iSoul In the beginning is reality.

Category Archives: Space & Time

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

Space Time Speed Pace and Mean

Legend: x-axis coordinate, x; length interval, Δx; unit of length, ; t-axis coordinate, t; time interval, Δt; unit of time, .

For scalars:

Rates Speed/Pace Symbols Space Time
Ordinary Time speed Δx(t)/ measure length/distance/ displacement given period/unit
Alternate Space speed t(x) given length/distance/unit measure interval/period /dischronment
Inverse Space pace Δt(x)/ given length/distance/unit measure interval/period /dischronment
Alt. Inverse Time pace x(t) measure length/distance/ displacement given period/unit
Point
Ordinary Instantaneous speed dx(t)/dt given length/distance/ displacement given instant
Alternate Punctaneous speed dx/dt(x)= 1/(dt/dx) given point given interval/period
Inverse Punctaneous pace dt(x)/dx given point given interval/period
Alt. Inverse Instantaneous pace dt/dx(t)= 1/(dx/dt) given length/distance/ displacement given instant

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Instantaneous time and space speed

This post relates to the previous one here.

Speed is the time rate of distance traversed. Pace is the space rate of elapsed time.

The time speed of a body is the distance traversed per unit of independent time without regard to direction, Δx/. The instantaneous speed is the time speed at a point in space and time: dx(t)/dt. It is the reciprocal of the instantaneous pace, 1/(dt(x)/dx).

The space speed of a body is the independent distance traversed per elapsed time, Δx/Δt. The puncstanceous speed is the space speed at a point in time and space: dx(t)/dt.

The space pace of a body is the traversal time per unit of independent distance without regard to direction, Δt(x)/. The puncstanceous pace is time pace at a point in time and space: dt(x)/dx.

The time pace of a body is the independent traversal time per distance traversed, Δt/Δx(t). It is the reciprocal of time speed. The instantaneous pace is the space pace at a point in space and time: dt/dx(t). It is the reciprocal of the puncstanceous speed 1/(dx(t)/dt).

The time velocity of a body is the displacement traversed per unit of independent time, Δx/. The instantaneous velocity is the time velocity at a point in space and time: dx(t)/dt. It is the reciprocal of the puncstanceous lenticity: 1/(dt(x)/dx).

The space velocity of a body is the independent displacement traversed per elapsed time at a location, Δx(t)/Δt. The puncstanceous velocity is the time velocity at a point in time and space: dx(t)/dt.

The space lenticity of a body is the dischronment per unit of independent length, Δt(x)/. The puncstanceous lenticity is the time lenticity at a point in time and space: dt(x)/dx.

The time lenticity of a body is the independent traversal time per distance traversed, Δtx(t). It is  the reciprocal of time velocity. The instantaneous lenticity is the space lenticity at a point in space and time: dt/dx(t). It is the reciprocal of the instantaneous velocity: 1/(dt(x)/dx).

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Adding and averaging rates

A rate is the quotient of two quantities with different, but related, units. A unit rate is a rate with a unit quantity, usually the denominator. A vector rate is a rate with a vector quantity, usually the numerator. Rates with the same units may be added, subtracted, and averaged.

Addition

Rates with the same units in their denominator are added using ordinary addition, which will be called arithmetic addition since addition takes place in the numerator. For example, if x1 and x2 are lengths, and t is a given time interval, then the time speed rates v1 and v2 are added by arithmetic addition:

v_{1}+v_{2}=\frac{x_{1}}{t}+\frac{x_{2}}{t} =\frac{x_{1}+x_{2}}{t} = v_{3}

If x1 and x2 are displacements, and t is a given time interval, then the time velocity rates v1 and v2 are added by arithmetic addition:

\mathbf{v}_{1}+\mathbf{v}_{2} =\frac{\mathbf{x}_{1}}{t}+\frac{\mathbf{x}_{2}}{t} =\frac{\mathbf{x}_{1}+\mathbf{x}_{2}}{t} = \mathbf{v}_{3}

where vector addition means addition of dimensions, i.e., parallelogram addition.

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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 (sometimes called the inverse speed).

~However, these two speeds are not the same. One difference is how the speeds are aggregated or averaged: time speed requires arithmetic addition and arithmetic averaging but space speed requires harmonic addition and harmonic averaging. The arithmetic average of speeds v1 and v2 is (v1 + v2)/2. The harmonic average of space 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.

One should always use mathematical variables and functions that mirror the physical variables and functions. A physically independent variable should always appear as the argument in a mathematical function. A physically dependent variable should always appear as a function, even if the functional relation is uncertain.

To continue the example above, the space speed should be stated mathematically as dx/dt(x) so that distance is shown as an independent variable and time is shown as a dependent variable. Harmonic addition (see harmonic algebra) is for adding quotients whose denominator is a function of the numerator.

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.

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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.

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.

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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 y}{\partial x^2}-k^{2}\frac{\partial^2 y}{\partial t^2}=0.

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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:

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