iSoul In the beginning is reality

Category Archives: Space & Time

Explorations of multidimensional space and time with linear and angular motion.

Motion vs. movement

The English words motion and movement are similar. They both have to do with “changing position or going from one place to another.” (Collins English Dictionary)

Then what’s the difference? Here are a few ways of putting it:

motion is used to describe physical properties, while movement is used to describe the qualities of motion. Ref.

motion doesn’t always imply a purpose, and movement usually does. Ref.

The difference is very fine. I would say that movement is déplacement d’un lieu à un autre [displacement from one place to another] whereas motion is le fait de ne pas rester immobile [not to stand still]. But usage and context are crucial. Ref.

People may not be consistent about it but for the purposes here they can be distinguished. Motion is the general term in kinetics, the study of motion. It says nothing about the purpose of a motion, or its origin and destination. Something just happens to change place.

However, movement includes some purpose, some origin and destination. A movement is a complete motion, from beginning to end. So movement would be preferred in the arts and social sciences and motion in the natural sciences.

Physics studies motion. Transportation studies movement. They may both speak about something changing position but there is a different perspective.

A movement is an entity, a thing, not just a change as a motion is. A motion can be studied abstractly but a movement is not fully abstract because it is an entity.

A body has its motion and a movement has its figure. A body is flesh-and-blood 3D, with motion only adding a thin 1D time perspective. A movement has 3D animation and life, with a figure only adding a thin 1D space perspective.

Bodies and motions

For kinematics and dynamics, one can begin with a body – something that takes space – and apply a motion to it. Or one can begin with a motion – something that takes time – and apply it to a body. In either case the result is a body and a motion – either a body with a motion or a motion with a body.

That, in a nutshell, is the difference between the spatiocosm and the tempocosm, that is, 3D space with 1D time and 3D time with 1D space. They are two ways of saying the same thing with a different emphasis.

Yes, a motion can be a “thing” – an entity of its own – the motion around a race track, for example. Applied to different contestants such a motion produces different timings. That is what the race is all about.

A body applies to different motions, too – a body such as a projectile, for example. When launched in different ways such a body travels different lengths, as in a contest for the longest length.

Compare the von Neumann computer architecture in which instructions and data share the same address space. Whether an address represents data or instructions depends only on the interpretation.

Different situations call for different perspectives. A race against time calls for a tempocosmic vew. A contest for distance calls for a spatiocosmic view.

Is there a combined perspective? Yes, in a relativistic sense, but it collapses to one or the other perspective as soon as a measurement is taken. If spatial position is measured, it’s a spatiocosmic view. If time and direction are measured, it’s a tempocosmic view.

Space-time duality

Space and time are dual concepts. They are complementary to one another as an inverse binary symmetry. They go together as space with time in classical physics or time with space or spacetime as in relativistic physics.

Parallel terms:

Space Time
3D space +1D time 3D time + 1D space
vector space + scalar time vector time + scalar space
spatiocosm tempocosm
reference frame reference timeframe
space position time position
particle moticle
a body with a motion a motion with a body
waypoint instant
odologe clock

For further details see here and other posts on this blog.

Linear, radial, and scalar space and time

In 3D space the distance between points is the magnitude of the displacement vector, which is a scalar length. In 3D time the duration between instants (time points) is the magnitude of the distimement vector, which is a scalar time.

Radial space is space with radial coordinates, 2D polar or 3D spherical, in which the distance from or to a point is the magnitude of the radial displacement. Radial time is time with radial coordinates, 2D polar or 3D spherical, in which the duration to an instant is the magnitude of the radial distimement. The radial space and time are “as the crow flies,” with the imaginary crow flying at a constant rate.

The distance between two points along a path (or curve) is the arc length (scalar space) between them. The duration between two instants along a path in time is the arc time (scalar time) between them. Each of these may be conceived of as one rectilinear dimension.

A measuring wheel can measure the arc length along the space path between points. An odologe can measure the arc time along the space path between events. An odologe is equivalent to a measuring wheel operating at a fixed rate. A stopwatch with a conversion speed is effectively an odologe.

