Time – EthnoPhysics

Outline

EthnoPhysics begins with the premise that we can understand time by describing sensory experience. Mysteries of the universe aside, we start by simply counting heartbeats and days. Then we assess Planck’s postulate as a feature of life on Earth. Time dilation is discussed. And the relations between cause and effect are linked to the notion of stability. Finally, the phase angle is introduced for describing small particles.

Counting Bundles

The frequency of observation, classification and rotation is suggested by this gear-like French egraving. — Lamarck, Jean-Baptiste (1744-1829). *Asterias*, Tableau Encyclopédique et Méthodique des Trois Règnes de la Nature, Paris 1791-1798. Photograph by D Dunlop.

To extend an earlier discussion about counting quarks, consider some particle P, that is characterized by a repetitive chain of events written as

$\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \, \ldots \, \right)$

Each repeated cycle $\mathsf{\Omega}$ is a bundle of quarks

$\mathsf{\Omega}^{\mathsf{P}} = \left( \mathsf{q}_{1}, \, \mathsf{q}_{2}, \, \mathsf{q}_{3} \, \ldots \, \mathsf{q}_{ N_{ \! \mathsf{q}} } \right)$

And the total number of quarks in the bundle is $N_{ \! \mathsf{q}} ,$ a finite positive integer. Depending on the level of objectification, this bundle $\mathsf{\Omega}$ might be thought of as a set of sensations or maybe a seed aggregate or perhaps some kind of spatial trajectory. But for any interpretation we can make a mathematical description of P just by counting bundles.

The chain $\Psi$ connects an indefinite number of cycles, there may be two or two-billion repetitions. But we can specify a definite quantity, written as $\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}} \, ,$ by making the description relative to seeing the Sun.

Let $\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}} \, \rule{0px}{6px}^{\mathsf{P}}$ be the number of bundles $\mathsf{\Omega}^{\mathsf{P}}$ that are experienced during one solar day. Or in other words, determine $\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}}$ by using the Sun as a clock while counting how often P is observed. This quantity is called the daily number of occurrences. The units of $\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}}$ are bundles-per-day or cycles-per-day. The Sun is presumably always available for this purpose because it is a reference sensation for EthnoPhysics.

Solar clocks are historically important, but not much used anymore. So consider evaluating $\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}}$ for an ordinary clock noted by $\mathbf{\Theta}$ . If this clock $\mathbf{\Theta}$ is calibrated so that its cycles are in seconds, then

$\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}} \rule{0px}{6px}^{\hspace{1px} \mathbf{\Theta}} =$ 86,400 (seconds per day)

This number comes to us from the Sumerian and Babylonian peoples of ancient Mesopotamia.¹George Sarton, A History of Science, page 74. Harvard University Press, Cambridge 1952. About four thousand years ago their astronomical observations and sexagesimal mathematics established what we mean by a second. Namely that one day is parsed as twenty-four hours of sixty minutes, each of sixty seconds. So $\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}} \rule{0px}{6px}^{\hspace{1px} \mathbf{\Theta}}$ is number established by convention. It is called the time units conversion factor.

Sensory interpretation: For EthnoPhysics, we also associate $\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}} \rule{0px}{6px}^{\hspace{1px} \mathbf{\Theta}}$ with the reference sensation of hearing a human heartbeat because the heart rate of human adults usually pulsates between forty and one hundred beats-per-minute when resting. So $\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}} \rule{0px}{6px}^{\hspace{1px} \mathbf{\Theta}}$ gives an order-of-magnitude account of the number of heartbeats-per-day for most people, thereby relating celestial and human-scale events.

Frequency

A temporal frequency is usually defined as some quantity of events happening per unit of time. But EthnoPhysics begins with the premise that time is known from sensory experience. So we have explicitly employed a solar clock to specify $\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}}$ as the number of daily occurrences for some particle. This tally is used to define the frequency as

$\nu \equiv \dfrac{\, \; \raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}} \; \; }{ \raisebox{-0.08cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}} \rule{0px}{6px}^{\hspace{1px} \mathbf{\Theta}} } N_{\! \mathsf{q}}$

This frequency $\nu$ describes a flux of quarks because $N_{\! \mathsf{q}}$ allows for the number of quarks in a bundle. So the frequency has units like quarks-per-second. Formally, frequency units are called hertz and abbreviated as (Hz).

