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

More thoughts on this:

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

I've found the answer in Schutz. When deriving the metric he says
 For fields that change only because the sources move with velocity v, we have that $\partial / \partial t$ is of the same order as $v \partial / \partial x$
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 $\partial / \partial t$ is of the same order as $v \partial / \partial x$?

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