The result of a measuring wheel operated throughout a region or network is a (spatial) linear referencing system (LRS or SLRS), which produces a set of physical wayposts or conceptual waypoints.

The result of an odologe operated throughout an event region or network is a temporal linear referencing system (TLRS), which produces a set of physical eventposts or conceptual eventpoints/eventicles. A visual or audible signal synchronized with an event would be an example of an eventpost. This might be a train conductor’s “all aboard!” or a signal at the stops of an automated transit system.


An odologe (o′∙do∙loje) a constant-rate length-measuring device symmacronized with a common waypoint. It is a new coinage from the Greek odo(s), way/path + (horo)loge, clock. In short, it is a clock that shows length or angle instead of time.

The simplest odologe takes time from a clock and multiplies it by a conversion speed to produce a length or angle. This is commonly done in relativity with c, the speed of light: ct equals time multiplied by the speed of light, resulting in a length. A device which output such a length would be an odologe.

Given a starting point the Moon could be considered an odologe that travels at the rate of 3,683 kilometers per hour, since that is its speed around the Earth. The Earth itself could be considered an odologe that travels at its orbital rate of 107,000 km/h. Another odologe is the distance of the Pioneer 10 spacecraft from the Earth and Sun, which is tracked here.

A circular analog clock is also an angular odologe of the angle made by each hand, as below:

Clock angles

Each minute is represented by 1/60 of a 360° circle, or 6° of arc. Each hour is represented by 1/12 of a circle, or 30° of arc.

There are 24*60 = 1440 minutes in a day, which equates to 360 = 1440/4 degrees, so the apparent mean speed of the Sun along the ecliptic is one degree every four minutes. The Sun provides an approximate angular odologe.

A kind of virtual odologe comes from using the typical speed for a vehicle in a metropolitan area to convert travel time into travel distance. So, for example, one might estimate that a traveler has progressed 20 kilometers since they have been gone a half hour and the typical speed is 40 km/h.

Another virtual odologe is an app that displays an odometer that increases at a constant rate, as illustrated below.

Why space and time are not different

Many differences are proposed between space and time. This post briefly indicates how all of them are a matter of convention, and so not real. For details, consult posts on this blog.

(1) There are three space dimensions but only one time dimension.

Directionality can be associated with either length (distance) or time (duration). 3D time is as legitimate as 3D space.

(2) The time is always changing (“flowing”) but space position need not change.

Time seems to “flow” because of its association with continual change, as of a river. But change can just as well be associated with length. Something floating down a river continually changes it length of travel. So a continual increase of length can be associated with change.

(3) There is an “arrow” of time, but not of space.

As in (2) above, if length is associated with change, then there is an “arrow” of length. Compare a vehicle’s odometer, which never decreases. An odologue shows length always increasing, like a clock.

(4) Time has a past, present, and future, but space does not.

Space includes where one was in the past, where one is now, and where one will be in the future.

(5) Travel takes place in space in all directions, but only in one direction in time.

Travel takes place in time in all directions since there is a travel time for each direction of travel. Space may be associated with the travel distance, which has one direction.

(6) Entropy grows with time, but not with space.

One can equally well define an entropy of space and show that it tends to increase with the distance of motion.

(7) Causality is ordered in time, not in space.

Causality is ordered in space as in time. What happens in one place is causally related to what happens in adjoining places, and on to other places.

(8) Causally connected events are timelike or lightlike, but cannot be spacelike.

In a 3D time and 1D space context, the properties of timelike and spacelike are interchanged.

Ten meanings of time

Carlo Rovelli’s “Analysis of the Distinct Meanings of the Notion of “Time” in Different Physical Theories” (Il Nuovo Cimento B, Jan 1995, Vol 110, No 1, pp 81–93) describes ten distinct versions of the concept of time, which he arranges hierarchically. Here are excerpts from his article:

We find ten distinct versions of the concept of time, all used in the natural sciences, characterized by different properties, or attributes, ascribed to time. We propose a general terminology to express these differences. p.81

… our aim is to emphasize the general fact that a single, pure and sacred notion of “Time” does not exist in physics. p.82

