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Tag Archives: Space & Time

Matters relating to length and duration in physics and transportation

From length to duration and back

Let’s start with one-dimensional, i.e., scalar, functions, f, g, h, and k. Say there is the following functional relation:

s = f(t) = f(h(s)) ≡ g(s) = t,

t = g(s) = g(k(t)) ≡ f(t) = s,

in which s and t are parameters with different units. By implication the functions are either f or its inverse:

s = f(t) = f(f-1(s)) = t,

t = f-1(s) = f-1(f(t)) = s.

Function f takes t-units into s-units, and function f-1 takes s-units into t-units. The vector versions are as follows:

s = f(t) = f(f-1(s)) = t,

t = f-1(s) = f-1(f(t)) = s.

Motion space is an ordered pair of vectors s and t: (s, t), resulting in their direct sum vector space. Addition is conducted by components: (r, w) + (s, t) = (r + s, w + t). Scalar multiplication is also by component: (a, b) (s, t) = (as, bt). To multiply a scalar and only one component requires the other component to be unity. Thus additive unity is (0, 0) and multiplicative unity is (1, 1).

There are two ways to mask an ordered pair of vectors: left mask (s, t) = (s, t) and right mask (s, t) = (s, t), where s = |s| and t = |t|. What was described here as expansion and contraction may now be shown more clearly as masking and unmasking. A parametric length vector function is converted to a parametric duration vector function as follows:

r(t) = masked r(t) ↑ unmasked r(t) ↔ (inverted) = unmasked t(r) ↓ masked t(r) = t(r).

r(t) = [r(t), θ(t), φ(t)] ↑ [(t´, χ´, ψ´), θ´(t´, χ´, ψ´), φ´(t´, χ´, ψ´)] ↔ ((r´, θ´, φ´), χ´(r´, θ´, φ´), ψ´(r´, θ´, φ´)) ↓ [t(r), χ(r), ψ(r)] = t(r).

Intentional and extensional causes

This post continues previous posts on causes, especially the one here.

Final and formal causes constitute top-down causality, which may lead to efficient and material causes. Material and efficient (mechanism) causes constitute bottom-up causality, which may lead to formal and final causes. Top-down is intentional. Bottom-up is extensional.

The Inverse Causality Principle states that top-down causality is inverse of bottom-up causality.

The Inverse Correspondence Principle states that intentional motion is the inverse of extensional motion and experimentation is the inverse of observation. Similarly, transmission is the inverse of reception, developmental is the inverse of empirical, and time is the inverse of space.

The goal of science is empirical theory. The goal of engineering is development of something practical.

Goal and action go together like form and content or matter.

Consider Galileo dropping two balls, one wooden and one metal, from the tower of Pisa. One observer says it’s a race to the ground. Another observer says it’s an experiment. What is the nature of the balls? Or what does Nature do?

Final and formal causes are the inverse of efficient and material causes.

Newton and Einstein compared

Isaac Newton expanded on what is now called the Galilean transformation (GT). The GT encapsulates a whole approach to physics. Length and duration are independent variables, and accordingly are universal, and may be measured by any observer. The length of a body is a universal value. The duration of a motion is a universal value. These values are independent of the control or condition of an observer.

Albert Einstein expanded on what is now called the Lorentz transformation (LT). The LT encapsulates a whole approach to physics. There are two universal constants: the speed of light in a vacuum and the orientation of reference frames. These constants are independent of any observer, though the speed of light may be measured by any observer. The orientation of reference frames is assumed to be the same universally, as if all are aligned with the fixed stars according to a universal convention.

Galileo described the relativity of speed, so that inertial observers do not have a universal speed but have speeds relative to other inertial observers. There is no universal maximum one-way speed. The two-way speed of light is a universal constant, but one leg of its journey may be instantaneous by convention, consistent with common ways of speaking. The orientation of reference frames is also relative, so that two frames view each others’ velocities as having the same direction.

Einstein described the relativity of length and duration, depending on their relative speed, which is always less than the speed of light in a vacuum. By convention, the mean of the two-way speed of light is assigned to every leg of its journey. Since the orientation of reference frames is the same, two frames view each other’s velocities as opposite in direction.

The strength of Newton’s vision is his mechanics and its continuity with common ways of speaking. The strength of Einstein’s vision is its continuity with Maxwell’s equations of electromagnetism.

