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With and between independent variables

This post continues the previous post here on independent and dependent variables.

The selection of a physical independent variable (or variables) applies to a context such as an experiment. Within that context all other variables are, at least potentially, dependent on the independent variable(s) selected. Functions with the physical independent variable as a functional independent variable have a common independent variable and form a kind of closed system.

Nevertheless, an independent variable in one context may be the same as a dependent variable in another context. The functions in both of these contexts may be related. That is, there may be relations between the functions of one independent variable in one context and the functions of another independent variable in another context.

For example, speed (the time speed) is defined as the change in distance per unit of time with time as the independent variable. This appears to be equal to the reciprocal of the pace, that is, the change in distance per unit of time with distance as the independent variable. This is called the space speed, spot speed, or inverse speed. We could call the space speed the quasi-speed, since it seems like the speed (the time speed).

However, these two speeds are not the same. One difference is how the speeds are aggregated or averaged: the speed uses arithmetic addition and averaging but the quasi-speed uses subcontrary (or harmonic) addition and averaging. The average of speeds v1 and v2 is (v1 + v2)/2. The average of quasi-speeds v1 and v2 is 2/(1/v1 + 1/v2).

A quasi variable is a dependent variable in the denominator as if it were an independent variable. If it is a first-order variable, one can take the reciprocal to put it into its proper dependent position, the numerator. Second and higher order variables need to use their inter-functional relation.

Distance as an independent variable

A previous post here gives “Motion as a Function of Distance” in which distance is a functional but not a physical independent variable. So distance is an independent variable of the inverse of the function of motion as a function of time. But this functional independence does not change the original independent variable of time.

In this post distance is the physical independent variable. At first it will be the functional independent variable, too, but then time will be the functional independent variable. We will see that the physical independent variable remains and does not allow the change of function to change its character.

We begin at Leutzbach’s section I.1.2, Distance-dependent Description, with one important change, the distance-dependent functions are the same (up to a conversion factor) as the time-dependent ones:

Distance-dependent Description

We define a new parameter as a function of distance, which is analogous to speed. This means that motion is represented in a t-x-coordinate system. This new parameter pace or lenticity (vector form) equals the change in time per unit distance [s/m] as a function of distance is defined as

f(x) = w(x) = dt(x)/dx

by analogy with

f(t) = v(t) = dx(t)/dt

with time, or duration, t, a function of distance t(x). The function f is the same in both cases, with a conversion constant between x and t supressed or equal to one.

Similarly

b(x) = dw(x) /dx = d²t(x)/dx²

by analogy with

a(t) = dv(t)/dt = d²x(t)/dt².

Retardation is the name given to b(x). The functions v(t) and w(x) are functionally the same, up to a conversion factor. They are not inverse functions, although pace and speed are reciprocals. They are different versions of the same functional relations. In each case the denominator represents the independent variable, which is the standard representation.

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Independent and dependent variables

There are two kinds of independent variables: (1) functional independent variables, and (2) physical independent variables. To avoid confusion an independent variable it is standard that a variable be of both kinds, since being of one kind does not imply being of the other kind.

A physical independent in an experiment remains the independent variable throughout the experiment. A function with a functionally independent variable that is also a physical independent variable remains a physical independent variable even if the function is changed into one with a different functional independent variable, as a non-standard case.

There are two ways of expressing an independent variable: (1) its value is fixed or controlled separately from measuring any dependent variable, or (2) its values are a pre-defined sequence of values within the experiment, but they may be imagined to continue indefinitely beyond the experiment. Once the independent variable is determined, then one or more dependent variables can be measured in relation to it.

Examples of the first way are specifying a time interval and then taking a measurement for the specified interval of time. One could also specify a distance, and then measure the elapsed time. It is important to note that if the distance is independent, it is absolute within the experiment, whereas time is relative.

The second way commonly makes time the independent variable, which is absolute within the experiment. Space in the form of distances (spaces) can also be the independent variable, which is called stance so that stance intervals are distances. In this case stance is absolute within the experiment, whereas time is relative.

If time is the independent variable, the universe of the experiment is spatio-temporal (dimensionally 3+1). If space (stance) is the independent variable, the universe of the experiment is temporo-spatial (dimensionally 1+3).

The independent variable is in the denominator of a rate. Otherwise, the rate must be inverted. For example, the spatio-temporal rate of motion is speed or velocity; the temporo-spatial rates are pace or lenticity. To add vectors one must have the independent variable in the denominator. So to add velocity or lenticity one adds them as vectors. However, velocity in a temporo-spatial context requires one must invert the velocity before adding. Similarly, lenticity in a spatial-temporal context requires one must invert the lenticity before adding. This is the reason that the harmonic mean is used to average velocities in a temporo-spatial context.

