Force of gravity in relativity

[SUROJL]

How can we calculate force of gravity between two objects using Einstein's relativity theory. Using Newton's method we can easily find attractive force between two objects by using formula

f=g*m1*m2/r^2

But how can we calculate the force using Einstein's equation?

 

[elfmotat]

Well, gravity isn't really a force in GR. What you do to figure out how things are going to move is this:

1. You figure out the energy-momentum distribution (i.e. a single spherical mass, a cloud of dust, etc.)
2. You solve the Einstein Field Equations for the metric tensor.
3. Assuming objects placed in this region of space are too small to change the spacetime very much, you use the geodesic equation (which you get from the metric) to calculate how they are going to move.

 

[pervect]

As elfmotat remarked, one generally doesn't calculate forces in General Relativity. It is possible, however, to calculate the reading on an accelerometer of an object that you hold "stationary" - using clocks and rulers local to said object. (You can use other clocks and rulers if you really want to, but the results are generally not very physically meaningful.)

Hopefully it's obvious why this is very similar to a force? It's rather like taking the reading of a scale for the "force" needed to hold the object stationary.

While in general it's hard to define what "hold stationary" really means, it's pretty easy in cases where you have a stationary metric - which more or less means in cases where none of the bodies is moving relative to any of the others.

If the reason you're interesting in calculating the force is to determine the equations of motion, it's worth nothing that the usual approach to calculating the equations of motion doesn't involve calculating forces or applying the Newtonian equation F=ma. Instead, one writes down the so-called "geodesic equations", which basically say that bodies move along extremal paths in curved space-time. "Extremal" paths are pretty much the shortest paths, though there are some subtle differences.

Another way of describing this is the principle of extremal aging (sometimes called maximal aging).

I hope this helps - I'm not sure of your background, so I tried to keep the answer very non-technical.