Note: The Galilean transformation is related to the Lorentz transformation in one of three ways: (1) as c → ∞, (2) as v → 0, or (3) as the simultaneity of the backward (or forward) light cone (i.e., c0 = ∞) [see here].

Dual dynamics equations

(1) Newton’s Second Law

Momentum is defined as the product of mass m and velocity v. The mass of a body is a scalar, though not necessarily a constant. Velocity is a vector equal to the time rate of change of location, v = ds/dt.

The time rate of change in momentum is dp/dt = m dv/dt + v dm/dt = ma + v dm/dt by the rules of differential calculus and the definition of acceleration, a.

If mass is constant, then v dm/dt equals zero and the equation reduces to dp/dt = ma. If we define F = dp/dt, then we get Newton’s famous F = ma.

The dual equation is derived similarly:

Fulmentum is defined as the product of etherance n and lenticity u. The etherance of a body is a scalar, though not necessarily a constant. Lenticity is a vector equal to the stance rate of change of chronation, u = dt/ds.

The stance rate of change in fulmentum is dq/ds = n du/ds + u dn/ds = nb + u dn/ds by the rules of differential calculus and the definition of retardation, b.

If etherance is constant, then u dn/ds equals zero and the equation reduces to dq/dt = nb. If we define R = dq/ds, then we get the dual of Newton’s second law, R = nb.

(2) Work and kinetic energy

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Mean speed of light postulate

Einstein stated his second postulate as (see here):

light is propagated in vacant space, with a velocity c which is independent of the nature of motion of the emitting body.

Since the one-way speed of light cannot be measured, but only the round-trip (or two-way) speed, let us modify this postulate to state:

The measured mean speed of light in vacant space is a constant, c, which is independent of the nature of motion of the emitting body.

This is the most that can be empirically verified. Then for convenience sake, let us adopt the following convention:

The final observed leg of the path of light in empty space takes no time.

Since the (harmonic) mean speed of light is c, the speeds of the other legs of light travel are at least c/2 such that the mean speed equals c. In this way, the Galilean transformation is preserved for the final leg. And interchanging length and duration leads to an alternate version of the Galilean transformation.

This accords with common ways of speaking. Even astronomers speak of where a star is now, rather than pedantically keep saying where it was so many years ago. Physical theory should be in accord with observation of the physical world as much as possible. This is an example of how amateur scientists can help re-integrate science and common life.

Half-duplex relativity

Galilean relativity requires the speed of light to be instantaneous (i.e., zero pace). Because the one-way speed of light is not known, it may be instantaneous as long as the mean speed of light is finite. Such a situation is possible if light is conceived as in half-duplex telecommunications: one direction at a time is observed or transmitted, but never both simultaneously.

Consider a light clock in this context:

light at restSaw-tooth light path

Let Δt be the time for one cycle of light at rest (top diagram). Let Δt’ be the time for one cycle of light traveling at relative velocity v (bottom diagram). The mean speed of light is c. Then

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Interchange of length and duration

Length and duration are independent measures of the extent of motion, which are measured by comparing the target motion to a uniform reference motion. Although uniform linear motion is simpler in theory, uniform circular motion is simpler in practice – especially for unstopped motion. With one addition, the classic circular clock with hands serves as a reference motion. The addition is to mark the circumference in length units along with the duration units of the angles between the hands and the vertical. See post on a metreloge here.

Galileo uses horizontal uniform linear motion to mark length and duration below (from his Dialogues Concerning Two New Sciences, Fourth Day):

Falling projectile

The horizontal uniform motion of a particle coming from the right at a-b is continued with b-c-d-e as the horizontal component of the particle descending with uniform acceleration b-o-g-l-n. The vertical component represents the dependent variable, which has the form of a semi-parabola. The uniform horizontal motion, which could be any in direction, is the independent variable in units of either length or duration.

To interchange length and duration in an equation with a parametric function of time requires four steps: (1) with a change of variables switch the independent and dependent variables, time and stance; (2) linearize the stance, that is, break its dependent relation; (3) bring time under a functional relation with the new parameter, stance; and (4) expand time to include angular components. Functions are inverted and the independent and dependent status of variables is switched. An inversion and a kind of re-inversion return to the same function.

In the example above, the horizontal uniform motion which was taken by Galileo to represent time is re-conceived to represent the independent length variable, stance. The constant acceleration of the vertical component is re-conceived to represent the dependent duration variable with constant lenticity. The quadratic sequence in units of length becomes a sequence in units of duration at a constant rate.

