If one were to consider all the evidence for the supposed clock time dilation, it would fall into the general category of alterations in rates associated with the spontaneous state transitions between energy levels in matter. It has been dramatically demonstrated by the phenomena of radioactive decay where half-lives of basic particles are substantially altered when their relative motions are increased with respect to the laboratory. The results are in precise agreement with Einstein’s formula.1 So if the half-life of the particle type were assumed to be a standard unit of clock time, the only legitimate conclusion would seem to be that time is indeed dilated in such cases. The same basic numerical agreement is obtained with atomic clocks, like the cesium clocks cited by Will,2 which involve an atomic resonance between energy levels as a standard unit of time.

It has been argued elsewhere, however, that clocks (and measuring rods) of relatively moving observers need not (and could in fact only inconsistently) be culpable in the case of there being unique values of time and space measurements of the occurrence of events obtained by relatively moving observers that are related by the Lorentz transformation. So, if such a claim of inconsistency were true as the author conjectures, why do timed state transitions with well-defined half-lives and resonances exhibit increases in the value of this ‘standard unit of time’ parameter exactly as would be predicted if time dilation were the correct interpretation of the temporal Lorentz Transformation equation? In other words, in the face of such convincing experimental data that seem to confirm time dilation, how could one rationally still maintain that there is no such thing?

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To begin this discussion, let us consider how handy it would be if the most basic building blocks of nature were to carry clocks by which we can verify the interpretations of our theories. But do they? Whatever the nature of these ‘clocks’, they were most certainly not *designed* specifically to check our theories, so we must investigate the degree to which the temporal quantities produced agree with the specified characteristics of clocks as defined in special relativity. In other words, to what extent do measured decay rate data represent standard time units generated by an ideal clock? This will obviously involve the issue of what constitutes such an ideal clock. Let us consider this.

Invariance of the measured time interval duration of a periodic mechanism is key. Precise periodicity is exhibited on earth by gravitational pendulums, astronomically by Keplerian motions and statistically at microscopic levels of reality by ensembles of radioactively decaying particles and resonating atoms. Of systems that have been used as clocks, the measured time intervals associated with resonance frequencies of atoms exhibit the highest degree of invariance. On the other hand, radioactive particles are the easiest to accelerate to extreme velocities and so they have typically been the ‘clocks’ most frequently selected for relativistic experimentation.

When such accelerated radioactive particles are moving at a constant velocity (as in a collimated beam) relative to laboratory apparatus, the distribution of the distances traveled prior to decay provides an accurate assessment of their half-lives. Half-life being determined as,

where T_v is the measured half-life, < d > is the average distance traveled and v is the velocity of the particles. Notice that this is merely an empirical formula for measuring half-life, *not* a theoretical parametrical derivation for determining the value of a particular particle half-life *a priori*.

In the case of the pendulum and Keplerian motion we have some understanding of the mechanisms or ‘workings’ of the clocks so that a theoretical *a priori* prediction *can* be obtained for sizes of time intervals between successive cycles as functions of parameters pertinent to the construction of the clock. For example, to a high order approximation, the differences in the cycle time of a pendulum placed on the moon and an identical version on earth would not be attributable to differences in the scale of time for observers situated on the moon and on earth. This is because the difference can be traced directly to a parametrical difference between the descriptions of the two clocks, namely the ratio of the mass of the moon relative to that of the earth that determines the force pulling the pendulum back to its null position. So although we have a measurable difference, it would be ridiculous to assume that clock time itself should be altered by this situation because we know a great deal about the mechanism of the clock and how the situation would produce the measured effect without altering our conception of time.

In the case of atomic and subatomic ‘clocks’, therefore, before we can attribute measurable differences in behavior to one cause or another, we must know something concerning the mechanism of radioactive decay. One could not otherwise discriminate between the half-life of radioactive particles being altered by a time scale difference affected by the relative motion as predicted by Einstein or by the decay process proceeding at a different rate after the energy of the particle has been increased by its having been accelerated. The amount of this difference has of course been demonstrated to depend upon the eventual relative velocity in accordance with the proscribed functionality of the time dilation formula. However, that might either be a coincidentally identical functionality through the energy content parameter of operation or, as typically accepted, a presumed change in the scale of time itself by an opportunistic interpretation of the temporal Lorentz equation. For example, if it were to be conjectured that time proceeds more slowly on the moon in accordance with the ratio of the masses of the earth and moon, the results of the pendulum experiment would make that absurd hypothesis somewhat difficult to disprove just because all data in the cited experiments would seem to confirm it. One would be forced to demonstrate that the peculiar functionality of a pendulum, and not the nature of time itself, has determined that behavior. So we are forced to attempt an understanding of the possible mechanism of particle decay, acknowledging nonetheless that the details of such a mechanism has *not* currently been identified – nor even anticipated as of such extreme significance.

