Question on General Relativity

[eoghan]

Hi!
I'm reading General Relativity by Wald. In chpater 4.4a about Newtonian limit of linearized gravity, it says:
"When gravity is weak, the linear approximation to GR should be valid. The assumptions about the sources (relative motion << c and material stresses << mass-energy density) then can be formulated more precisely as follows: there exists a global inertial coordinate system of $\eta_{ab}$ such that:
$$T_{ab} \approx \rho t_a t_b$$
where $t_a=(\frac{\partial}{\partial x^0})_a$ is the "time direction" of this coordinate system."

I've always thought that
$$T_{ab} \approx \rho v_a v_b$$
where v is the velocity of the observer (or in other words, the relative velocity between the source and the observer). So, how can I say that
$$v_a=(\frac{\partial}{\partial x^0})_a$$
?

 

[haushofer]

That means that the components $v^{\mu}$ of the velocity vector are given by

$$v^{\mu} = \frac{dx^{\mu}}{d\tau} \approx (1,0,0,0)$$

This is the same as saying

$$\frac{dx^0}{d\tau} > > \frac{dx^i}{d\tau}$$

which is the non-relativistic limit.

 

[eoghan]

Thank you!