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Duality as a convention

Is color an absorption phenomena or an emission phenomena? The answer is that it’s both. But absorption works subtractively whereas emission works additively. The question then is whether color is subtractive or additive. Again the answer is that it’s both. Color is a duality.

Does an artist work with subtractive colors or additive colors? Here the answer is one or the other. A painter works with pigments that are considered subtractive, whereas a glass artist works with stained glass that is considered additive. Even though absorption and emission are operating in both cases, we pick one or the other as our convention.

A simultaneity convention can also be a duality. What has been called apparent simultaneity is the convention that the backward light cone is simultaneous. But it is possible to adopt a complementary convention in which the forward light cone is simultaneous (see here). Either of these is something of a combination of Newton’s and Einstein’s physics.

One could recover Newtonian physics by adopting a combination of the backward and forward light cone simultaneity conventions. For an absorption event the backward light cone is simultaneous. For an emission event the forward light cone is simultaneous. This is like half-duplex communication (push to talk, release to listen).

Such a duality convention recovers Newtonian physics because it is as if the speed of light is instantaneous in all directions.

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 vass n and legerity u. The vass of a body is a scalar, though not necessarily a constant. Legerity is a vector equal to the base rate of change of chronation, u = dt/ds.

The base 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 expedience, b.

If vass 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. They are measured by comparing them with a uniform reference motion. Although uniform linear motion is simpler in theory, uniform circular motion is simpler in practice – and essential 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.

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

Galileo parabola

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. Because the horizontal motion is uniform, it can represent either length or duration of the motion. The vertical component represents the dependent variable, which has the form of a parabola.

To interchange length and duration in an equation with a parametric function of time requires five steps: (1) replace length components with their radius, which becomes the base; (2) switch time and base, that is, switch the independent and dependent variables; (3) linearize base, that is, break its dependent relation; (4) bring time under a functional relation with the new parameter, base; and (5) 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, base. The constant acceleration of the vertical component is re-conceived to represent the dependent duration variable with constant legerity. 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 base. In that sense it would be best to abstract a functional representation that applies to both length and duration, time and base.

Galileo’s reciprocity

From Galileo’s Dialogue Concerning the Two Chief World Systems, translated by Stillman Drake (UC Press, 1967):

Salv. Now imagine yourself in a boat with your eyes fixed on point of the sail yard. Do you think that because the boat is moving along briskly, you will have to move your eyes in order to keep your vision always on that point of the sail yard and to follow its motion?

Simp. I am a sure that I should not need to make any change at all; not just as to my vision, but if I had aimed a musket I should never have to move it a hairsbreadth to keep it aimed, no matter how the boat moved.

Salv. And this comes about because the motion the ship confers upon the sail yard, it confers upon you and also upon your eyes, so that you need not move them a bit in order to gaze at the top of the sail yard, which consequently appears motionless to you. [And the rays of vision go from the eye to the sail yard just as if a cord were tied between the two ends of the boat. Now a hundred cords are tied at different fixed points, each of which keeps its place whether the ship moves or remains still.] p.249-250

Galileo is portraying motion as viewed by a human observer. The implication is that the observer in another ship would be observing the same kinds of things. Then two observers in motion with respect to one another who observe one another must face one another. That is, they are positioned opposite one another, effectively each turned 180º from the other.

This is the Galilean Reciprocity Principle, the convention that an observed frame has the opposite orientation of the frame from which it is observed, which ensures that corresponding velocities are equal.

Relativity of orientation

The Principle of Relativity states that the laws of physics are the same in all inertial frames of reference (IRF). Since a frame of reference includes an orientation, that is, a convention as to which rectilinear semi-axes are positive (and so which are negative). Since there is no preferred frame of reference, each frame has its own orientation, not the orientation of a particular frame. That means IRF orientations are what is called “body-fixed” orientations.

A frame of reference is called “body-fixed” if it is conceptually attached to a rigid body, such as a vehicle, watercraft, aircraft, or spacecraft. Body-fixed frames are inertial frames if the body to which the frame is affixed is in inertial motion. The body is usually referenced in anthropomorphic terms, such as its left, right, face, or back, although some craft have their own terms, notably, ships with port, starboard, fore, and aft.

Consider the following scenario of cars in five lanes, oriented so that their forward direction is positive, with their unsigned speeds shown relative to the two parked cars in the middle lane:Six cars in five lanesCompare the direction of cars B, C1, C2, and D according to the frames attached to the five cars:

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Velocity reciprocity clarified

This is a follow-on to posts here and here.

It is common to derive the Lorentz transformation assuming velocity reciprocity, which seems to say that if a body at rest in frame of reference is observed from a frame of reference S that travels with relative velocity +v, then a body at rest in frame of reference S will be observed from the frame of reference to be traveling with velocity –v. But that’s not the case.

Consider the typical scenario in which a person standing on the earth (embankment, station) with frame of reference S observes a person sitting in a railway car with frame of reference . Say they are both waving their right hands and their frame of reference follows a right-hand orientation: the positive direction is toward their right.

Person waves to train

The first illustration shows the scenario from behind the observer standing on the earth in frame S, who observes the passenger sitting in the train moving to their right with velocity +v. The scenario is typically presented from only this perspective, that of an observer at rest in frame A, even if the perspective of an observer at rest in frame is described.

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Galilean relativity defended

Galilean relativity is a relational theory of motion as a function of time, which leads to the Galilean transformation. Here is a defense of Galilean relativity from two postulates:

(1) The Galilean principle of relativity, which states that the laws of mechanics are invariant under a Galilean transformation.

(2) A convention that rectilinear coordinates for frames of reference follow the right-handed rule: the unit vectors i, j, and k are related as i × j = k.

The Galilean transformation for constant motion on the x axis is x´ = xvt,  and t´ = t. Postulate (2) means if the extended right-hand thumb points to the positive X axis and the extended right-hand first finger points to the positive Y axis, then the right-hand middle finger points orthogonally to the positive Z axis.

The standard configuration for derivations of the Lorentz transformation consists of two inertial frames of reference moving relative to each other at constant velocity, with Cartesian coordinates such that the x and x′ axes are collinear facing the same direction:

Axes with same orientation

In this case the velocity of S´ relative to S is +v and the velocity of S relative to S´ is –v. This is called the principle of velocity reciprocity.

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