iSoul Time has three dimensions

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Coordinate transformations

Coordinate Transformations with t

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


Galileian transformation

speed: = rvt and t′ = t,         pace: = rt/u and t′ = t.

Co-Galileian transformation

speed: = tr/v and r′ = r,       pace: = tur and r′ = r.


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

speed: r = ct or r/c = t and r′ = ct′, or r′/c = t′,

pace:  çr = t or r = t/ç and çr′ = t′ or r′ = t′/ç.


Lorentz transformation

speed: γ = (1 – v2/c2)–1/2 with r′ = γ (r − vt) and t′ = γ (trv/c²),

pace:  γ = (1 – ç2/u2)–1/2 with r′γ (rt/u) and t′γ (trç²/u),

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


Co-Lorentz transformation

speed: λ = (1 − c2/v2)–1/2 with t′λ (t − r/v) and r′λ (rt (c2/v)),

pace:  λ = (1 – u2/ç2)–1/2 with t′λ (tur) and r′ λ (rt (u/ç²)),

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

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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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Space-time exchange postulate

Rates of motion are almost always expressed as a ratio with respect to time. For example, the average speed of a body is the travel distance of the body divided by the travel time. This makes the independent variable time and distance the dependent variable.

However, there is no physical dependency of motion on time rather than distance. One could just as well express the average rate of motion as the travel time of the body divided by the travel distance. The ratios are equally valid.

This is a general result. There is a binary symmetry between space and time. Travel distance and travel time are interchangeable as far as the equations of physics are concerned. J. H. Field has expressed this as a postulate for space-time exchange (STE):

(I) The equations describing the laws of physics are invariant with respect to the exchange of space and time coordinates, or, more generally, to the exchange of the spatial and temporal components of four vectors. (A four-vector has three components of length and one of distime.)

He avoids the question of 3D time by limiting the STE to the direction of inertial motion. Here we generalize the STE postulate to include 3D time:

(II) The equations describing the laws of physics are invariant with respect to the exchange of space and time coordinates, or, more generally, to the exchange of the spatial and temporal components of six-vectors. (A six-vector has three components of length and three of time.)

Field found the STE to violate Galileian symmetry, but this is incorrect because time is three dimensional, and there is a co-Galileian transformation symmetric to the Galileian transformation.

The STE postulate affirms the complete symmetry of space and time, which is built on the symmetry of length and duration. As distance is the metric of space, a kind of length, so distime is the metric of time, a kind of duration. The metric of space or time may be used to organize events linearly, with equivalence classes defined for events at the same position in the order.

Rest in space and time

Rest means no motion, or at least no motion detected by an observer.

We know what rest in space means: staying in the same place. That is, rest means no change of position, no travel distance, no length of motion. So at rest the numerator of the speed is zero.

Yet clocks tick on. The denominator of speed is not zero. The speed of rest then equals zero, that is, a zero length of motion divided by a non-zero quantity of time. Speed v = Δxt = 0/Δt = 0.

What is rest in time? It means staying at the same time. That is, rest means no duration of motion, no travel time. So at rest the numerator of the pace is zero.

In this case, is the length of motion zero, too? No. For pace the length is the independent quantity. It doesn’t depend on the motion. It depends on the given length or unit of length. So the pace of rest is zero, that is, a time of zero divided by a non-zero length. Pace u = Δtx = 0/Δx = 0.

Yet a zero pace seems to say one gets a change of place with no lapse of time. What gives?

Length of motion in the pace ratio is the independent variable. Whether length is conceived to be continually increasing, as if it were a clock, or just a quantity of length for comparison, it is independent of the motion measured. The numerator, the time, is what is measured and compared with a quantity of length to determine the pace.

It is similar with speed. Whether or not there is a clock ticking away, the denominator is a quantity of time compared with a quantity of length. All the clocks in the world could be broken, yet the denominator of speed, the change in time, would still be non-zero.

Consider a vehicle with an odometer and a stopwatch that is running whenever the vehicle is in motion. Both the odometer and the stopwatch would record no additional time for a vehicle at rest. This could not be represented as a ratio since 0/0 is not a valid ratio. Such a state has an indeterminate rate of motion.

Relativity alone

In a paper titled Nothing but Relativity (Eur. J. Phys. 24 (2003) 315-319) Palash B. Pal derived a formula for transformations between observers that is based on the relativity postulate but not a speed of light postulate. In a paper titled Nothing but Relativity, Redux (Eur. J. Phys. 28 (2007) 1145-1150) Joel W. Gannett presented an alternate derivation with fewer implicit assumptions. Here we’ll use Pal’s approach to derive the time-space version.

Consider two inertial timeframes S and , where the second one moves with legerity u, along the t-axis, with respect to the first one. There are two other time axes. The coordinates and radial distance in the S-timeframe will be denoted by t and x, and in the timeframe will be denoted with a prime. The time-space transformation equations have the form:

= T(t, x, u) and = X(t, x, u),

and out task is to determine these functions. A few properties of these functions can readily be observed. First, the principle of relativity tells us that if we invert the legerity in these equations, we must obtain the same functional forms:

t = T(t´, x´, –u) and x = X(t´, x´, –u).

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Circular orbits

*** Revised from June 2017 ***

With circular motion there is a radius and circumference that may be measured as distance or duration. Call the spatial circumference S, and the temporal circumference T, which is known as the period. Distinguish the spatial and temporal versions of the radius: R, for space, and Q, for time. Then S = 2πR and T = 2πQ. Also, R = Qv, and Q = Ru, with speed, v, and pace, u.

Circular orbits are a convenient entry into Kepler’s and Newton’s laws of planetary motion. Copernicus thought the orbits were circular, and most planetary orbits are in fact nearly circular. We have then three propositions from the perspective of the Sun toward each orbiting planet:

  1. Each planet orbits the Sun in a circular path with radius R in 3D space.
  2. The Sun is at the center of mass of each planet’s orbit.
  3. The speed of each planet is a constant, v.

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