 That is, space-time is a structure whose automorphisms—the Galilean transformations that relate one inertial frame to another—are affine transformations: they take straight lines into straight lines, and parallel lines into parallel lines. The former condition implies that an inertial motion in one frame will be an inertial motion in any other frame, and likewise for an accelerating or rotational motion. The latter implies that uniformly-moving particles or observers who are relatively at rest in one frame will also be relatively at rest in another. An inertial frame can be characterized as a family of parallel straight lines “filling” space-time, representing the possible trajectories of a family of free particles that are relatively at rest. See Figure 5. Therefore, to assert that an inertial frame exists is to impose a global structure on space-time; it is equivalent to the assertion that space-time is an affine space that is flat. 