Deriving force of gravity from energy-momentum tensor

Question

Lets say there is a mass $m_{1}$ which produces the energy momentum-tensor $T^{1}_{\mu\nu}$ at some point in space and there is a mass $m_{2}$ which produces the energy momentum-tensor $T^{2}_{\mu\nu}$ at some other point in space. How does one derive the force of gravity just by using these two energy-momentum tensors ($T^{1}_{\mu\nu}$ and $T^{2}_{\mu\nu}$)?

Answer 1

There isn't a simple relationship between the stress-energy tensors and the gravitational acceleration. Well, there is in the Newtonian limit but that's just Newton's law of gravity.

Starting from the stress-energy tensor we have to solve the Einstein field equations to get the spacetime geometry. If we then pick a set of coordinates we can calculate the Christoffel symbols and from these we can calculate the proper acceleration for an observer stationary in our coordinates. That gives us the gravitational acceleration i.e. the force per unit mass experienced by the stationary observer.

If you're interested I explain how to calculate the gravitational acceleration in How does "curved space" explain gravitational attraction? This is a bit math heavy but you can ignore the details and the general approach should still be clear.

The problem is that except for a few special cases where symmetry helps us, we cannot solve the Einstein equations analytically. For any general stress-energy tensor there isn't a nice simple spacetime geometry. Even the case of two spherically symmetric masses that you describe can't be solved analytically. If it could then describing black hole mergers would be a lot simpler than it is.