It is well-known that, according to special relativity, there is an absolute “speed limit” on objects traveling in space-time: nothing can travel faster than light. It turns out that an object’s acceleration is also limited by the geometry of space-time, but in a more complex manner. For objects viewed as points (negligible spatial extent), special relativity imposes no particular constraints on the magnitude of their acceleration. For objects that have spatial extent, however, it turns out that the geometry of space-time does impose limits. The case we are considering here is what has been defined as “rigid motion” (Born [1], Franklin [2]). This is motion in which an object’s speed is changed in such a way that it is neither stretched nor compressed. All of our discussion is limited to a single spatial dimension plus time (a moving rod). We assume that acceleration is applied all along the rod’s length with no assumptions required about its rigidity. Nor do we include such dynamic physical effects as momentum or elasticity. It turns out that speed changes cannot be uniform along the length of the rod if it is to remain in rigid motion. Franklin [2] derived a formula relating the required accelerations of various points along the rod. His derivation was for the special case in which acceleration is constant over time. Here we show that Franklin’s key formula (Equation 14 in [2]) applies to acceleration that is non-constant as well. Franklin’s formula reveals an interesting property of space-time: If the rod’s acceleration exceeds a fixed, finite bound the rod must experience distortion -- stretching or compressing in the direction of the acceleration. Furthermore, if a rod is accelerated at this bound, in order to maintain rigid motion, its trailing end must accelerate instantaneously (infinite acceleration), while its leading end accelerates at a finite constant rate. The rod’s trailing end will acquire its new speed in zero time, while the leading end takes a finite time. That is, the leading end ages, during this acceleration, over the trailing end.