Sensory interpretation: The frequency describes flow strength in a stream of consciousness. It is proportional to the quantity of Anaxagorean sensations experienced per second.

Planck’s Postulate

In 1900 the German physicist Max Planck asserted that the energy of a particle is directly proportional to its frequency in a fixed ratio that is now called Planck’s constant.²The Theory of Heat Radiation by Max Planck. Translated into English by Morton Masius and published by P. Blakiston’s Son & Co. in Philadelphia, 1914. Here is a plausibility argument for the postulate that is based on understanding some particle P as a repetitive bundle of $N_{ \! \mathsf{q}}$ quarks written as

$\mathsf{\Omega}^{\mathsf{P}} = \left( \mathsf{q}_{1}, \, \mathsf{q}_{2}, \, \mathsf{q}_{3} \, \ldots \, \mathsf{q}_{ N_{ \! \mathsf{q}} } \right)$

Let P be characterized by $E$ its mechanical energy. Then we can define a number $\mathrm{S}$ called the action of P by

$\mathrm{S} \equiv \dfrac{\; \raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}} \rule{0px}{6px}^{\hspace{1px} \mathbf{\Theta}} }{\, \; \raisebox{-0.08cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}} \; } E$

To expound on the action recall that $\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}} \rule{0px}{6px}^{\hspace{1px} \mathbf{\Theta}}$ is a constant, and that $\raisebox{-0.04cm}{\it{N}} ^{\hspace{-.15cm}^{\mdsmblkcircle}}$ is the number of bundles observed during one day. So if $E$ is dispersed over all these bundles, then $\mathrm{S}$ is directly proportional to the daily-average energy of one bundle. Roughly speaking, the action is like the energy of a typical bundle. $\mathrm{S}$ is also used to define the action of an average individual quark as

$\widetilde{\mathrm{S}} \equiv \mathrm{S} / N_{\! \mathsf{q}}$

So in terms of the frequency $\nu ,$ the action due to this typical quark is

$\widetilde{\mathrm{S}} = \dfrac{\, E \dot{N}^{\mathbf{\Theta}} \, }{ N_{\! \mathsf{q}} \dot{N} } = \dfrac{E}{\nu}$

Thus $\widetilde{\mathrm{S}}$ describes the action of a typical quark in a typical bundle. And

$E = \widetilde{\mathrm{S}} \nu$

Consider evaluating $\widetilde{\mathrm{S}}$ for a special case where P is the Earth. The number $\widetilde{\mathrm{S}} ^{ \, \mathsf{Earth}}$ is presumably a constant because the Earth is so large and permanent. Quark distributions are at least as stable as rock formations that change on geological time scales. This number is called Planck’s constant. It is traditionally noted by $h$ so we write

$h \equiv \widetilde{\mathrm{S}}^{ \, \mathsf{Earth}}$

Now let P be some other big terrestrial particle. That is, let $N_{\! \mathsf{q}}^{ \mathsf{P}}$ be large enough so that the statistical law of large numbers takes effect. This implies that a measurement of $\widetilde{\mathrm{S}}^{ \mathsf{P}}$ is close to $\widetilde{\mathrm{S}}^{ \mathsf{Earth}}$ and that they tend to get closer as $N_{\! \mathsf{q}}^{ \mathsf{P}}$ rises. So if P is large enough, we may say it satisfies the large particle condition written as

$\widetilde{\mathrm{S}} ^{ \mathsf{P}} = \; \widetilde{\mathrm{S}} ^{ \mathsf{Earth}}$

Thus for large particles we can combine the last three equations to obtain

$E = h \nu$

This is the conventional statement of Planck’s postulate. It is an experimental fact that $h$ is well known to about one part in a billion, and apparently not much changed over the last century. So we make vigorous use of Planck’s postulate for particles that contain many quarks.

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 a Greek philosopher who founded one of the most famous and influential schools of antiquity. His physics was taught at Athens around 300 BCE. According to Lucretius³Titus Lucretius Carus, De Rerum Natura, translated by R. E. Latham, page 40. Penguin Books 1951.

