2.9: S10. A Newtonian Homogeneous Expanding Universe - SOLUTIONS
Exercise 10.1.1
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Since \(\vec x(t) = a(t) \vec x_{\rm c}\) with \(\vec x_{\rm c} \) a constant, we have \(\vec v = dx(t)/dt = \dot a \vec x_{\rm c} = (\dot a/a) \vec x\) which is Hubble's law with the proportionality between velocity and distance given by \( \dot a/a \). For the second part, note that \( \vec \nabla = \hat x \partial/\partial x + \hat y \partial/\partial y + \hat z \partial/\partial z \) and \(\vec v = (\dot a/a)(x\hat x + y\hat y+ z\hat z)\) so \(\vec \nabla \cdot \vec v = (\dot a/a)\times (1+1+1) = 3 (\dot a/a)\). (Here we've used e.g. \( \hat x\) as the unit vector in the \(x\) direction.)