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2.9: S10. A Newtonian Homogeneous Expanding Universe - SOLUTIONS

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Nov 8, 2022
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- 1.37: Redshifts
- Back Matter

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( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

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Exercise 10.1.1

Answer: 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.)

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Exercise 10.1.1

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