... 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 $p ,$ a particle’s momentum. Next “their restful immobility” is represented by $m ,$ the rest mass. And finally these numbers are combined to define the orbital period $\widehat{\tau}$ as

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

This expression can be simplified using the modern notion of mechanical energy. Recall that $E \equiv \sqrt{ c^{2}p^{2} + m^{2}c^{4} \; } .$ So the period can always be written as

$\widehat{\tau} = h / E$

Furthermore, for particles containing many quarks, we can use Planck’s postulate which states that $E = h \nu$ where $\nu$ is the frequency. So for large particles the period can be put simply as

$\widehat{\tau} = 1 / \nu$

To relate this orbital period to time, let some particle P be characterized by a repetitive chain of events written as $\Psi = ( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2} \; \ldots \; \mathsf{\Omega}_{k} \; \ldots )$ where each orbit $\mathsf{\Omega}$ is described by its period $\widehat{\tau} .$ The time of occurrence of event $\mathsf{\Omega}_{k}$ is then defined by

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

where $\epsilon_{t}$ notes the direction of time. Let $\Psi$ be historically ordered then $\epsilon_{t} \! = \! 1$ and the time-of-occurrence is given by a straightforward sum of periods. The value of the original event $t_{0}$ is arbitrary. In general, the time-of-occurrence may also be referred to as the time or a time coordinate.

The time $t$ is an important parameter for describing repetitive sensations. But recall that each repetition $\mathsf{\Omega}$ is a set of quarks written as

$\mathsf{\Omega} = \left( \mathsf{q}_{1}, \, \mathsf{q}_{2}, \, \mathsf{q}_{3} \, \ldots \, \mathsf{q}_{ N_{ \! \mathsf{q}} } \right)$

where $N_{ \! \mathsf{q}}$ is a finite integer. So the most detailed description of sensation could expand into an account of individual quarks. Then the smallest change in $t$ would be due to the effect of a single quark. This little step is called an increment of time. It is written as $d \! t$ and defined by

$d \! t \equiv \widehat{\tau} / N_{\!\mathsf{q}}$

Finally, we remark that the time-of-occurrence is a relative characteristic. The orbital period $\widehat{\tau}$ 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 coordinate is frame-dependent too. If $p = 0$ then $t$ is called the proper time.

Elapsed Time

Let P be represented by a historically ordered chain-of-events written as

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

And let each event $\mathsf{\Omega}$ be described by its time-of-occurrence $t$ . Then consider an arbitrary pair of events $\mathsf{\Omega}_{i}$ and $\mathsf{\Omega}_{f}$ where $i < f$ . Since $\Psi$ is in historical order we call them initial and final events. The elapsed time between these two events is defined by

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

If P is isolated and the frame of reference is inertial then the mass 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

$\Delta t = \left( f-i \rule{0px}{9px} \right) \widehat{\tau}$

Sensory interpretation: The elapsed time while experiencing one bundle of sensation $\mathsf{\Omega}$ , is the period $\widehat{\tau}$ . This period is some fraction of a day. So the elapsed-time is based on the reference sensation of seeing the Sun. However, the direction of time $\epsilon_{t}$ is defined by thermal similarity to the Earth. And thermal descriptions are ultimately referred to touching steam and touching ice.

Telling Time

According to Albert Einstein time is what a clock tells.⁴To 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 rudimentary method for determining $\Delta t$ using a clock. Consider some particle P that is represented by a chain of events written as

$\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}^{\mathsf{P}}, \, \mathsf{\Omega}_{2}^{\mathsf{P}} \; \ldots \; \mathsf{\Omega}_{a}^{\mathsf{P}} \; \ldots \; \mathsf{\Omega}_{b}^{\mathsf{P}} \; \ldots \; \right)$

The elapsed time $\Delta t^{\, \mathsf{P}}$ between some pair of events $\mathsf{\Omega}_{a}^{\mathsf{P}}$ and $\mathsf{\Omega}_{b}^{\mathsf{P}}$ depends on the frame-of-reference F used to determine P’s momentum. Let this frame include a clock $\mathbf{\Theta} .$ And let F be represented by a chain-of-events written as

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

Since F is employed as a reference frame, presumably every measurement of P is accompanied by an observation of F. That is, we assume that events of F and P can be closely associated in pairs such as

