"Proper" accelerations and rigid bodies. In Special Relativity, the term "proper" in front of a property means, as I understand it, the value of that property as measured by an observer with test equipment at rest relative to the object being measured. Thus "proper length" is the length determined by an observer in the same frame as the object and the "proper time" is the time according to a clock that is carried with the observer throughout the relevant period. However, in textbooks, articles and research papers in SR, I frequently come across the statement that "the "proper acceleration" of the rear end of an accelerated rigid rod must be greater than that of the front end." This is often illustrated by a Minkowski diagram showing two differently curved hyperbolae for the trajectories of the ends. This is very strange since it is clearly the accelerations of the rod ends estimated by an un-accelerated, inertial observer that are being represented - quite contrary to the established meaning of the term "proper". From consistency of meaning "proper acceleration" should refer to accelerations measured by co-moving observers. Since co-moving observers verify a constant "proper" length for the rod, and an observer at each end will record the same "proper" time, then by definition the "proper" acceleration must surely in fact be identical at each end of the rod, whether they are measured by inertial accelerometers or by reference to the background inertial frame.