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The clock-hypothesis is the fundamental assumption in the theories of relativity that duration, measured by clocks, is proportionate to the length of their respective world lines. Over the years, there have been contributions both, theoretical and experimental in nature, either confirming or questioning this hypothesis. We give an elementary model of two classes of clocks, which turn out to be relativistic clocks, and by doing so also offer a basis to see the limitations of the clock-hypothesis. At the same time, we find support for a hypothesis of L. de Broglie, regarding the existence of an internal clock of electrons. Our aim is to give a precise, yet accessible account of the subject.

In his seminal work of 1905 [^{1}; an assumption, which is known today as the clock-hypothesis. This hypothesis underlies much of the geometric structure of the theories of relativity [

In this short note, we will use a simple model, using elementary tools only, to describe two classes of clocks: moving particle-based clocks, especially electrons, and atomic clocks. We will show that they are indeed relativistic clocks and will rediscover a hypothesis by L. de Broglie [

Let’s first consider a quantum system represented by a wave function

By the time-energy inequality and a result in [

In (2) ^{2}. By (2) it produces in an incremental time-step

To describe the first class of clocks based on (3) we think of the system

The energy E represents kinetic and inner energy, defining the time-component of the four momentum

The expression

There holds along any (time-like) world line

For electrons, satisfying

Therefore, by integration, we get for any distance

Equation (8) confirms that electrons can indeed act as relativistic clocks, ticking with an internal frequency of a multiple of

a result, which fits well in the discussion in e.g. [

^{3}The expectation value is taken with respect to the state

Another class of clocks consists of atomic devices, where ^{3}. By (2) and (3) we get in the local rest-frame

The most direct ansatz for a covariant formulation of (10) is to simply chose the length s of the world-line of the clock as the parameter. We get

If (11) is correct, then atomic clocks are indeed relativistic clocks as well. Let’s gather evidence for it and assume first that the clock (atom) is in transversal uniform motion relative to an observer at rest. We have

Therefore, we get for the frequency ν in the frame of the observer

where as usual

With

Hence

The effect of gravitational red-shift on atomic clocks (15) has indeed been observed as well [

Experiments with the two kinds of clocks support the clock-hypothesis and our elementary model gives an accessible theoretical framework to explain it. The basis was that we managed to establish a relation (3) between the internal evolution of a system and the length of the world line it is supposed to pass through. While, of course, we compromised in principle by choosing a semi-classical treatment, it seems that, compared to moving particles, atomic clocks are even less ideal devices though, since they are not point-like, and Equation (11) only holds neglecting any real clock components other than the photon-emission/absorption. There might be effects on the photon by other parts of the clocks [

It is not obvious to see how systems, which cannot be brought into a covariant form of type (3)

could support the clock-hypothesis. Special relativity is the result of a conceptual merger of (classical) particle dynamics with electrodynamics. It is interesting to notice, that the clock-hypothesis is indeed best confirmed by devices, which are based on either of the two pillars: particle motion or electromagnetism. Systems, whose internal evolution base on other forces, like thermal energy, seem more resistant to confirm the hypothesis [

More work will be done on the question of clocks in order to tackle the deeper issue, namely the one of the true nature of time.

Schlatter, A. (2016) A Brief Note on the Clock-Hypothesis. Jour- nal of Modern Physics, 7, 2098-2102. http://dx.doi.org/10.4236/jmp.2016.715183