$\left\{ \, \mathsf{\Omega}_{a}^{\mathsf{P}} \; , \; \mathsf{\Omega}_{j}^{\mathsf{F}} \, \rule{0px}{13px} \right\}$

and

$\left\{ \, \mathsf{\Omega}_{b}^{\mathsf{P}} \; , \; \mathsf{\Omega}_{k}^{\mathsf{F}} \, \rule{0px}{13px} \right\}$

To make a generic measurement of elapsed time, first calibrate the frame’s clock $\mathbf{\Theta}$ so that its period $\widehat{\tau}^{ \, \mathbf{\Theta}}$ is a known quantity. Then observe events to determine the numbers $j$ and $k$ by counting clock cycles. Report the result as

$\Delta t^{\, \mathsf{P}} = \left( k-j \right) \widehat{\tau}^{\, \mathbf{\Theta}}$

This approach is good enough for the large slow particles of everyday experience. But if P is very small or fast then the assumption of being “closely associated” with F may require further elaboration.

Time Dilation

Let $\mathbf{\Theta}$ be a material clock that might be moving. The clock’s rest mass is written as $m .$ And its mechanical energy is noted by $E .$

We now evaluate how the elapsed time, as told by $\mathbf{\Theta} ,$ is affected by motion. An earlier discussion of the Lorentz factor $\gamma ,$ related mass and energy by $E = \gamma m c^{2} .$ So the clock’s period may be expressed as

$\widehat{\tau}^{\, \mathbf{\Theta}} = \dfrac{h}{E} = \dfrac{h}{\gamma mc^{2}}$

Then the elapsed time between some arbitrary initial and final events is given by

$\Delta t = \left( f-i \rule{0px}{9px} \right) \widehat{\tau}^{\, \mathbf{\Theta}} = \dfrac{h(f-i)}{\gamma mc^{2}}$

For comparison, set $\gamma \! = \! 1$ to define the elapsed time that would be indicated if $\mathbf{\Theta}$ was at rest. This number is noted by $\Delta t^{\ast}$ and called the proper elapsed time

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

The two quantities are related as $\Delta t^{\ast} = \gamma \Delta t .$ But the Lorentz factor of a moving particle is always greater than one. So a moving particle always experiences less elapsed time than a stationary particle. Thus $\Delta t \leq \Delta t^{\ast}$ and this effect is called time dilation.

Cause and Effect

Consider some particle P that is described by its mean life $\tau ,$ its orbital period $\widehat{\tau}$ and a historically ordered chain of events written as

$\Psi^{\mathsf{P}} = \left( \mathsf{P}_{i} \; \ldots \; \mathsf{P}_{\! f} \right)$

where $\mathsf{P}_{i}$ and $\mathsf{P}_{\! f}$ are the initial and final events in a history of P. They represent P’s formation and decay. So the difference in their occurrence times is P’s lifetime

$t_{f} - t_{i} = \tau$

Furthermore let P be isolated. Then for an inertial frame of reference, the period of P is a constant, and the elapsed time $\Delta t$ between initial and final events is just

$\Delta t = \left( f-i \rule{0px}{10px} \right) \widehat{\tau}$

But by definition $\Delta t \equiv t_{f} - t_{i} \, .$ So for P, eliminating the time coordinates and rearranging gives

$\tau = \left( f-i \rule{0px}{10px} \right) \widehat{\tau}$

If $\Psi$ is a long chain containing very many events we write $1 \ll f-i \, .$ Then substitution implies that for long chains, P’s mean-life is much longer than its orbital period

$\widehat{\tau} \ll \tau$

For long chains it is probable that P will not decay during any specific cycle. One orbit will almost invariably follow another with dependable regularity. So a period that is negligible compared to the lifetime implies permanence, steadiness or stability.

Stability and predictability are very useful, so we give particles like P a special name. When the orbital period $\widehat{\tau}$ is much smaller than the mean life $\tau ,$ then P satisfies the stability condition that $\widehat{\tau} \ll \tau$ and we say that P is a stable particle. The initial event $\mathsf{P}_{i}$ is called the cause. And the final event $\mathsf{P}_{f}$ is called an effect of the cause.

