- Published 2005

1.a The rate of change of a vector v flowing along the integral curves of u is given by v̇ = u∇bv = v∇bu = Bbv = 1 3θv + σbv + ωbv for Bb := ∇bu where we have used the fact that the vector fields u and v commute. It follows from this that the length of v changes by d dλ |v| = 1 |v|(3θv + σabvv) = 3θ|v|+ σabvv/|v|. Now suppose that v is one of a set of three spacial vector fields spanning the t = constant sections in a Robertson-Walker spacetime, i. e. the co-moving frame. By isotropy, the shear for the flow of this vector must vanish, for if it did not, a round sphere at t = t0 would evolve into a squashed sphere for t > t0 which is certainly not invariant under rotations about its center. Hence d dλ |v| = 3θ|v|. Now the metric for a Robertson-Walker spacetime can be put in the form

@inproceedings{Linch2005Homework3W,
title={Homework 3 William D . Linch},
author={William D. Linch},
year={2005}
}