If one maps the variables, then the independent variable should be the background map that the dependent variables are indicated on. For example, a map of the local geography forms the background for indicating the location of various dependent variables in the foreground. A temporo-spatial map has a time scale in the background with the chronation of various dependent event variables indicated on the foreground.

Temporo-spatial rest

Speed is the travel distance per unit of duration (or time interval). Rest in space is a speed of zero. That is, there is no change in location per unit of time. A body does not change location (relative to an inertial observer) while time continues.

But rest in time seems different. It cannot be zero pace because that would mean it takes no time to go a positive distance, right? No, that is not what zero pace means.

Pace is the travel time per unit of distance (or stance interval). Time is the dependent variable and distance is the independent variable.

Consider a race that is about to begin. The runners are in place waiting for the signal to start. The official timer is set to begin. In terms of motion, the runners are at rest with speed of zero. They are not making any distance, but time continues as usual.

What is the pace of the runners in that case? There is no change on the official timer. But the stance continues as usual. For example, if stance is related to the distance from the Sun of a Voyager spacecraft (see here), it continues to increase as usual.

A map with a time scale shows a point for a pace of zero. Despite the distance made by an odologe, a body with a pace of zero remains in the same place in time. It is at rest in time.

Runner about to start

What about an infinite value for pace in time? The Galilean transformation implicitly has an infinite speed of information in space, which makes information spatially ubiquitous since it travels an infinite distance in a finite time. The symmetric Galilean transformation implicitly has an infinite pace of information in time, which makes information temporally ubiquitous since it takes an infinite time to travel a finite distance.

Space and time as frames

An observer is a body capable of use as a measurement apparatus. An inertial observer is an observer in inertial motion, i.e., one that is not accelerated with respect to an inertial system. An observer here shall mean an inertial observer.

An observer makes measurements relative to a frame of reference. A frame of reference is a physical system relative to which motion and rest may be measured. An inertial frame is a frame in which Newton’s first law holds (a body either remains at rest or moves in uniform motion, unless acted upon by a force). A frame of reference here shall mean an inertial frame.

A rest frame of observer P is a frame at rest relative to P. A motion frame of observer P is a frame in uniform motion relative to P. Each observer has at least one rest frame and at least one motion frame associated with it. An observer’s rest frame is three-dimensional, but their motion frame is effectively one-dimensional, that is, only one dimension is needed.

Space is the geometry of places and lengths in R3. A place point (or placepoint) is a point in space. The space origin is a reference place point in space. The location of a place point is the space vector to it from the space origin. Chron (3D time) is the geometry of times and durations in R3. A time point (or timepoint) is a point in chron. The time origin is a reference time point in chron. The chronation of a time point is the chron vector to it from the time origin.

A frame of reference is unmarked if there are no units specified for its coordinates. A frame of reference is marked by specifying (1) units of either length or duration for its coordinates and (2) an origin point. A space frame of observer P is a rest frame of P that is marked with units of length. A time frame of observer P is a motion frame of P that is marked with units of duration.

Speed, velocity, and acceleration require an independent motion frame. Pace, lenticity, and retardation require an independent rest frame. These independent frames are standardized as clocks or odologes so they are the same for all observers.

Let there be a frame K1 with axes a1, a2, and a3, that is a rest frame of observer P1, and let there be a motion frame K2 with axes 1, 2, and 3, that is a motion frame of P1 along the coincident a1-a´1 axis. See Figure 1.

Two frames

Figure 1

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Proper and improper rates

The independent quantity in a proper rate is the denominator. The independent quantity in an improper rate is the numerator. If a rate is multiplied by a quantity with the units of the independent quantity and the result has the units of the dependent quantity, it is proper. Otherwise, it is improper.

A proper rate becomes improper if the proper rate is inverted. An improper rate becomes proper if the improper rate is inverted. If two or more improper rates are added, each must first be inverted. The result of adding proper rates must be inverted again to return to the original improper rate. This is subcontrary addition:

\frac{b}{a_{1}}+\frac{b}{a_{2}} \Rightarrow \left (\frac{a_{1}}{b}+\frac{a_{2}}{b} \right)^{-1}

If the addition of improper rates is divided by the number of addends so that it is the average or arithmetic mean of the inverted rates, then the result inverted is the harmonic mean:

\frac{1}{2}\left (\frac{b}{a_{1}}+\frac{b}{a_{2}} \right ) \Rightarrow \left ( \frac{1}{2} \left (\frac{a_{1}}{b}+\frac{a_{2}}{b} \right ) \right)^{-1}

Time speed is the speed of a body measured by the distance traversed in a known time, which is a proper rate because the independent quantity, time, is the denominator. Space speed is the speed of a body measured by the time it takes to transverse a known distance, which is an improper rate because the independent quantity, distance, is the numerator. Space speeds are averaged by the harmonic mean and called the space mean speed. The time mean speed is the arithmetic mean of time speeds.

