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Time

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 important relations between cause and effect are linked to the notion of stability. Finally, the phase angle is introduced for quantized descriptions.

Counting Bundles

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)

where each repeated cycle  \mathsf{\Omega} is a bundle of  N quarks written as

\mathsf{\Omega} = \left( \mathsf{q}^{1}, \, \mathsf{q}^{2}, \,  \mathsf{q}^{3} \, \ldots \, \mathsf{q}^{N} \right)

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

Depending on the level of objectification, each bundle \mathsf{\Omega} may be thought of as a set of sensations, or an orbit, or an aggregation of seeds, or perhaps a compound quark. But for any interpretation we can make a mathematical description of P just by counting bundles.

Solar clocks are like this radiant icon.

The chain  \Psi is a sequence of an indefinite number of bundles, there may be two or two-billion of them. But we can specify a definite quantity  N_{\mathsf{\Omega}} by making the description relative to a reference sensation provided by seeing the Sun. Let  N_{\mathsf{\Omega}}^{\mathsf{P}} be the number of P’s bundles observed during one solar day. This quantity has units of bundles-per-day or cycles-per-day. Solar clocks are historically important, but not much used anymore. So consider evaluating  N_{\sf{\Omega}} where P is an ordinary clock noted by  \mathbf{\Theta}. If this clock  \mathbf{\Theta} is calibrated so that its cycles are in seconds, then

N_{\mathsf{\Omega}}^{\mathbf{\Theta}} = 86,400 (seconds per day)

This number comes to us from the Sumerian and Babylonian peoples of ancient Mesopotamia1George 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.

Oddness is illustrated by this heartbeat icon for binary somatic sensations.

For EthnoPhysics, we also associate  N_{\mathsf{\Omega}}^{ \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  N_{\mathsf{\Omega}}^{ \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. The foregoing tallies of bundles are used to define the angular frequency of P as

\omega \equiv \dfrac{\, 2\pi N_{\mathsf{\Omega}}^{\mathsf{P}} \, }{N_{\mathsf{\Omega}}^{\mathbf{\Theta}}}

This angular frequency has units like bundles-per-second. As descriptions are objectified, we associate 2 \pi radians with each bundle and speak more generally of radians-per-second. And since all physical particles are supposedly made of quarks, we can also define a very general expression of the frequency as

\nu \equiv N \dfrac{\omega}{2\pi}

where  N is the number of quarks in  \mathsf{\Omega}. This generic frequency characterizes the flux of quarks associated with P. It has units like quarks-per-second, or more generally hertz, and is abbreviated by (Hz). This frequency describes the 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 any particle P, is directly proportional to its frequency, in a fixed ratio 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 P as a repetitive chain of events written as

Planck's postulate concerns the statistical distribution of sensations like the highly structured chromatic variations in this hand-woven textile from China.
Baby Collar, Dong people, 39 x 17 cm. China, Yunnan province, 20th century. Photograph by D Dunlop.

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

Let P be characterized by its angular frequency  \omega and mechanical energy  E . Then we can specify a number called the action of P as

X \equiv \dfrac{2\pi E}{\omega}

Also let each repeated bundle  \mathsf{\Omega} be composed from  N quarks so that \mathsf{\Omega} = ( \mathsf{q}_{1}, \, \mathsf{q}_{2} \; \ldots \; \mathsf{q}_{N} ) . Then the action associated with a typical quark is

\widetilde{X} \equiv \dfrac{X}{N} = \dfrac{2\pi E}{N \omega}

Recall that the generic frequency  \nu of any particle is given by \nu \equiv N \omega / 2 \pi. So the action for some average quark in P can be written in terms of the frequency as

\widetilde{X} = \dfrac{E}{\nu}

Earthiness is illustrated by this planet icon for the sensation of touching the earth.