The result of this interchange process is that the equations of motion for length and duration are interchangeable without functional change. All of the equations of physics in terms of parametric functions of time may be adopted as parametric functions of stance. In that sense it would be best to abstract a functional representation that applies to both length and duration, time and stance.

Length and duration

Although it has seemed natural to speak of “space and time”, that is a confused designation of length and duration, as well as their metrics, stance and time (see glossary here). Space is the space of motion, so length space is only one side; the other side is duration space. Speaking consistently is challenging but a must in order to bring clarity to a confused subject.

Here is a diagram of the combinations of length and duration, stance and time (click to enlarge):

length-duration diamond

The extent of motion is measured by length and duration. The space of motion has three dimensions. Length space and duration space therefore have three dimensions as well. The space of motion is  represented by an ordered pair of length space and duration space coordinates. If there is absolute inter-convertibility between length and duration, then the space of motion is a six-dimensional space of either length units or duration units.

The metric of length is distance and the metric of duration is distime (though often called time). Stance variable is the distance of a point from the origin in length space. Time variable is the distime of an instant from the origin in duration space. Events are located in length space and chronated in duration-space.

As one can speak of distance in three-dimensional length space, one can speak of distime in three-dimensional duration space. In this sense, time is a three-dimensional concept. There are three dimensions of duration rather than three dimensions of time per se, but time has a three-dimensional aspect.

Two measures of motion

By common experience, we know there are three dimensions of motion. That is, space, which is the space of motion, is three dimensional. To measure the extent of motion requires comparing one motion with another, of which there are two ways: length and duration. The length of a motion is measured by comparing it with synstancial but not necessarily synchronous motion. The duration of a motion is measured by comparing it with synchronous but not necessarily synstancial motion.

Length of motion considered by itself forms a length space, which is space with a metric of length. Duration of motion considered by itself forms a duration space, which is space with a metric of duration. Since there are three dimensions of motion, length space and duration space are both three dimensional metric spaces. By convention, both are Euclidean. The length metric is called distance. The duration metric may be called distime.

Each point in length space has a length position (LP) vector that begins with the length origin. Each point in duration space has a duration position (DP) vector that begins with the duration origin. The magnitude of a length position vector is called the stance. Every point in length space that is equidistant from the origin has the same stance. The magnitude of a duration position vector is called the time. Every point in duration space that is an equal distime from the origin has the same time.

Stance and time are vector magnitudes, with their direction ignored. Stance is a radius from the origin of length space. A unit of length is the absolute value difference between two stances, that is, between the radii of two length vectors with unit difference. Time is a radius from the origin of duration space. A unit of duration is the absolute value difference between two times, that is, between the radii of two duration vectors with unit difference.

The rate of motion measured by the length of motion per unit of duration is called speed. The rate of motion measured by the duration of motion per unit of length is called pace. Note that a faster speed is a larger ratio, whereas a faster pace is a smaller ratio. Also, the ratio of a slower speed to a faster speed is less than one but the ratio of a faster pace to a slower pace is less than one.

The vector rate of change in the length vector per unit of duration is called velocity. The vector rate of change in the duration vector per unit of length is called lenticity. The vector rate of change in velocity per unit of duration is called acceleration. The vector rate of change in lenticity per unit of length is called retardation.

The length position vector of a trajectory evolves as a function of the time. The duration position vector of a trajectory evolves as a function of the stance. These functions are inverses of one another.

Newton’s laws and their duals

The following is based on Classical Mechanics by Kibble and Berkshire, 5th ed., Imperial College Press, 2004, with the dual version indented and changes italicized.

p.2 The most fundamental assumptions of physics are probably those concerned with the concepts of space and time. We assume that space and time are continuous, that it is meaningful to say that an event occurred at a specific point in space and a specific instant of time, and that there are universal standards of length and time (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements).

The most fundamental assumptions of physics are probably those concerned with the concepts of length and duration. We assume that length and duration are continuous, that it is meaningful to say that an event occurred at a specific point in length space and a specific instant of duration space, and that there are universal standards of length and duration (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements).

In ‘classical’ physics, we assume further that there is a universal time scale (in the sense that two observers who have synchronized their clocks will always agree about the time of any event), that the geometry of space is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all positions and velocities.

In dual ‘classical’ physics, we assume further that there is a universal length scale (in the sense that two observers who have synstancized their clocks will always agree about the stance of any event), that the geometry of time is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all chronations and lenticities.

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