Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 15E / i / Solution

Let a comoving test particle have physical radius $R=a(t)\chi$, where $\chi$ is fixed. Assume Newtonian gravity, spherical homogeneity, negligible pressure, and no mass flux through this sphere, so its enclosed mass $M=(4\pi /3)\rho R^3$ is constant. The shell theorem gives a conserved specific energySubstitute $R=a\chi$, divide by $R^2/2$, and write $2\mathcal{E}/{\chi }^2=-k{c}^2$ to obtainHomogeneity makes the energy scale as ${\chi }^2$, leaving the same $k$ at every comoving radius. The sign of $k$ encodes the binding-energy sign. This Newtonian derivation applies to pressure-free matter; interpreting $k$ as spatial curvature and extending the equation to relativistic pressure components requires general relativity, rather than conservation of this Newtonian shell energy alone.