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hw3 - HW#3 Phys776Spring2005 Due Tuesday March 8 before...

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HW#3 —Phys776—Spring2005 Prof. Ted Jacobson Due Tuesday, March 8, before class. Room 4115, (301)405-6020 [email protected] 1. Raychaudhuri equation & cosmology The Raychaudhuri equation for a timelike geodesic congruence parameterized by proper time t is dt = - 1 3 θ 2 - σ ab σ ab + ω ab ω ab - R ab u a u b , (1) where θ , σ ab , and ω ab are the expansion, shear, and twist, respectively, and u a is the 4-velocity. Consider a a homogeneous isotropic universe (i.e. a Robertson-Walker spacetime, possibly with spatial curvature) with scale factor a ( t ), comoving energy density ρ , and pressure p . (a) Use the Raychaudhuri equation for the cosmological geodesic flow ( u a orthogonal to the homogeneous isotropic surfaces) to derive the second order Friedmann equation, ¨ a/a = - (4 π/ 3)( ρ + 3 p ). Use the Einstein equation only to evaluate R ab u a u b in terms of ρ and p . [Note that a positive pressure adds to the attraction of energy density, while a negative pressure substracts. If p < - ρ/ 3 the universe is accelerating. Vacuum
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