The real line is a traditional metaphor for the idea of time. Time is frequently represented by a variable t in R. The structure of R corresponds to an ensemble of attributes that we naturally associate to the notion of time. These are the following:
a) The existence of a topology on the set of the time instants, namely the existence of a notion of two time instants being close to each other, and the characteristic “one dimensionality” of time;
b) The existence of a metric. Namely the possibility of stating that two distinct time intervals are equal in magnitude. We denote this possibility as metricity of time.
c) The existence of an ordering relation between time instants. Namely, the possibility of distinguishing the past direction from the future direction;
d) The existence of a preferred time instant, namely the present, the “now”. p.83

In the natural language, when we use the concept of time we generally assume that time is one-dimensional, metrical, external, spatially global, temporally global, unique, directed, that it implies a present, and that it allows memory and expectations. The concept of time used in Newtonian physics is one-dimensional, metrical, external, spatially global, temporally global, unique, but it is not directed and it does not have a present. In thermodynamics, time has the additional property of being directed. Proper time along world line in general relativity is one-dimensional, metrical, temporally global but it is not external, not spatially global, not unique; on the other side, the time determined by a matter clock is one-dimensional, metrical, but not temporally global, an so on. p.87

… the notion of present, of the “now” is completely absent from the description of the world in physical terms. This notion of time can be described by the structure of an affine line A. p.88

… our list does not include the possibility of considering a non-metric but directional notion of time. p.89

Table I. [without the fourth column]

Time concept Attributes Example
time of natural language memory brain
time with a present present biology
thermodynamical time directional thermodynamics
Newtonian time uniqueness Newton mechanics
special relativistic time being external special relativity
cosmological time space global proper time in cosmology
proper time time global world line proper time
clock time metricity clocks in general relativity
parameter time 1-dimensional coordinate time
no time none quantum gravity

… our hypothesis concerning time is that the concepts of time with more attributes are higher-level concepts that have no meaning at lower levels. p.91

If this hypothesis is correct, then we should deduce from it that most features of time are genuinely meaningless for general systems. p.91

… we suggest that the very notion of time, with any minimal characterization, is likely to disappear in a consistent theory that includes relativistic quantum-gravitational systems. p.91

… the concept of time, with all its attributes, is not a fundamental concept in nature, but rather that time is a progressively more specialized concept that makes sense only for progressively more special systems. p.92

Space and time involution

J. C. C. McKinsey, A. C. Sugar and P. Suppes (hereafter MSS) wrote “Axiomatic foundations of classical particle mechanics”, (Journal of Rational Mechanics and Analysis, v.2 (1953) p.253-272), which is also described in Suppes’ Introduction to Logic (Van Nostrand, New York, 1957), pp.291-322 (see here). It is only a partial axiomatization of Newtonian mechanics but is sufficient to present an involution of mechanics below.

An involution in mathematics is a function that is its own inverse, which means if it is repeated the output is the input, that is, f(f(x)) = x. The involution here is the interchange of spatial and temporal quantities along with the inversion of mass. We start with MSS:

MSS system has six primitive notions: P, T, m, s, f, and g. P and T are sets, m is a real-valued unary function defined on P, s and g are vector-valued functions defined on the Cartesian product P × T, and f is a vector-valued function defined on the Cartesian product P × P × T. Intuitively, P corresponds to the set of particles and T is to be physically interpreted as a set of real numbers measuring elapsed times (in terms of some unit of time, and measured from some origin of time). m(p) is to be interpreted as the numerical value of the mass of p ∈ P. sp(t), where tT, is a 3-dimensional vector which is to be physically interpreted as the position of particle p at instant t. f (p, q, t), where p, qP, corresponds to the internal force that particle q exerts over p, at instant t. And finally, the function g(p, t) is to be understood as the external force acting on particle p at instant t. (Anna & Maia p,9)

We define MSS´ by the following involution: interchange P and Q; p and q; T and S; m and n=1/m; s and t; f and k; g and ; particle and moticle; instant and point; mass and vass. Then the explanation is:

