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ps5 - Physics 523 Introduction to Relativity Homework 5 Due...

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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 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 ) dr dt 2 + 1 r dr dr 2 + r 2 dr 2 - dr dr + d 2 α dr 2 exp ( - 2 β ) 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 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 0 = ( i T ) ij = ( i T ) ir = ∂T rr ∂r + ( Γ t tr T rr + Γ r rr T rr + Γ θ θr T rr ) + ( Γ r tt T t + Γ r rr T rr + Γ r
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