iSoul In the beginning is reality.

# 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.

# 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.

# Traffic flow in time and space

The following is based on Wilhelm Leutzbach’s Introduction to the Theory of Traffic Flow (Springer, 1988), which is an extended and totally revised English language version of the German original, 1972, starting with page 3 (with a few minor changes):

I.1 Kinematics of a Single Vehicle

I.1.1 Time-dependent Description

I.1.1.1 Motion as a Function of Time

Given any trajectory then, in the time-dependent case:

x(t) = distance: as a function of time [m];

f(t) = v(t) = dx/dt = speed: as a function of time [m/s];

a(t) = dv/dt = d²x/dt² = acceleration: as a function of time, the change of speed per unit time [m/s²];

If the initial conditions are denoted, respectively, by t0, x0, v0, a0, etc., the following equations of motion result, with integrals from t0 to t:

x(t) = x0 + ∫ v(t) dt   (I.1)

v(t) = v0 + ∫ a(t) dt   (I.2)

x(t) = x0 + ∫ v0 dt + ∫∫ a(t) dtdt   (I.3)

etc.

# Dual Galilean transformation

The Galilean transformation is based on the definition of velocity: v = dx/dt, which for constant velocity leads to

x = ∫ v dt = x0 + vt

So for two observers at constant velocity in relation to each other we have

x′ = x + vt

with their time coordinates unchanged: t′ = t if their origins coincide.

The dual Galilean transformation is based on the definition of lenticity: w = dt/dx, which for constant lenticity leads to

t = ∫ w dx = t0 + wx

So for two observers at constant lenticity in relation to each other we have

t′ = t + wx

with their length coordinates unchanged: x′ = x if their origins coincide.

These transformations reinforce the proposition that time is not necessarily the independent variable, and so is best understood as measured by a stopwatch rather than a clock.

# Changing coordinates for the wave equation

The following is based on section 3.3.2 of Electricity and Magnetism for Mathematicians by Thomas A. Garrity (Cambridge UP, 2015). See also blog post Relative Motion and Waves by Conrad Schiff.

The classical wave equation is consistent with the Galilean transformation. Reflected electromagnetic waves are also consistent with classical physics using the dual Galilean transformation, in which linear location is the independent variable. The dual Galilean transformation for motion in one dimension is: x′ = x; t′ = twx = tx/v, where w = 1/v is the relative pace of the moving observer, which is equivalent to their inverse speed.

Suppose we again have two observers, A and B. Let observer B be moving at a constant pace w with respect to A, with A’s and B’s coordinate systems exactly matching up at location x = 0. Think of observer A as at rest, with coordinates x′ for location and t′ for time, and of observer B as moving to the right at pace w, with location coordinate x and time coordinate t. If the two coordinate systems line up at location x = x′ = 0, then the dual Galilean transformations are

x′ = x and t′ = t + wx,

or equivalently,

x = x′ and t = t′wx.

Suppose in the reference frame for B we have a wave y(x, t) satisfying the wave equation

$\frac{\partial^2&space;y}{\partial&space;x^2}-k^{2}\frac{\partial^2&space;y}{\partial&space;t^2}=0.$

# Michelson-Morley experiment

This post relates to a previous post here.

The Michelson-Morley experiment is a famous “null” result that has been understood as leading to the Lorentz transformation. However, an elementary error has persisted so that the null result is fully consistent with classical physics. Let us look at it in detail:

The Michelson-Morley paper of 1887 [Amer. Jour. Sci.-Third Series, Vol. XXXIV, No. 203.–Nov., 1887] explains it using the above figures:

# Temporo-spatial light clock

This post builds on the post about the Michelson-Morley experiment here.

One “Derivation of time dilation” (e.g., here) uses a light clock, pictured below:

The illustration on the left shows a light clock at rest, with a light beam reflecting back and forth between two mirrors. The distance of travel is set at the beginning with the separation of the mirrors. It is like a race: one lap is a round-trip between the two mirrors. So the independent variable is distance, and the dependent variable is time. This is key to properly understanding the experiment.

In the stationary frame the round trip longitudinal distance between the two mirrors is 2L, the speed is c, and the time of one cycle is T = 2L/c.

The illustration on the right shows an observer moving with lenticity w transversely to the light clock. In this case there are two components of motion: the longitudinal axis and the transverse axis. These components are independent of one another since they are in different dimensions. The transverse axis is the same as the stationary case above: the total distance is 2L and the total time is 2L/c. The mean speed is 2cT/(2T) = c.

The longitudinal axis is simply the Galilean transform: t′ = t + wx since motion in different dimensions is independent. If the light clocks coincide at t = 0, this is t′ = wx, which is what was given.

Thus the distance, time, and mean speed are the same for both observers and independent of their relative velocity, which is what the Michelson-Morley experiment found.

# Michelson-Morley re-examined

There are many expositions of the famous Michelson-Morley experiment (for example here) but they all assume the independent variable is time, which is not the case. As we shall see, distance is the independent variable, and so the experiment is temporo-spatial (1+3). Let us examine the original experiment as it should have been done:

The configuration diagrammed above is as follows: the apparatus is presumed to travel with pace w relative to the aether. In it a light source travels with pace k = 1/c to a beam splitter, whereupon part of it travels a distance L longitudinally and is reflected back, whereas another part travels a distance transversely and is reflected back. Part of the light is sent to an observer, who looks for an interference pattern.

Since the distance L is fixed, distance is the independent variable. In the stationary frame the round trip longitudinal distance between the beam splitter and the mirror is 2L. Let the time that light travels longitudinally from the beam splitter to the mirror be T1, and let the time for the return be T2. Then

# Principle of relativity

The relativity of uniform motion was stated by Galileo in the 17th century, though it was known to Buridan in the 14th century. Galileo’s statement of the principle of relativity is in terms of ships in uniform motion:

… so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still [Galileo Galilei, Dialogues Concerning the Two Chief World Systems (February 1632), Stillman Drake tr. (University of California Press, Berkeley, 1962, pp 186-8.]

This has been applied to constant speeds or zero accelerations, but it could just as well be applied to constant paces or zero retardations. In any case, if the speed is constant, so is its inverse, the pace. Let’s see how this operates.