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I can only perform the demonstration from the much simpler $E = mc^2$.

Take as given the Einstein field equation:

$G_{\mu\nu} = 8 \pi \, T_{\mu\nu}$

... can it be proved that Newton's formulation of gravitational and mechanical force (e.g. $F = ma$) corresponds to Einstein's in the limit when masses are small and speeds are relatively slow?

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Newton's gravitational theory is a weak-field, low-velocity limit of general relativity but the precise map between equations and limits is different than you think.

In general relativity, one must use both Einstein's equations as well as the condition that freely falling objects are moving along geodesics – time-like world lines that maximize the proper time on them, i.e. satisfy $$\delta\int {\rm d}t_{\rm proper}=0$$ and these two equations – Einstein's equations describing the gravitational field as created by the sources of gravity; and the geodesic equations describing how probes react to the gravitational field – may be used to derive $$ \frac{GMm}{r^2} = m\vec{\ddot x }$$ or similar classical equations (please add the unit vector to the left hand side above). Of course, when one does so, he should know the natural description of classical gravity in terms of the gravitational potential, the Poisson equation it obeys, and other things. In the Newtonian limit, $g_{00}$ component of the metric tensor largely depends on the gravitational potential $\Phi$ as $g_{00}=1+2\Phi/c^2$, as can be seen by simplifying Einstein's equations in the non-relativistic limit where they reduce primarily to the Poisson equation, and this influences geodesics etc.

One may also study the motion of non-freely-falling objects in general relativity and derive the equations for objects influenced by many forces although the formalism may look "advanced" in the language of general relativity and one recycles some concepts of classical physics, anyway.

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That freely falling objects move on geodesics follows from the field equations if one assumes a delta-function as a source and makes use of the Bianchi-identities. Thus, strictly speaking one must not use both. –  WIMP Jan 22 '12 at 7:40
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Good point, WIMP. Still, deriving mechanics out of field theory of solitons is a bit advanced way for a beginner who wants to see his or her old equations of mechanics... –  Luboš Motl Jan 23 '12 at 11:20

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