MSS´ system has six primitive notions: Q, S, n, t, k, and . Q and S are sets, m is a real-valued unary function defined on Q, t and  are vector-valued functions defined on the Cartesian product Q × S, and k is a vector-valued function defined on the Cartesian product Q × Q × S. Intuitively, Q corresponds to the set of moticles and S is to be physically interpreted as a set of real numbers measuring scalar space (in terms of some unit of length, and measured from some origin point). n(q) is to be interpreted as the numerical value of the vass of qQ. tq(s), where sS, is a 3-dimensional vector which is to be physically interpreted as the position of moticle q at point s. k(q, p, s), where q, pQ, corresponds to the internal surge that moticle p exerts over q, at point s. And finally, the function (q, s) is to be understood as the external surge acting on moticle q at point s.

The corresponding axioms are as follows:

A1 Q is a non-empty, finite set.
A2 S is an interval of real numbers.
A3 If qQ and sS, then tq(s) is a 3-dimensional vector (tq(s) ∈ℜ³) such that d²tq(s)/ds² exists.
A4 If qQ, then n(q) is a positive real number.
A5 If p, qQ and sS, then k(p, q, s) = −k(q; p; s).
A6 If p, qQ and sS, then tq(s) × k(p, q, s) = –tp(s) × k(q, p, s).
A7 If p, qQ and sS, then n(q) d²tq(s)/dt² = ΣpQ k(p, q, s) + ℓ(q, s).

These axioms generate a dual to Newtonian mechanics. A5 corresponds to a weak dual version of Newton’s Third Law: to every surge there is always a countersurge. A6 and A5, correspond to the strong dual version of Newton’s Third Law. A6 establishes that the direction of surge and countersurge is the direction of the line defined by the coordinates of moticles p and q. A7 corresponds to the dual of Newton’s Second Law.

MSS show that mass is independent of the remaining primitive notions of their system. Because of this, its dual could be defined differently. It was thought best to take the inverse of mass, called vass, for the involution.

Centers of motion

Bodies in space-time orbit by gravitation around their barycenter, the center of mass. The word barycenter is from the Greek βαρύς, heavy + κέντρον, center. The barycenter is one of the foci of the elliptical orbit of each body.

For the two-body case let m and M be the two masses, and let r and R be vectors to m and M respectively. Then the center of mass or barycenter is

(mr + MR) / (m + M).

Define the reduced mass μ = mM/(m + M). Then the orbit is as if the orbiting body has reduced mass μ and there is a stationary central body with mass equal to the total mass (m + M). That is, the two bodies mutually orbit the center of mass.

Let’s reconsider the orbit in relation to the vasses, the mass inverses, orbiting by levitation. For the two-movement case let  and L be the two vasses, and let r and R be vectors from an origin to and L respectively. Then the center of vass is

(ℓr + LR)/( + L) = (Mr + mR) / (m + M).

Define the reduced vass Λ = ℓL/( + L). Then the orbit is as if the orbiting movement has reduced vass λ and there is a stationary central movement with vass equal to the total vass ( + L). That is, the two movements mutually orbit the center of vass.

The result for vass is the same except that the roles of the movements are reversed. One could call the center of vass the elaphracenter after ελαφρά, light (weight) + κέντρον, center.

Speed vs. velocity

For some background, see here and here.

Velocity is defined as: “The time rate of change of position of a body; it is a vector quantity having direction as well as magnitude.” Speed is defined as: “The time rate of change of position of a body without regard to direction; in other words, the magnitude of the velocity vector.” (McGraw-Hill Dictionary of Physics, 3rd ed.)

However, it’s not that simple. A common example shows the problem:

When something moves in a circular path (at a constant speed …) and returns to its starting point, its average velocity is zero but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle. This is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Wikipedia

So the average speed is not the magnitude of the velocity (which is zero in this case) but something else – the travel distance divided by the travel time.

The question is whether the speed over a finite interval should be the magnitude of the displacement divided by the time interval or the arc length divided by the time interval (i.e., the integral of the norm of the velocity function over the time interval). The answer should be the latter, although the former is implied by the common definition of speed.

It is better to define speed as the ratio of the arc length (travel distance) divided by the arc time (travel time). In short, speed is that which is measured by a speedometer.