For terrestrial particles made from lots of quarks, the statistical law of large numbers guarantees that   \widetilde{X} has a definite value determined by the distribution of quarks on Earth. Moreover, this value is presumably constant because the quark distribution is at least as stable as rock formations that change on geological time scales. This constant is called Planck’s constant, and noted by  h. We can write

h \equiv \widetilde{X} \! \left( \mathsf{Earth} \right) \cong \widetilde{X}^{\, \mathsf{P}}

Then if  N is large enough, the foregoing result that \widetilde{X} = E / \nu implies that

E = h \nu

This is the conventional statement of Planck’s postulate. It is an experimental fact that the ‘constant’ is well known to about one part in a billion, and apparently unchanged over the last century. So we make vigorous use of Planck’s postulate for particles that contain many quarks.

Solar clocks are like this radiant icon.

Sensory interpretation: The angular frequency  \omega is proportional to the number of sensory bundles observed per day. So if the mechanical energy of P is equally shared between these bundles, then the action  X represents the energy in a typical bundle. And  \widetilde{X} is like the energy of a typical quark in a typical bundle. If we assume that Planck’s postulate applies to P, then the mechanical energy  E is proportional to the daily flux of Anaxagorean sensations associated with P. Any variation in this stream of sensory consciousness can be expressed using   \delta \! E and the signal to noise ratio.

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 Lucretius3Titus 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 > 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.

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 generic description of how to determine  \Delta t using a clock.

The 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}} .

XXX

Sensory interpretation: 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.

Cause and Effect

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

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

Let these events be historically ordered. Then  \mathsf{P}_{i} and  \mathsf{P}_{f} are called the initial and final events of \Psi. And if the particle is perfectly isolated then  \mathsf{P}_{i} and  \mathsf{P}_{f} are established by the formation and decay of P. With an inertial frame of reference, the period does not vary. Then the elapsed time  \Delta t between initial and final events is P’s lifetime, and all these quantities are related as

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

Let \Psi be a chain of very many events so that

i + 1 \ll f

For this case the lifetime must be much greater than the orbital period

1 \ll f-i = \dfrac{ \; \tau \; }{\widehat{\tau}}

and we say that P is stable. Then it is probable that P will not decay during any specific cycle. P is steady and consistent. One orbit will almost invariably follow another with dependable regularity. This predictability is useful, so we give the events of particles like this special names. If

\widehat{\tau} \ll \tau

then the initial event of \Psi is called the cause, and the final event is called the effect of the cause. We may say that P’s events are causally linked to each other. For example consider the proton in its ground-state. It has a period  \widehat{\tau} of about 10-24 seconds, so in principle we could assign a very precise time of occurrence to any events in the history of a proton.

For protons, the lifetime  \tau is more than 1036 seconds, so  \widehat{\tau} \ll \tau . And for electrons, the period is about 10-20 seconds with a lifetime of more than 1034 seconds. So both the proton and electron are extremely 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

Let particle P be described by an ordered chain of events written as

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

that is repetitive so that  \Psi may also be written as

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

where each orbital cycle  \mathsf{\Omega} is composed of  N sub-orbital events

\mathsf{\Omega} = \left( \mathsf{P}_{1}, \, \mathsf{P}_{2}, \, \mathsf{P}_{3} \, \ldots \, \mathsf{P}_{N} \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 on a statistical basis. So we also discuss the order of events using a phase angle defined by

\theta_{k} \equiv \theta_{\mathsf{o}} - \delta_{z} \dfrac{2\pi k}{N}

where  \theta_{\mathsf{o}} is arbitrary and  \delta_{z} is the helicity of P. The ground-states and many excited-states of atoms have spin-down orientations, then  \delta_{z}=-1 . Also, the change in  \theta during one sub-orbital event is called the phase angle increment. It is defined by  d \! \theta \equiv 2 \pi \! / \! N , so usually

\theta_{k} = \theta_{\mathsf{o}} + k \hspace{1px} d \! \theta

EthnoPhysics uses a finite categorical scheme of binary distinctions to describe sensation. So the number of sub-orbital events  N 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 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 between sub-atomic events regardless of their quark content.

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.

Next we discuss EthnoPhysics faviconBumpy Space.

References
1George Sarton, A History of Science, page 74. Harvard University Press, Cambridge 1952.
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.
3Titus Lucretius Carus, De Rerum Natura, translated by R. E. Latham, page 40. Penguin Books 1951.
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.