The answer, of course, does indeed depend on what one means by “move around,” and in particular the comments refer to the notion of a “reference frame.” I can think of at least three different things one might mean by that phrase. First there is the idea of a “global reference frame.” The second notion is that of an “inertial reference frame.” Inertial frames are actually a subset of all possible global frames; in particular, they are the global frames in which free (unaccelerated) particles appear to move on straight lines. Finally, we have the more flexible notion of a “coordinate system.” Unlike a global frame or the even-more-restrictive inertial frame, a coordinate system can be set down throughout space in any old way, so long as it assigns unique coordinates to each point.