bantilan TOV Star in 2+1 Dimensions Consider a static circularly symmetric spacetime in 2+1 dimensions, so that the metric takes the form g ij dx i dx j =- exp (2 α ) dt 2 + exp (2 β ) dr 2 + r 2 dθ 2 where α = α ( r ) and β = β ( r ), and include perfect fluid matter 1 with energy-momentum tensor compo- nents T ij = ( ρ + P ) U i U j + Pg ij . a. We are to write down the TOV equation in 2+1 dimensions. We begin by computing the Einstein tensor, which gives us G ij dx i dx j = 1 r exp (2 α- 2 β ) dβ dr dt 2 + 1 r dα dr dr 2 + r 2 dα dr 2- dα dr dβ dr + d 2 α dr 2 ! exp (- 2 β ) dθ 2 . This calculation, along with the perfect fluid energy momentum tensor T ij dx i dx j = ρ exp (2 α ) dt 2 + P exp (2 β ) dr 2 + r 2 Pdθ 2 implies that tt and rr components of the Einstein equations G ij = 8 πT ij read, dm dr = 2 πrρ ( r ) and dα dr = 8 πrP ( r ) 1- 8 m ( r ) respectively, where m ( r ) = (1- exp (- 2 β )) / 8 is such that exp (2 β ) = 1 / (1- 8 m ( r )). Instead of in- specting the θθ component, we equivalently apply the second Bianchi identity ∇ R = 0 to the Einstein equations, giving = ( ∇ i T ) ij = ( ∇ i...
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This note was uploaded on 02/02/2012 for the course PHY 523 at Princeton.