The stability condition is easily satisfied for particles that have a temperature near or below zero (K) because by definition their mean-life will be at least as long as the lifetime of a proton. Measurements of the proton’s mean-life have determined⁵K.A. Olive et al. Particle Data Group Review of Particle Physics, Chin. Phys. C, 38, 090001 (2014). that it is more than ~10³⁶ seconds. So for these cold particles $\tau$ is exceedingly large.

The orbital period of a proton is given by $\widehat{\tau} = h / E$ where $E \simeq m c^2 .$ The mass of a proton is well known and used to find $\widehat{\tau}$ as ~10^-24 seconds. That is, the lifetime of a proton is at least 10⁶⁰ times larger than its period. And for electrons, the lifetime-to-period ratio is at least ~10⁵⁴. So both the proton and the electron are exceptionally stable particles. This gives them starring roles in narratives connecting cause and effect.

Cause and effect are suggested by the images of life and death in this Indonesian weaving. — Tampan, Paminggir people. Lampung region of Sumatra, Kota Agung district, circa 1900, 38 x 37 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.

Phase Angle

Consider some particle P, that is characterized by a repetitive chain of events written as

$\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \, \ldots \, \mathsf{\Omega}_{k} \, \ldots \, \right)$

where each repeated cycle $\mathsf{\Omega}$ is a bundle of $N_{ \! \mathsf{q}}$ quarks noted by

$\mathsf{\Omega} = \left( \mathsf{q}_{1}, \, \mathsf{q}_{2}, \, \mathsf{q}_{3} \, \ldots \, \mathsf{q}_{ N_{ \! \mathsf{q}} } \right)$

If P contains just a few quarks, then the time of occurrence may not be a useful parameter for describing these events because Planck’s postulate is plausibly justified only if P is large. So we also discuss events using a phase angle defined by

$\theta_{k} \equiv \theta_{\mathsf{o}} + 4\pi k$

where $\theta_{\mathsf{o}}$ is arbitrary. The units of $\theta$ are called radians and abbreviated by (rad). This definition associates $4\pi$ radians with each repetition of $\mathsf{\Omega} \, .$ That is, one complete spatial rotation of $2\pi$ radians for each phase-component of P.

The phase angle $\theta$ is an important parameter for describing repetitive sensations. But remember that each repetition $\mathsf{\Omega}$ is a set of quarks. So the most detailed description of sensation could possibly expand into an account of individual quarks. Then the smallest change in $\theta$ would be due to the effect of a single quark. This little step is called the phase angle increment. It is written as $d \! \theta$ and defined by

$d \! \theta \equiv 4\pi / N_{ \! \mathsf{q}}$

The phase-angle increment may be combined with the increment of time to define $\omega$ the angular speed as

$\omega \equiv \dfrac{ d \! \theta }{ d \! t } = \dfrac{4\pi}{\widehat{\tau}}$

EthnoPhysics uses a finite categorical scheme of binary distinctions to describe sensation. So the number of quarks $N_{\!\mathsf{q}}$ may be large but not infinite. This requirement can be relaxed later to make a continuous approximation, thereby allowing the use of calculus. But in principle $N_{\!\mathsf{q}}$ is finite and accordingly changes in $\theta$ may be small, but not infinitesimal. For isolated particles the increment in the phase angle does not vary and so there is an equipartition of $\theta$ regardless of quark type.

Phase angle dependence on quark distributions is suggested by this colorful German engraving of algae. — Melethallia (detail), Ernst Haeckel, Kunstformen der Natur. Chromolithograph 32 x 40 cm, Verlag des Bibliographischen Instituts, Leipzig 1899-1904. Photograph by D Dunlop.

Bumpy Space

Space is defined from sensation, it is not continuous. Spatial isotropy and homogeneity are analyzed. Empty space is not defined.

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References
1	George Sarton, A History of Science, page 74. Harvard University Press, Cambridge 1952.
2	The Theory of Heat Radiation by Max Planck. Translated into English by Morton Masius and published by P. Blakiston’s Son & Co. in Philadelphia, 1914.
3	Titus Lucretius Carus, De Rerum Natura, translated by R. E. Latham, page 40. Penguin Books 1951.
4	To 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.
5	K.A. Olive et al. Particle Data Group Review of Particle Physics, Chin. Phys. C, 38, 090001 (2014).