iSoul In the beginning is reality.

Tag Archives: Length & Duration

Matters relating to length and duration in physics and transportation

Motion as a parade

Have you ever seen a parade? Have you ever been in a parade?

parade

A parade is basically a linear (1D) motion. It begins at a point in space and time and ends at a point in space and time. It is planned to progress in a certain order.

The view from the side sees the parade pass by. The parade participants change in time but the location does not. The parade represents the diachronic perspective of motion in time.

The bird’s-eye view from above (as of a camera on a drone) sees the parade as a whole. The time keeps changing but the general location does not. The parade represents the synchronic perspective of motion in time.

The view from a parade participant sees the streetscape pass by. The parade watchers change in space but the chronation does not. The parade represents the diachoric perspective of motion in space.

The view from the plan for the parade sees the parade as a whole. The stance keeps changing but the chronology does not. The parade represents the synchoric perspective of motion in space.

Independent uniform motion

This continues posts here and here.

The extent of a motion is measured by a reference motion, just as a length is measured by a reference length. The reference motion used to measure other motions is a uniform motion. Galileo’s definition of uniform motion is the following:

By steady or uniform motion, I mean one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal. [Galileo’s Two New Sciences, Third Day]

Because equality is a symmetric relation, this could also be expressed as follows:

A steady or uniform motion is one in which the travel times of the moving particle during any equal intervals of space, are themselves equal.

Another way of saying this is that for a uniform motion the intervals of space and the corresponding intervals of time are proportional. That is, a uniform rate of motion is constant.

There are two measures of the extent of a motion, length and duration. Applied to the reference motion, these two measures produce two scalars, a measure of length called stance, and a measure of duration called time. Given a reference starting point, the length since the start point is called the stance, and the duration since the start instant is called the time. Because of the proportionality of uniform motion, given knowledge of stance or time along with the uniform rate, one may deduce the other measure.

The reference motion must also be an independent motion, not dependent on other motions, and it must continue indefinitely, so that any other motion would be at some point simultaneous or simulstanceous with the reference motion. Because of this, any motion may be a function of the reference motion.

This independent, uniform reference motion is commonly represented by a clock, which registers uniform motion continually. Even if the clock’s motion is a uniform circular motion, it represents a uniform linear motion as the numbers increase linearly. The reference motion may equally well be a metreloge, which is a uniform motion that registers length continually. As a clock may be a uniform angular motion whose angles register durations, a metreloge may be a uniform angular motion whose arcs register lengths.

Measures of the target motion may then be considered as a function of one of the reference measures, which acts as a parameter of the target motion. Parametric differential equations and geometry may then be used to represent the course of a motion.

Simultaneity and simulstanceity

Max Jammer’s book Concepts of Simultaneity (Johns Hopkins UP, 2006) describes the significance, meaning, and history of simultaneity in physics. Here are a few excerpts from his Introduction:

… Einstein himself once admitted: “By means of a revision of the concept of simultaneity in a shapable form I arrived at the special relativity theory.” p.3

That not only temporal but also spatial measurements depend on the notion of simultaneity follows from the simple fact that “the length of a moving line-segment is the distance between simultaneous positions of its endpoints,” as Hans Reichenbach … convincingly demonstrated. Having shown that “space measurements are reducible to time measurements” he concluded that “time is therefore logically prior to space.” p. 4-5

P. F. Browne rightly pointed out that all relativistic effects are ultimately “direct consequences of the relativity of simultaneity.” p.5

One might give the dual to the second statement as: That not only spatial but also temporal measurements depend on the notion of simulstanceity follows from the simple fact that “the duration of a moving line-segment is the time interval between simulstanceous chronations of its endpoints. Space is therefore logically prior to time.

In the next chapter, Terminological Preliminaries, Jammer clarifies the relevant concepts. It is ironic that he gives an early example of the metonym “of spatial terms to denote temporal relations that is frequently encountered both in ancient and in modern languages.” (p.9) Space has priority in language.

Read more →

Independent variable dimension

This continues the series of posts, see here.

Let’s begin with Galileo’s figure for uniform motion and uniform accelerated motion:

Falling projectile

Let the horizontal uniform motion be situated in a 2D x, y coordinate system:

2D uniform motion

The dependent uniform acceleration moves in an additional dimension, z, and so has 3D coordinates.

