smoothoperator said:
I'm having a difficulty in defining what a static spacetime is
The technical definition is that a static spacetime has a timelike Killing vector field ("static" means the field is also hypersurface orthogonal, but I don't think we need to go into that part here). Now I'll translate that into English. ;)
If a spacetime has a timelike Killing vector field, physically this means that there is some family of worldlines in this spacetime along each of which the metric doesn't change. We can therefore use these worldlines to define "points in space", such that each point in space has an unchanging spacetime geometry in its vicinity. In principle we can then have a family of observers in this spacetime each of whom always sees the same spacetime geometry (each observer follows one of the worldlines in the family), so we can think of each such observer as "standing still" at a particular point in space.
Note that the spacetime geometry can still change if we move from one point in space to another. See below.
smoothoperator said:
if objects with motion curve space time in a way that they change the curvature along their path, does this also imply that free-falling objecs also curve space time in the same way?
You appear to be making an incorrect distinction between "objects with motion" and "free-falling objects". You also are confusing "test objects", the objects that follow particular worldlines in a given spacetime, from objects that act as "sources" of spacetime curvature; the two are different.
Let me give a concrete example to illustrate what I mean with the first point (this will also illustrate what "moving from one point in space to another" means in what I said above): Schwarzschild spacetime, the solution of the Einstein Field Equation that describes the vacuum spacetime around a spherically symmetric gravitating object, is static. That means there is a family of worldlines along each of which the metric doesn't change. An observer following one of these worldlines will be "hovering" at a constant altitude above the gravitating object, and will not be revolving about the object at all. This observer is static, i.e., at the same point in space for all time. Note that such an observer is
not freely falling; he must either use rockets to maintain altitude, or be standing on something (like a platform, or indeed the surface of the gravitating body), and in either case he will have nonzero proper acceleration and will feel weight.
Any other observer in this spacetime is "moving" from one point in space to another. However, although any freely falling observer must be moving (because, as above, observers who are static cannot be freely falling), not all moving observers will be freely falling. An observer who uses rockets or stairs or a ladder to climb from one altitude to another is not freely falling, but is moving (not staying that the same point in space). An observer who jumps off a high platform, or out of his rocket, and falls toward the gravitating body is both moving and freely falling, of course. All of these moving observers see a changing spacetime geometry as they move; but note that they themselves do not cause the changes, they just observe them (see below).
(Note also that an observer can be freely falling without changing altitude, if he is in a circular orbit about the gravitating body. In this particular case, a moving observer will not see any change in spacetime geometry. That is because this particular spacetime has an additional symmetry, spherical symmetry, over and above being static. A static spacetime does not have to have any additional symmetry, although I don't have a handy simple example of one that doesn't.)
Now for the second point: In the example above, the spherically symmetric gravitating object is the "source". The other objects (i.e., all the different observers I described) are all test objects and are assumed to have a mass that is so small, compared to the mass of the source, that they don't affect the spacetime geometry. So test objects, whether they are freely falling or not, don't curve spacetime (more precisely, they don't curve it enough to be significant in the scenario under discussion).