Time measurement with accelerating light-clocks

Abstract

The clock hypothesis in relativity states that the rate of time as measured by any clock is determined by its Minkowskian proper-time, regardless of the nature of its motion; in particular independent of its acceleration, depending only on its instantaneous velocity. However, a unique proper-time may be assigned to an accelerating clock, as to any physical system, only in the limit of being point-like. But clocks, by their very nature, must be spatially extended systems, to allow an internal periodical mechanism. Therefore the question, How does the internal structure of the clock affect the clock hypothesis?The simplest model to examine the clock hypothesis is the so-called ‘light clock’, consisting of two mirrors with a light signal reflected between them. So far, such examinations were carried out mainly in the limits of point-like clocks and/or constant acceleration. Here the clock hypothesis is theoretically examined for spatially extended linearly accelerated light-clocks, parallel and vertical relative to the direction of motion, with arbitrarily varying accelerations. Using the rapidity of the clock as its evolution parameter, a Lorentz covariant analysis is neatly performed. Taking into account the spatial extension of the clock, differences between externally measured Minkowskian proper-times and the time-scale determined by the internal periodical mechanism of the clock are computed.Although these differences are practically very minute – of order aL/c 2 for characteristic acceleration a and spatial dimension L of the clock – theoretically they cannot be ignored. They indicate inherent inconsistency between the externally measured proper-time – the physical time-line which consists of moments-of-time – and the intrinsically defined age – the internal time-line, consisting of durations, intervals-of-time.