If the independent uniform motion is measured by time, then time has 2D coordinates. The coordinates are proportional to one another, and so may just as well be replaced by a scalar of the signed magnitude. The same can be done if the uniform motion has 3D coordinates. The scalar time is proportional to the common measure of time, with the appropriate rate of motion, it is the same as scalar time.

One can say the same if the independent uniform motion is measured by length, then length has 2D (or 3D) coordinates. This is nothing new, but because of the uniform motion, the coordinate lengths are proportional to one another, and so may just as well be replaced by a single scalar of the signed magnitude. In this way, length becomes a scalar, called here the stance.

Motion ordered by time or stance

This continues posts about Galileo such as here. Look again at this figure from Galileo:

Falling projectile

The horizontal line represents the independent variable, which is the horizontal component of a projectile with an initial speed that falls with uniform acceleration. The independent variable need not be horizontal, so let us represent it vertically with the vertical component of motion:

 

Four parallel lines

The first two lines represent the independent uniform motion but are measured differently, (1) by stance and (2) by time. The second and third lines show the correspondence between the independent variable (time) and the dependent variable (length). The first and fourth lines have the same relation: stance is the independent variable and duration is the dependent variable. Lines 1 and 2 are proportional, as are lines 3 and 4. As the units change, the functional relationship does not: the dependent variable is proportional to a square of the independent variable.

Time is the measure of duration of a uniform motion that is the reference motion, which means it is the independent variable. Similarly, stance is the measure of length of a uniform motion that is the reference motion, in which case it is the independent variable. The dependent variable for stance is duration in three dimensions, whereas for time the dependent variable is length in three dimensions.

Timeframes of reference

A kinematic frame of reference is a mathematical method to determine the position of points in abstract 3D space and scalar time. An inertial frame of reference is a physical method to measure the position of bodies in physical 3D space and scalar time. The latter is often envisioned as three mutually-perpendicular rigid rods attached at a common point, or a lattice of such rigid rods. In addition, there is envisioned a clock at every node of the lattice, which are all synchronized, which requires a method to synchronize them. The common point is called the origin point.

Such a frame of reference assigns coordinates in 3D space and 1D time to every event.

A lattice in all directions

A kinematic timeframe of reference is a mathematical method to determine the position of points in abstract scalar space and 3D time. An inertial timeframe of reference is a physical method to measure the position of bodies in physical 3D space and scalar time. The latter may be envisioned as three mutually-perpendicular rigid monorails attached at a common point, or a lattice of such rigid monorails, with a regular succession of small monorail vehicles traveling at constant velocity. In addition, there is envisioned a metreloge at every node of the lattice, which are all synstancized, which requires a method to synstancize them. The start of monorails leaving the common point is the origin event.

Such a timeframe of reference assigns coordinates in 1D space and 3D time to every event.

monorails

 

 

Places, spaces, and times

Time is like a river that flows on indefinitely, as observed from a place on its bank. The flow of time is downstream. Place does not change in this way but the time keeps changing.

Space is like a river that flows on indefinitely, as observed from a platform floating down the river. The flow of space is upstream, as places on the bank recede from view. Time does not change in this way but the place keeps changing.

Places have spaces between them. Spaces are distances measured as lengths (length of space). Places are also called stations, as in railroad stations, if they are places along a route (stance and station are related etymologically). Spaces are located by the places at their beginning and end points. “What station is it here?” could be asked by a passenger in a train at a stop.

Times have time intervals between them. Time intervals are distances measured as durations (length of time). Times are chronated (positioned) in 3D time. Time intervals are chronated by the times at their beginning and end instants. “What time is it now?” could be asked in many contexts.

Spacetime is a place-based metric. Timespace is a time-based metric.

In classical physics there is a conversion factor between space and time that is adopted as a convention by all observers and is measured by a uniform motion relative to each observer. In relativity physics there is a uniform motion that is absolute, that is, the same as measured by every observer, and functions as a conversion factor between space and time.

From spacetime to space and time

This relates to the post here.

There are three dimensions of motion with two measures of the extent of motion, which makes a total of six metric dimensions of motion. But these six metric dimensions collapse into two structures of one and three dimensions as the conversion factor approaches infinity.

The invariant proper length, , is:

dσ² = dr²dt²/ç² = dr1² + dr2² + dr3² – dt²/ç² = dr² – dt1²/ç² – dt2²/ç² – dt3²/ç² = dr1² + dr2² + dr3² – dt1²/ç² – dt2²/ç² – dt3²/ç².

