Newtonian limit of covariant derivative of stress-energy tensor(schutz ch7)

Mmmm

More thoughts on this:

If [itex]\phi[/itex] is a funtion of t, then [itex]g_{\alpha \beta}[/itex] is a function of t and so [itex]p_t =energy[/itex] is not conserved.
do we need energy conserved in newtonian limit and so [itex]\phi[/itex] independent of t?

Mmmm

I've found the answer in Schutz. When deriving the metric he says
This means for a stress-energy tensor independent of t, the t partial derivatives have v to 1st order. so v multiplied by a t derivative is second order and so very small and a second t derivative is also of second order and so very small.

This sorts out the problem

but why is [itex]\partial / \partial t [/itex] is of the same order as [itex]v \partial / \partial x [/itex]?

I can see that [itex]d / dt = v^\alpha \partial / \partial x^\alpha [/itex]
Is it because of this?