general relativity - How to get space component of weak field (linearized) metric?

For Minkowski space with a weak gravitational field the metric takes the form $$ ds^2 = (1+2\phi)dt^2 -(1-2\phi)(dx^2+dy^2+dz^2), $$ where $\phi$ is the Newtonian gravitational potential.

You can get the $(1+2\phi)$ factor in front of the $dt^2$ by starting with the geodesic equation and going to the "Newtonian limit" of slow speeds and a slowly changing field.

But is there a way to get the $(1-2\phi)$ factor for the spatial part of the metric by a similar procedure? What about some clever thought experiment?