Einstein field equations seem very nonlinear of second order derivative. Expressing LHS of Einstein field equations in purely metric tensor and derivatives, it consists somewhat of second-order derivatives of metric tensor. RHS of course contains stress-energy tensor.
This seems to mean that when calculating metric tensor at some spacetime, it depends on future metric tensor (at that spacetime point and surrounding points) as well.
But geodesic equation needs metric tensor to derive the least action path for matter fields/particles.
The first question: So when calculating metric tensor, are Einstein field equations and geodesic equations coupled together for calculation?
The second question: Suppose we wish to compute metric tensor at some given spacetime. Only the form of stress-energy tensor is given. Do we need entire form of stress-energy tensor at every spacetime to compute metric tensor at some given spacetime, or can we localize stress-energy tensor and use only that information?