For Minkowski space with a weak gravitational field the metric takes the form ds2=(1+2ϕ)dt2−(1−2ϕ)(dx2+dy2+dz2), where ϕ is the Newtonian gravitational potential.
You can get the (1+2ϕ) factor in front of the dt2 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 (1−2ϕ) factor for the spatial part of the metric by a similar procedure? What about some clever thought experiment?
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