The equation governing a FLRW universe is: $$\dot{\rho}+3H(\rho+p)=0$$ Where $\rho$ is the matter density, $H$ Hubble's constant and $p$ the pressure. The continuity equation derived from the divergence of the energy stress tensor for a perfect fluid is: $$u_a\nabla^a\rho+(\rho+p)\nabla^a u_a=0$$ As far as I know in the first equation the derivative is wrt to coordinate time and in the second equation because we are contracting the 4-velocity with the gradient the derivative in this equation is wrt to proper time. The two equations seem to look very similar, but are they actually the same? If not how can I convert the first equation into derivatives wrt to proper time. My guess is that one has to use the chain rule and apply it using the metric. But I am not sure how exactly to do that.

  • $\begingroup$ The first equation is the continuity equation for a perfect fluid in an FRW spacetime. Simply use the gradient $\nabla^a = (\partial/c\partial t, -{\bf \nabla})$ where $t$ is the coordinate time to derive it from the divergence of the stress tensor. $\endgroup$ – bapowell Nov 6 '18 at 15:59

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