Can we calculate the speed and time period of Earth's revolution around the Sun using general relativity? From General relativity we know that the sun's mass causes the spacetime to dip, so can we theoretically calculate the exact the time period and speed of earth's revolution around the sun using a mathematical model of curved spacetime and known masses of earth and sun? 
 A: In theory one certainly can calculate the orbital parameter's of Earth around the Sun using General Relativity. The framework provides a mathematically consistent way to describe how the Earth orbits the Sun. Practically speaking though, it's much more difficult to work with general relativity than it is with Newtonian mechanics. So what we usually do is something like use Newtonian mechanics to get a rough idea of the orbits, and then we include some perturbations from General Relativity to get corrections to those orbits to higher and higher accuracy.
Take as a concrete example, the 2 body problem. We consider the Sun and the Earth as 2 bodies which mutually attract each other. This is one approximation step better than the 1 body problem where we make the simplification that the Sun is so much more massive than the Earth that we treat the Earth as a negligible mass. In Newtonian mechanics, the 2 body problem is exactly solvable. We find that the 2 bodies orbit each other in conic sections (closed orbits = ellipses) and we can get exact parametric solutions for the orbits of both bodies. In general relativity this 2 body problem has no known analytical solution. This doesn't mean a solution doesn't exist in general relativity, only that the problem is so complex and intractable that we haven't been able to analytically solve it. So if we want to use GR to solve this problem, we can really only do so perturbatively. 
A: Yes.  One simply takes two independent orbital parameters of Earth's orbit (say, the distance to the sun, and the speed relative to the sun), and then uses them as initial conditions for Earth's geodesic equation, and solves the latter.  It's the same procedure one would use for Newtonian gravity, just a different set of equations.
