2.9: S10. A Newtonian Homogeneous Expanding Universe - SOLUTIONS
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Exercise 10.1.1
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Since →x(t)=a(t)→xc with →xc a constant, we have →v=dx(t)/dt=˙a→xc=(˙a/a)→x which is Hubble's law with the proportionality between velocity and distance given by ˙a/a. For the second part, note that →∇=ˆx∂/∂x+ˆy∂/∂y+ˆz∂/∂z and →v=(˙a/a)(xˆx+yˆy+zˆz) so →∇⋅→v=(˙a/a)×(1+1+1)=3(˙a/a). (Here we've used e.g. ˆx as the unit vector in the x direction.)