hw3 - HW#3 Phys776Spring2005 Prof. Ted Jacobson Due...

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Unformatted text preview: HW#3 Phys776Spring2005 Prof. Ted Jacobson Due Tuesday, March 8, before class. Room 4115, (301)405-6020 jacobson@physics.umd.edu 1. Raychaudhuri equation & cosmology The Raychaudhuri equation for a timelike geodesic congruence parameterized by proper time t is d dt =- 1 3 2- ab ab + ab ab- R ab u a u b , (1) where , ab , and 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 , 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 / 3)( + 3 p ). Use the Einstein equation only to evaluate R ab u a u b in terms of and p ....
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hw3 - HW#3 Phys776Spring2005 Prof. Ted Jacobson Due...

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