iSoul Time has three dimensions

Tag Archives: Space & Time

Matters relating to space and time in physics and transportation

Symmetric relativity

Although there are many experimental methods available to measure the speed of light, the underlying principle behind all methods [is] the simple kinematic relationship between constant velocity, distance and time given below:

c = D / t                     (1)

In all forms of the experiment, the objective is to measure the time required for the light to travel a given distance. (Ref.)

From the perspective of the experimenter, light is an object whose speed is to be determined. Even though the distance traversed is fixed, it is placed in the numerator because this speed is to be compared with the speeds of other objects. For the same reason the quantity to be measured, time, is placed in the denominator.

But if we take the perspective of the experiment, of what is measured, then the fixed distance is the independent variable, which is placed in the denominator. The dependent variable is the time, which is placed in the numerator, so the pace of light is measured:

= t / D                     (2)

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Doppler effect in spacetime and timespace

We follow the presentation of Guy Moore, formerly of McGill University, online here.

The Doppler effect is the phenomenon we have all noticed, that a sound produced by a moving source, or which you hear while you are moving, has its perceived frequency shifted.

Spacetime (3+1)

If a source of sound makes a sudden bang (pressure pulse) at time 0, then the location in space of the pressure pulse in relation to time will look like successive concentric circles. For a source moving to the right, letting out a series of “bangs,” the location of the successive pressure peaks will be closer together on the right and further apart on the left.

Now remember that a sound is just a series of pressure peaks, which are tightly separated in time, which is the period of the wave. Therefore, instead of thinking of the sound waves as due to “bangs,” you can think of them as pressure peaks in a periodic sound wave. In front of the moving object the peaks are closer together. That means that the wave length is shorter, which means it is a higher frequency wave. Behind, the peaks are farther apart, meaning that it is a longer wavelength sound, at a lower frequency. This is the gist of the Doppler effect.

Let us now actually calculate the size of the effect. Suppose a sound source is moving right at you, at velocity v. At time 0, it emits a pressure peak. At time Δt, it emits a second pressure peak. If its distance from you at time 0 was x, its distance from you at time Δt was xvΔt (it is nearer, since it is moving towards you).

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Mean speed and pace

Speed of a motion is the time rate of length change, that is, the length interval with respect to a timeline interval without regard to direction. Pace of a motion is the space rate of time change, that is, the time interval with respect to a locusline interval without regard to direction.

The symbol for speed is v = Δst and for pace is u = Δts. Instantaneous speed is ds/dt. Punctaneous pace is dt/ds.

There are two kinds of mean speed or pace: the time mean and the space mean. The time mean is the arithmetic mean if the denominators are a common time interval. The space mean is the arithmetic mean if the denominators are a common space interval. The time mean is the harmonic mean if the denominators are a common space interval. If the denominators are a common time interval, the space mean is the harmonic mean.

The time mean speed (TMS) is the arithmetic mean of speeds with a common time interval. The time mean pace (TMP) is the harmonic mean of paces with a common time interval. For example, the travel distance for vehicles on a highway during a time period is measured. The time mean speed or pace may then be calculated.

The space mean pace (SMP) is the arithmetic mean of paces with a common space interval. The space mean speed (SMS) is the harmonic mean of speeds with a common space interval. For example, the travel time for vehicles over a length of highway is measured. The space mean speed or pace may then be calculated.

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Coordinate transformations

Coordinate Transformations with t

x = space coordinate, t = time coordinate, v = velocity, u = pace


Galileian transformation

speed: = xvt and t´ = t,         pace: = xt/u and t´ = t.

Co-Galileian transformation

speed: = tx/v and x´ = x,       pace: = tux and x´ = x.


Light:   c := speed of light,        := pace of light.

speed: x = ct or x/c = t and = ct´, or x´/c = ,

pace:  c´x = t or x = t/c´ and c´r′ = or = t´/c´.


Lorentz transformation

speed: γ = (1 – v2/c2)–1/2 with x´ = γ (x − vt) and = γ (txv/c²),

pace:  γ = (1 – 2/u2)–1/2 with γ (xt/u) and t´γ (txc´²/u),

which applies only if |v| < |c| or |u| > ||.


Co-Lorentz transformation

speed: λ = (1 − c2/v2)–1/2 with λ (t − x/v) and λ (xt (c2/v)),

pace:  λ = (1 – u2/2)–1/2 with λ (tux) and λ (xt (u/c´²)),

which applies only if |v| > |c| or |u| < ||.

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Synchronic and diachronic time

Synchrony or diatopy means 3D space with simultaneous events in order of increasing time. Space-time is synchronic or diatopic.

Diachrony means 3D time with simulocuous events in order of decreasing distance. Time-space is diachronic.

Chronicles and histories are diachronic. Models and theories are synchronic or diatopic.

Knowledge of the distances between objects is important in order to understand their motions. Knowledge of the modal durations between subjects is important in order to understand their movements. In history travel time matters more than travel distance. In science travel distance matters more than travel time.