However quantum mechanics is based on experimental evidence of phenomena that fall into the category of energy dependent state transitions. There is a large body of data and an accepted theory that confirm that the likelihood of a system transitioning to a ‘lower’ energy state is in actual fact directly dependent on the difference in the energy content realized by the two states. This is especially true of the types of particles whose decay is assumed pertinent to oft cited time dilation measurements – *mu* mesons (muons) in particular. For example, Jackson3 states that:

“Since the rate of decay depends sensitively on the energy release, [difference between energy levels]…tightly bound negative *mu* mesons exhibit a considerably slower rate of decay than unbound ones…”

Now, it can be shown as a direct consequence of the Lorentz transformation (with no presumptions of scale differences in the units of measure) that the mass of a particle moving at velocity, v, relative to the observer increases with respect to its rest mass, m_o. This increase is given by the formula,

By the well-known related formula, the total energy of a particle is shown to be,

where Ev and mv indicate, respectively, the energy and mass of the particle when it is moving at the velocity v relative to the observer. So that in general we have the relation between the energy of a relatively stationary and a moving particle as follows:

Radioactive decay formulas are characterized by exponentially decreasing quantities as explicit functions of time: They have the form:

where N(t) is the number of particles which have not yet decayed after a time, t, if there were No particles originally. If k is large, decay is rapid; if it is small decay is slow. Since the half-life, To, of the particles can be determined by measuring the amount of time, t = To, required to reduce N(t) to one half its original value as follows:

We have from the exponential decay formula that:

This is merely an empirical formula, of course, for fitting the exponential decay formula to the actual decay distribution data. Refer to figure A.1 for an example of the data.

However, a derivation of the parameter k using heuristics is immediately forthcoming from quantum mechanical considerations of binding energy similar to what was used by Gamow, Condon and Gurney in deriving alpha particle emission rates in radioactive elements in 1928 that were in excellent agreement with the empirical data.4 Clearly, decay rate data is highly dependent on the binding energy of the specific types of particle employed in pertinent half-life experiments as indicated in Jackson’s comment above for muon decay. As stated, the higher the energy the less likely is decay. Let us posit, in particular, therefore, the following reasonable relationship:

Then we obtain from the expression for k above that:

Thus, we should expect that in general,

for a moving particle, from which it follows that:

At first glance, this might appear to be in complete agreement with Einstein’s prediction since the half-life for a stationary particle is predicted to be less than that for a moving particle in precisely that proportion. But it most certainly is *not* in agreement with that hypothesis! In this derivation, we have predicted that from *all perspectives* the particles will decay more slowly. Period. We have not attempted to take into account that the rate of ticking of some abstract ‘clock’, supposedly residing within the particle, might in some obscure sense have ticked off Tv seconds while laboratory clocks were ticking off To seconds. The times reflect merely the amount of time as measured on some (any) standard clock for decay to occur for unaccelerated particles and alternatively for accelerated ones.

In fact, we have just demonstrated the dependence of these amounts of time to be precisely like the analogy of how long it would take identical pendulums on the earth and on the moon to perform a complete swing. The difference in the generated time to decay has already been determined as dependent upon the functionality of matter transitioning from one energy state to another. Like in the analogy, the measured time interval whether observed directly on the moon or remotely via cameras back to earth, on the particle or in the laboratory, would agree that a pendulum swing or particle decay had occurred after the same time interval according to either of the clocks. Particle decay behavior is slower for accelerated particles *because* they possess more energy – period. It is in fact (in both analogous phenomena) the effect of a mass/energy difference that accounts for the difference in time interval – not on the scale of time chosen by two independent observers. When one ships a pendulum to the moon or accelerates muons to half the speed of light in a laboratory, their timing mechanisms are altered so that their operation is no longer commensurable with mechanisms that have not been so altered. And they can no longer be considered to represent a standard for time itself.

Suppose there is an argument put forward that, “Sure, but that just independently confirms the effect of relative motion on the operation of clocks.” According to that argument the value Tv would be the time read on a relatively moving clock on the particle However, if we were to additionally take into account the supposition that the scale of time is affected as suggested by Einstein (and virtually every physicist with any credentials on this subject since), we would obtain:

So the meaning of the temporal Lorentz transformation equation can not be that the scales of clocks must be transformed so as to compensate observed differences. The time intervals to corresponding events must *actually* differ according to that equation without the caveat, “It’s actually the same amount of time, but his clock is dilated.” The nature of that correspondence between transformed events now becomes the key issue of any viable theory of special relativity since the events that are being correlated by the Lorentz equations cannot be identical in any real sense without introducing inconsistency.

An explanation of why observed events correlate as they do for observers in relative motion has indeed eluded some of the finest minds for many decades. But one does not need an alternative explanation in order to reject inconsistent logic. That is the role of intelligence. It is, perhaps, a legitimate role of faith to allow one to survive periods without answers.

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4 Eisberg, pp. 238-239.

3 Jackson, p. 358.

1 Schwinger, pp. 55-58.

2 Will, pp. 54-57.

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Bibliography:

1. Robert Eisberg, *Fundamentals of Modern Physics*, John Wiley & Sons, 1961.

2. John Jackson, *Classical Electrodynamics*, John Wiley & Sons, 1962.

3. Julian Schwinger, *Einstein’s Legacy – The Unity of Space and Time*, Scientific American Books, Inc., 1986. 58.

4. Clifford Will, *Was Einstein Right? Putting General Relativity to the Test*, Basic Books, Inc., 1986.

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