As the conversion factor, ç, the pace of light, approaches infinity, this becomes

dσ² = dr² = dr1² + dr2² + dr3².

That is, the time coordinates separate from the invariant length, which becomes the Euclidean distance of three dimensional space. Time is left as an invariant scalar called the time.

The invariant proper time, , is:

dτ² = dσ²/c² = dr²/c² – dt² = (dr1² + dr2² + dr3²)/c² – dt² = dr²/c² – dt1² – dt2² – dt3² = (dr1² + dr2² + dr3²)/c² – dt1² – dt2² – dt3².

As the conversion factor, c, the speed of light, approaches infinity, this becomes

dτ² = – dt² = – dt² = – dt1² – dt2² – dt3².

That is, the length coordinates separate from the invariant time, which becomes the Euclidean distime of three dimensional time. Space is left as an invariant scalar called the stance.

The result is that six dimensional spacetime collapses into 3D space with scalar time or 3D time with scalar space.

Terminology contexts

This post continues the one here. While I avoid coining new terms or new definitions, some have been necessary. To have a consistent vocabulary, I try to imagine contexts in which they easily fit.

Some words are simply variations of words in use: distime is like distance; dischronment is like displacement; chronation is like location; vass is like mass; levitation is the opposite of gravitation; and oldtons are the units for rush, analogous to newtons for force. Metreloge is like horologe, which is a clock.

One context is racing. The term pace is used, particularly in running and (bi)cycling to mean the time interval per unit distance, which is the inverse of speed. The direction is ignored or assumed to follow the course of the race so a new term is needed to indicate the vector version of pace. For this I have chosen legerity, which is an old literary term for lightness of movement.

The second context is transport, such as package delivery. Consider an order to expedite a delivery. That means to reduce the time of transport, analogous to acceleration. A package stamped with “RUSH” gets a greater effort to reduce the time of delivery. Rush is analogous to a force applied. To hustle means to apply a rush over a distance, analogous to a force applied over time (which is called impulse). Surge is a rush applied over a dischronment, which is the inverse of work. Reserve is the capacity for surge, which is analogous to energy.

Ratios of length and duration

This post relates to others such as this.

Consider Galileo’s figure (see his Dialogues Concerning Two New Sciences, tr. Crew & De Salvio p.249 Fig. 108 or Drake’s translation p.221):

Falling projectile

A projectile moves with uniform velocity horizontally to the left and begins to descend at point b. Galileo used the sequence a-b-c-d-e to represent time and the sequence b-o-g-l-n to represent the height of the projectile above the Earth. The sequence b-i-f-h represents the parabolic path of the falling projectile.

Any uniform motion can serve as a reference motion. There are two uses of a reference variable: (i) as a parametric variable, or (ii) as a measurement variable. A parametric variable is an independent variable that provides ordered input for any dependent variable. A measurement variable is a variable that is dependent on the independent variable being measured. In the figure above the parametric variable is the time (duration) of the uniform motion on the horizontal axis, and the measurement variable is the height (length) of the uniform acceleration on the vertical axis.

Combine this with the two measures of motion, length and duration, and there are four possible cases: (1) independent duration variable with dependent length variable; (2) independent length variable with dependent duration variable; (3) independent length variable with dependent length variable; and (4) independent duration variable with dependent duration variable.

The figure above is an example of case (1). Its complement is case (2). Cases (3) and (4) include only one measure, length or duration, and so cannot express a rate of motion. Galileo expresses case (1) as a proportion between ratios of the variables at different times: s1 : s2 :: t12 : t22, which avoids combining different units in a single ratio, consistent with Eudoxian proportionality.

Consider case (2) in which the independent variable is length. This variable is a stanceline for locating other motions, which is like a timeline except that it expresses an independent length as the order parameter. The dependent reference variable in this case is duration, which measures any independent variable, in this case projectile height. This could be expressed as a proportion between ratios of the variables at different times: t1 : t2 :: s12 : s22, avoiding different units in a single ratio.

Case (1) enables multiple length variables dependent on one independent variable, the timeline. Case (2) enables multiple duration variables dependent on one independent variable, the stanceline. Rates of motion in case (1) are in units of the independent timeline, which is duration. Rates of motion in case (2) are in units of the independent stanceline, which is length.