Here is the Newtonian equation used to determine the value of G using a torsion balance. It is taken from the Instruction manual of a Pasco torsion balance model AP-825.

where,

= torsion constant for the band

= angle through which the band is twisted

= distance between centers of two masses

= length of lever arm of pendulum bob crosspiece

My question: Can the Einstein field equations be used in a similar manner to determine the value of G? It does not have to be a torsion balance. It can be any method. The point is that the Einstein field equations are used instead of the Newtonian equations.