For the study of history, 3D time is more significant then 3D space. What matters more is not the distance between places, but the transit time. The reason is that time is the measure of effort to go from one place to another.

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Past, present, and future

This continues the post on the Arrow of tense.

Past, present, and future are characteristics of time. But they are also characteristics of places, of things, or events, etc.

Yesterday was in the past, today is in the present, and tomorrow is in the future.

Past places are remembered, present places are experienced, and future places are imagined.

Some things were in the past, some things are in the present, and some things will be in the future.

Some events happened in the past, some events are happening in the present, and some events will be happening in the future.

So past, present, and future are not time itself. They are characteristics that apply to various things.

Past, present, and future are tenses, which refer to order, not time per se. One can order events, experiences, objects, places, and periods of time in various ways. One can study which order is the right physical order.

And time has order. But order is not time.

Time is duration.

New terms for 3D time

This post highlights several recent terms or definitions added to the glossary above.

The distance is the metric of space, the shortest length between two points in space. Similarly, the distime is the metric of time, the shortest duration between two points in time.

A timeline is a linear ordering of events by distime from or to a reference event. A locusline is a linear ordering of events by distance from or to a reference event.

A clock shows the present instant in the local timeline. An odologe (o′∙do∙loje) is an app or device that shows the present point in the current locusline.

Instantaneous events occur in an instant of time. Punctaneous events occur in a point of space.

Simultaneous events occur at the same time. Simulocus events occur at the same place.

Synchronous motions are parallel in time, as in having the same period. From Greek syn+chron+ous. Symmacronous motions are parallel in space, as in having the same orbit. From Greek sym+macron+ous.

Pseudo-length is measured by time and expressed as length, as with multiplying time by the free-flow speed. Pseudo-duration is measured by length and expressed as time, as with multiplying length by the free-flow pace.

Inertia (linear) is the resistance of an object to any change in its state of motion. Facilia (linear) is the nonresistance of a subject to a change in its state of movement.

Immediate motion

I recently wrote about rest in space and time here. This post is about the opposite: immediate motion, arriving at a destination instantly.

Immediate motion means an infinite speed in space. An infinite speed results in an immediate change of place: something moves from one location to another in an instant. It’s here and there at the same time. The departure and arrival are simultaneous.

A body at infinite speed is at two places at the same time, but a speed ratio has a finite time interval. If it’s the same time, how can there be a finite time interval?

For speed the time interval is fixed as the length changes. If the speed approaches infinity, then the travel length in the numerator approaches infinity, so the time interval in the denominator becomes a smaller and smaller proportion and the ratio approaches infinity. The body is at two places sinultaneously.

Immediate motion also means a zero pace in time. A zero pace results in an immediate change of time: something moves from one time to another in an instant. It’s now and then at the same location. The departure time and arrival time are at the same location. I’m calling this simulocus.

But wait, two times at the same location seems like no motion at all. What gives?

For pace the length interval is fixed as the time changes. If the pace approaches zero, then the travel time in the numerator approaches zero, so the length interval in the denominator becomes a larger and larger proportion and the ratio approaches zero. The body is at two times simulocusly.

Does immediate motion exist? Not under the Lorentz transformation, in which there is a finite maximum speed. But the Galileian transformation implicitly uses an infinite speed of light. And the co-Galileian transformation implicitly uses a zero pace of light.

Three relativity transformations

Two transformations of inertial reference frames are well-known: the Galileian and the Lorentz transformations. There is a third transformation as well, which will be called the co-Galileian transformation. Below is a derivation of all three transformations, closely following the paper Getting the Lorentz transformations without requiring an invariant speed by Andrea Pelissetto and Massimo Testa (American Journal of Physics 83 (2015), p.338-340). Their approach is based on the work of von Ignatowsky in the early 20th century.

We wish to characterize the transformations that relate two different inertial frames. Let us consider two inertial observers K and K′. Let r = (x, x2, x3) and w = (t t2, t3) be space and time coordinates for K and = (x´, x2´, x3)´ and = (t´, t2´, t3´) be the corresponding quantities for K′.

In order to simplify the argument, we will restrict our considerations to the subgroup of transformations involving x and t only, setting x2´= x2, x3´ = x3, t2´ = t2, and t3´ = t3. This is equivalent to choosing coordinates so that K and K′ are in relative motion along the x and t directions in K and the x′ and t´ directions in K´.

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Simultaneity without clocks

Watches didn’t always exist. Neither did clocks that were transportable or manufactured in large quantities. I mention this because one way to determine the simultaneity of events is to have synchronized clocks transported to multiple locations – even an endless number of locations in theory.

How can an observer determine the simultaneous events from their frame of reference? Answer: simultaneous events are observed simultaneously by an observer. But how can this be reconciled with other observers who may observe the same events as non-simultaneous?

That is the point of relativity: applying transformations to coordinates from different frames of reference so that the equations of physics are the same in all reference frames. But relativity requires a convention of simultaneity (or a demonstration of what events are simultaneous events). Since I have defined time in terms of stopwatches rather than clocks, how can simultaneity be determined?

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