According to general relativiy, why does earth revolve around the sun? Does the earth follow geodesics in spacetime caused by sun? I know according to classical theory, its centrifugal force.
Also, I need to assure that mass curves spacetime, not space.
Now, if mass curves spacetime, how can an apple which follows geodesic in curved spacetime (due to earth) fall straight onto the
ground while the moon which follows geodesic in curved spacetime (due to earth) go around the earth in an elliptic path.
 A: In good old Newtonian mechanics the equation of motion of an object is:
$$ \frac{d^2x}{dt^2} = \frac{F}{m} $$
where the left hand side of the equation is just the acceleration. In general relativity the analogous equation is called the geodesic equation and it's:
$$ {d^2 x^\mu \over d\tau^2} = - \Gamma^\mu_{\alpha\beta} {dx^\alpha \over d\tau} {dx^\beta \over d\tau} $$
The left hand side of this equation is the four-acceleration, just as in the Newtonian case the left hand side is the (coordinate) acceleration. The right hand side has two bits. The symbols $\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols and they describe the spacetime curvature. The $dx^\alpha/d\tau$ is the four velocity of the object whose path we are calculating.
The key point is that the four acceleration isn't just dependant on the spacetime curvature, it's also dependant on the four velocity. That means if I throw a ball fast the four-acceleration will be different to if I throw it slowly, and the two balls will travel different paths even though both paths are geodesics.
