Epicurus, 341~270 BCE. Drawn from a damaged Roman copy of a Greek original now in the Palazzo Massimo alle Terme, Rome. |

"… 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 …"^{1}

The WikiMechanics 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** $\hat{\tau}$ as

$\begin{align} \hat{\tau} \equiv \frac{ h }{ \sqrt{ c^{2}p^{2} + m^{2}c^{4} \vphantom{\sum^{2}} \; } } \end{align}$

Recall that $\begin{align} E \equiv \sqrt{ c^{2}p^{2} + m^{2}c^{4} } \end{align}$ 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

$\begin{align} \hat{\tau} = \frac {h }{ E } = \frac {1 }{ \, \nu \, } \end{align}$

Now let particle $\sf{P}$ be characterized by some repetitive chain of events $\Psi ^{\sf{P}} = \left( \sf{\Omega}_{1}, \sf{\Omega}_{2} , \sf{\Omega}_{3} \; \ldots \; \sf{\Omega}_{\it{k}} \; \ldots \; \right)$ where each orbit $\sf{\Omega}$ is described by its period $\hat{\tau}$. Definition: The **time of occurrence** of event $\sf{\Omega}_{\it{k}}$ is

$\begin{align} t_{k} \equiv t_{0} + \epsilon_{t} \! \sum_{i=1}^{k} \hat{\tau}_{i} \end{align}$

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 directly given by a sum of periods. The period depends on a particle's momentum, which in-turn depends on whatever frame of reference is used to describe a particle's motion. So the time 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 some repetitive chain of events that are arranged in historical order

$\Psi ^{\sf{P}} = \left( \sf{\Omega}_{1}, \sf{\Omega}_{2}, \sf{\Omega}_{3} \; \ldots \; \sf{\Omega}_{\it{i}} \; \ldots \; \sf{\Omega}_{\it{f}} \; \ldots \; \right)$

And let each orbital event $\sf{\Omega}$ be described by $t$ so that $\sf{P}$ can also be described by an ordered set of occurrence times. Consider a pair of events $\sf{\Omega}_{\it{i}}$ and $\sf{\Omega}_{\it{f}}$ where $\it{i} < \it{f}$. Since $\Psi$ is in historical order we call them the initial and final events of the pair. Definition: the **elapsed time** between these events is

$\Delta t \equiv t_{f} - t_{i}$

If $\sf{P}$ is isolated and the frame of reference is ideal then the mechanical energy and momentum do not change from event to event along the chain $\Psi$. For these conditions, the period $\hat{\tau}$ is constant too. But the time coordinate is defined by a sum of periods, so the elapsed time can be written as$\Delta t = \left(\, f-i \right) \hat{\tau}$

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 rest mass. These are related by $E = \gamma m c^{2}$ where $\gamma$ is the Lorentz factor. Then the period is given by

$\begin{align} \hat{\tau} \, = \frac{h}{E } = \frac{ h }{ \gamma m c^{2} } \end{align}$

So in terms of the mass

$\begin{align} \Delta t = \frac{ h \, (\, f-i \, ) }{ \gamma m c^{2} } \end{align}$

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

$\begin{align} \Delta t ^{\ast} \equiv \frac{ h \, (\, f-i \, ) }{ m c^{2} } \end{align}$

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

$\begin{align} \Delta t^{\ast} = \gamma \Delta t \end{align}$

The Lorentz factor for a particle in motion is always greater than one, $\gamma ≥ 1$. So a moving particle always experiences less elapsed time than a stationary particle, $\Delta t ≤ \Delta t^{\ast}$. This effect is called **time dilation**.

## Measuring Elapsed Time

^{2}So here is a generic description of how to determine elapsed time using a clock. Let particle $\sf{P}$ be characterized by some repetitive chain of events noted as

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

Consider measuring $\Delta t$ between events $\sf{\Omega}_{\it{i}}$ and $\sf{\Omega}_{\it{f}}$. This quantity depends on the frame of reference $\sf{F}$ which is represented by another chain of events

$\Psi ^{\sf{F}} = \left( \sf{F}_{1}, \sf{F}_{2} \; \ldots \; \sf{F}_{\it{j}} \; \ldots \; \sf{F}_{\it{k}} \; \ldots \; \right)$

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

$\left\{ \sf{P}_{\it{i}}, \sf{F}_{\it{j}} \right\}$ and $\left\{ \sf{P}_{\it{f}}, \sf{F}_{\it{k}} \right\}$

To make a laboratory measurement of elapsed time first select some clock $\mathbf{\Theta}$ that is presumably part of the frame of reference $\sf{F}$. Let this clock be calibrated so that its period $\; \hat{\tau}^{ \mathbf{\Theta}}$ is a known quantity. Observe events to determine the numbers $j$ and $k$ by counting clock cycles. Report the result as

$\Delta t ^{\sf{P}} = \left( k-j \right) \hat{\tau}^{ \mathbf{\Theta} }$