= Newtonian fluid derivation of the Raychaudhuri equation
{c}
For a homogeneous pressureless fluid in comoving coordinates, the <Euler equations for an inviscid fluid>[Euler equation] gives
$$
\nabla\bar\Phi=-a\ddot a\,\mathbf x,
$$
while the <Poisson equation> gives
$$
\nabla^2\bar\Phi=\frac{4\pi G}{c^2}\bar\rho a^2.
$$
Taking the <divergence> of the first relation and comparing them yields
$$
\frac{\ddot a}{a}=-\frac{4\pi G}{3c^2}\bar\rho.
$$
This local fluid derivation describes a homogeneous infinite universe without choosing a physical centre of expansion.
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