Velocity normalized by the speed of light is proper because the invariant speed of light is independent. The speed of light divided by a velocity is improper and must be added as subcontraries. Lenticity normalized by an hypothesized maximum pace is proper, but if the lenticity is divided by the pace of light, it is improper.

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Subcontrary conversion of space and time

As noted here, there are two kinds of mean rates: the time mean and the space mean. If the denominators have a common time interval, the time mean is the arithmetic mean and the space mean is the harmonic mean. If the denominators have a common space interval (stance), the space mean is the arithmetic mean and the time mean is the harmonic mean.

For example, light reflected back from a mirror at known distance forms two successive trips whose mean rate is the space mean pace. Several measurements with the same apparatus have a time mean pace. The mean speed is the inverse of the mean pace, which is a kind of subcontrary of speed.

The general principle is that quantities with independent time (such as velocity) and a common time interval use ordinary algebra but such quantities with a common space interval use subcontrary algebra. Alternately, quantities with independent space (stance) such as lenticity and a common space interval use ordinary algebra but such quantities with a common time interval use harmonic algebra.

In other words, quantities over the same interval with independent variables use ordinary algebra but quantities with different independent variables use subcontrary algebra to convert between them.

For example, addition of velocities with a common time interval use ordinary vector addition (e.g., u + v) but addition of velocities with a common space interval use subcontrary vector addition (e.g., ((1/u) + (1/v))−1 ≡ ((u+v)/u·v)−1 with u, v, u·v, u+v0.

The relativity parameter γ is based on a (3+1) spatial frame. The parameter γ in a temporal frame with a common time interval (k ≡ 1/c and ≡ 1/v) is:

γ² = (1 − v·v/c²)−1 ⇒ (1 − ℓ·ℓ/k²)−1.

The parameter γ in a temporal frame with a common space interval (stance) is:

γ² = (1 − v·v/c²)−1 ⇒ H(1 − ℓ·ℓ/k²)−1 = ((1 − k²/ℓ·ℓ)−1)−1 = (1 − k²/ℓ·ℓ) ≡ (1 − v·v/c²) = 1/γ².

Interchanging space and time

The space-time exchange invariance, as stated by J. H. Field (see here) has an implicit second part. In addition to (1) the exchange (or interchange) of space and time coordinates, there is (2): the exchange of linear and subcontrary algebra for ratios. Subcontrary algebra is described here.

This is seen in the different averaging methods for velocities that differ spatially vs. velocities that differ temporally. If two vehicles take the same route, their average velocity is their arithmetic mean (u + v)/2, but if one vehicle has velocity u going and velocity v returning, then the average velocity is their harmonic mean 2/(1/u + 1/v). However, if one vehicle has pace u going and pace v returning, then the average pace is their arithmetic mean, but if two vehicles take the same route, their average pace is their harmonic mean.

Space and time are related to each other as covariant and contravariant components. If space is covariant, then time is contravariant, and if time is covariant, then space is contravariant.

The equations of space-time (3+1) and time-space (1+3) physics are symmetric to one another with the interchange of space and time dimensions. The equations of spacetime (4D) physics are self-symmetric. The interchange of space and time dimensions produces equivalent 4D equations.

To interchange the space and time coordinates, take these steps: For the equations of classical physics, (1) ensure either space or time is a parameter, (2) interchange one dimension with the parameter, and (3) expand the single dimension into three dimensions. For the equations of relativistic physics, (1) ensure there is a symmetry between space and time dimensions, (2) interchange one space and time dimension but leave dimensionless quantities unchanged, and (3) expand the single dimension into three dimensions.

These steps reflect the difference between Galileo’s and Einstein’s relativity. Galileo transforms one frame into another frame but does not combine frames as Einstein’s does. For example, Einstein requires all frames to have the same orientation, but Galileo accepts frame-specific orientations such as the right-hand rule.

The Galilean transformation represents the addition and subtraction of velocities as vectors. The dual Galilean transformation represents the addition and subtraction of lenticities as vectors. The Lorentz transformation represents the combination of Galilean and dual-Galilean transformations, as previously shown.

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 distance keeps changing but the chronology does not. The parade represents the synchoric perspective of motion in space.

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.