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Time of Occurrence

Time was analyzed the ancient Greek philosopher Epicurus who is pictured here.
Epicurus, 341~270 BCE.

Titus Lucretius Carus was a Roman poet best known for proclaiming the ideas of Epicurus who founded one of the most famous and influential philosophical schools of antiquity. His physics was taught at Athens around 300 BCE. According to Lucretius1Titus Lucretius Carus, De Rerum Natura, translated by R. E. Latham, page 40. Penguin Books 1951.

An icon indicating a quotation.... time by itself does not exist; but from things themselves there results a sense of what has already taken place, what is now going on and what is to ensue. It must not be claimed that anyone can sense time by itself apart from the movement of things or their restful immobility ...

The EthnoPhysics approach to understanding time follows this Epicurean recipe. First we consider “things themselves” by carefully defining particles. Then we characterize “the movement of things” from a particle’s momentum  p. Next “their restful immobility” is represented by the rest mass  m. And finally these numbers are combined to define the period  \widehat{\tau} as

\widehat{\tau} \equiv \dfrac{h}{\sqrt{  c^{2}p^{2} + m^{2}c^{4} \, \rule{0px}{9px} }}

Recall that   E \equiv \sqrt{ c^{2}p^{2} + m^{2}c^{4} \; } is the mechanical energy, and that Planck’s postulate asserts that E =h \nu where  \nu is the frequency. So the foregoing definition implies that

\widehat{\tau} = \dfrac{h}{E} = \dfrac{1}{\, \nu  \,}

Now let some particle P be characterized by a repetitive chain of events  \Psi = ( \mathsf{\Omega}_{1}, \mathsf{\Omega}_{2} \ldots \mathsf{\Omega}_{k} \ldots ) where each repetition  \mathsf{\Omega} is described by its period  \widehat{\tau}. The time of occurrence of event  \mathsf{\Omega}_{k} is defined as

\displaystyle t_{k} \equiv t_{0} + \epsilon_{t} \! \sum_{i=1}^{k} \widehat{\tau}_{i}

where  \epsilon_{t} notes the direction of time. The value of the original event  t_{0} is arbitrary. Let  \Psi be historically ordered, then  \epsilon_{t}=1 and the time of occurrence is given by a simple sum of periods. The period depends on a particle’s momentum, which in-turn depends on whatever frame of reference is used to describe the particle’s motion. So the time of occurrence is frame-dependent too. If  p = 0 then  t is called the proper time. Please notice that this quantity has been entirely established by a systematic description of sensation.

Time Dilation

Let P be represented by the historically-ordered chain-of-events

\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}^{\mathsf{P}}, \mathsf{\Omega}_{2}^{\mathsf{P}} \; \ldots \; \mathsf{\Omega}_{i}^{\mathsf{P}} \;  \ldots \; \mathsf{\Omega}_{f}^{\mathsf{P}} \; \ldots \; \right)

And let each event  \mathsf{\Omega} be described by its time of occurrence  t. Then consider a pair of events \mathsf{\Omega}_{i} and \mathsf{\Omega}_{f} where i < f. Since  \Psi is in historical order we call them the initial and final events of the pair. The elapsed time between these initial and final events is \Delta t \equiv t_{f} - t_{i}. If P is isolated and the frame of reference is inertial then the energy and momentum do not change from event to event along the chain  \Psi. For these conditions, the period  \widehat{\tau} is constant too. But the time coordinate is defined from a sum of periods, so the elapsed time can be written as \Delta t = \left( f-i \rule{0px}{9px} \right) \widehat{\tau}. We evaluate this quantity for a material particle that is considered to be a clock in motion. Let P be described by  E its mechanical energy and  m its mass. These are related by  E = \gamma m c^{2} where  \gamma is the Lorentz factor. Then the period is given by

Time dilation was developed by Albert Einstein pictured here in Vienna 1921.
Albert Einstein, 1879—1955.

\widehat{\tau} = \dfrac{h}{E} = \dfrac{h}{\gamma mc^{2}}

So in terms of the mass

\Delta t = \dfrac{h(f-i)}{\gamma mc^{2}}

If P is a clock, this is the elapsed time that it would indicate between events. For comparison, set \gamma = 1 to define

\Delta t^{\ast} \equiv \dfrac{h(f-i)}{mc^{2}}

This is the elapsed time that would be recorded if P was at rest, it is called the proper elapsed time. The two quantities are related as \Delta t^{\ast} = \gamma \Delta t. The Lorentz factor for a particle in motion is always greater than one, \gamma \ge  1. So a moving particle always experiences less elapsed time than a stationary particle, \Delta t \le \Delta t^{\ast}. This effect is called time dilation.

Measuring Elapsed Time

According to Albert Einstein time is what a clock tells.2To be more precise, this quotation has been translated by Robert W. Lawson as “… we understand by the time of an event the reading (position of the hands) of that one of these clocks which is in the immediate vicinity (in space) of the event. In this manner a time-value is associated with every event which is essentially capable of observation.” From Relativity, the Special and the General Theory, page 24. Published by Methuen & Company, London 1936. So here is a generic description of how to determine  \Delta t using a clock. This elapsed time between initial and final events depends on the frame of reference F which is represented by another chain of events

\Psi^{\mathsf{F}} = \left( \mathsf{\Omega}_{1}^{\mathsf{F}}, \mathsf{\Omega}_{2}^{\mathsf{F}} \; \ldots \; \mathsf{\Omega}_{j}^{\mathsf{F}} \;  \ldots \; \mathsf{\Omega}_{k}^{\mathsf{F}} \; \ldots \; \right)

Since F is employed as a reference frame, we assume that every report about P is accompanied by an observation of F, so that events of F and P can be associated in pairs like

\left\{ \, \mathsf{\Omega}_{i}^{\mathsf{P}} \; , \; \mathsf{\Omega}_{j}^{\mathsf{F}} \, \right\}

and

\left\{ \, \mathsf{\Omega}_{f}^{\mathsf{P}} \; , \; \mathsf{\Omega}_{k}^{\mathsf{F}} \, \right\}

To make a laboratory measurement of elapsed time first select some clock  \mathbf{\Theta} that is presumably part of the frame of reference F. Let this clock be calibrated so that its period \widehat{\tau}^{ \, \mathbf{\Theta}} is a known quantity. Observe events to determine the numbers  j and  k by counting clock cycles. Report the result as \Delta t^{\, \mathsf{P}} = \left( k-j \right) \widehat{\tau}^{\, \mathbf{\Theta}}.

Sensory Interpretation

XXX

The elapsed time while experiencing one bundle of sensation  \mathsf{\Omega}, is the period  \widehat{\tau}. This period is the reciprocal of the frequency, which is proportional to the number of bundles observed per solar day. So the period can be interpreted as some fraction of a day. And the elapsed-time is based mostly on the reference sensation of seeing the Sun. However the direction of time  \epsilon_{t} is defined from thermal sensations, that are ultimately referred to the sensations of touching steam and touching ice.

References
1Titus Lucretius Carus, De Rerum Natura, translated by R. E. Latham, page 40. Penguin Books 1951.
2To be more precise, this quotation has been translated by Robert W. Lawson as “… we understand by the time of an event the reading (position of the hands) of that one of these clocks which is in the immediate vicinity (in space) of the event. In this manner a time-value is associated with every event which is essentially capable of observation.” From Relativity, the Special and the General Theory, page 24. Published by Methuen & Company, London 1936.