On Einstein's Newtonian Limit Calculation
Updated: Jan 30, 2019
In Chapter 7 of my Relativity without Spacetime, Section 126.96.36.199 (p. 157), I observe that for "any Newtonian weak field reduction" of Einstein's gravitational field equation, whether vacuum or non-vacuum, all metric components other than g44 drop out. Of course, for a weak/static field reduction in general, the g11, g22, and g33 metric components do not drop out. In his 1916 paper Einstein makes an additional stipulation, though, which accords with what he calls the "second point of view of approximation": that in the equation of geodesic motion for the Newtonian limit we are dealing solely with velocities small as compared with the velocity of light. With this stipulation in force, for a first approximation we need consider the metric component g44 alone.
My overarching point in this discussion remains the same: If the general relativistic equation of motion reduces to the Newtonian for a weak field (assuming Einstein's other stipulations), then we know that the inverse-square relation holds; and from that result we can recover the non-vacuum version (G44=kρ) of the field equation for the Newtonian limit. The stress-energy tensor in reality plays no role in Einstein's derivation and, indeed, we have no reason to regard mass density in the derivation as the "T44 component" of a stress-energy tensor. After all, if mass density is a gravitational source, it should not be preferentially paired with G44 as opposed to any of the other Gμν. What Einstein has really derived, then, is G44=0 at the Newtonian limit. That implies an inverse-square relation, and we know from Newtonian gravity that for an inverse square relation an interior field must be described by G44=kρ.
A good general discussion of the incoherence of the stress-energy tensor is Vishwakarma, "On the Relativistic Formulation of Matter." Astrophysics and Space Science 340: 373-379.