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Newtonian fluid derivation of the Raychaudhuri equation

For a homogeneous pressureless fluid in comoving coordinates, the Euler equation gives

\nabla \overset{ˉ}{Φ} = - a \overset{a}{¨} x,

while the Poisson equation gives

\nabla^{2} \overset{ˉ}{Φ} = \frac{4 π G}{c ^{2}} \overset{ρ}{ˉ} a^{2} .

Taking the divergence of the first relation and comparing them yields

\frac{a ¨}{a} = - \frac{4 π G}{3 c ^{2}} \overset{ρ}{ˉ} .

This local fluid derivation describes a homogeneous infinite universe without choosing a physical centre of expansion.

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