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We define "world line" in Minkowski Space to be a curve whose velocity is time-like. Then we define "proper time" to be its arc length (with respect to the metric). The given question is

"why is proper time between two events of the curve the time between these events as measured by an observer moving along that curve?".

The answer found in some books is that you can approximate the time as measured by a non-inertial observer by the sum of n inertial ones and then take the limit as n approaches infinity.
The hypothesis required is
Statement (1): "an instantaneous change in velocity results in an instantaneous change in time at max proportional to the square of the change in velocity", so their sum tends to zero as n gets arbitrarily large. Other books state the Clock Hypothesis straight ahead, claiming that the answer to the given question is "yes" by experimental facts.
Another equivalent statement is Statement (2): "the rate of a clock does not depend on its acceleration but only on its instantaneous velocity".

Statement (2) and the Clock Hypothesis can be proven equivalent. Statement (1) implies the Clock Hypothesis for smooth world lines and the Clock Hypothesis implies a much stronger version of statement (1), that is that under the conditions of statement (1), the change in time is zero. What are equivalent statements resulting to the Clock Hypothesis? Is there a statement as fundamental as statement (1) but in the same time avoiding the introduction of discontinuities in proper time?

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  • $\begingroup$ I wonder if a separate axiom is really necessary here? How about looking at the evolution, with just the ordinary laws of special relativity, of, say, an harmonic oscillator attached to an accelerated box? A "clock" is not a magical device after all, its evolution can be computed (a mechanical clock would be easier to model than the full body of the observer to compute its "age", but in principle the latter could also be done...). $\endgroup$
    – Luzanne
    Commented May 11, 2017 at 12:58
  • $\begingroup$ Related/possible duplicate: physics.stackexchange.com/q/53334/50583 $\endgroup$
    – ACuriousMind ♦
    Commented May 11, 2017 at 13:07
  • $\begingroup$ The role of a "clock" in this question is that of the fundamental to relativity device that is attached to any observer so that time can be measured. $\endgroup$
    – T.T.
    Commented May 11, 2017 at 13:10
  • $\begingroup$ My point is that you don't need such a separate, abstract/fundamental notion of time. At the end of the day, physics is about predicting the result of actual measurements, and these can be entirely computed from the basic laws (of, say, relativity+EM+QM whenever you can neglect gravity & high-energy physics), no need for extra postulates. In particular, experimental time is what concrete clocks measure. $\endgroup$
    – Luzanne
    Commented May 11, 2017 at 13:36
  • $\begingroup$ Given a particle destined to "live" a given amount time when at rest, the Clock Hypothesis is necessary for any conclusions to be made on how long it will "live" in the system of the laboratory when it moves on some arbitrary world line. $\endgroup$
    – T.T.
    Commented May 11, 2017 at 13:54

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