iSoul In the beginning is reality.

Newtonian mechanics in time-space

We follow the treatment by David Tong of Cambridge University in his Classical Dynamics.

A transicle is defined as a moving object of insignificant size. The motion of a transicle of vass n at the chronation t is governed by Newton’s Second Law for time-space, R = nb or, more precisely,

R(t; t′) = h′           (1.1)

where R is the release which, in general, can depend on both the chronation t as well as the lenticity t′, and h = nt′ is the fulmentum. Both R and h are 3-vectors which we denote by the bold font. A prime indicates differentiation with respect to stance x. Equation (1.1) reduces to R = nb if n′ = 0. But if n = n(x), then the form with h′ is correct.

General theorems governing differential equations guarantee that if we are given t and t′ at an initial stance x = x0, we can integrate equation (1.1) to determine t(x) for all x (as long as R remains finite). This is the goal of classical dynamics.

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Composition order

Written compositions organized by temporal order are narratives. Items such as descriptions of people, places, or objects are organized as they occur to the narrator, for example, as the narrator takes apart an object or walks through a building or meets various people. This is a common method of composition but there are others.

Spatial order is another method of composition. Items such as descriptions of people, places, or objects are organized by their physical or spatial positions or relationships, for example, starting at the top and proceeding downward. Explanations of a geopolitical matter might proceed in geographic order.

Travel can be described temporally or spatially. An itinerary is usually arranged temporally but telling about it afterwards might be more interesting if arranged spatially. There are other principles of organization such as climactic order (order of importance) and topical order.

In science the independent variable determines the type of organization. If the independent variable is time, the organization is temporal. If the independent variable is space or distance, the organization is spatial. The stance in spatial organization corresponds to the time in temporal organization.

The values of the independent variable are the index to the order of the composition. If the independent variable is time, then the times indicate the steps in the order. If the independent variable is space or distance, then the stances indicate the steps in the order. Once the step is indicated, the composition may be the same: whether it’s Tuesday, so the tour is in Paris or it’s Paris, so the tour is on Tuesday makes no difference.

Equations of Motion Generalized

This is an update and expansion of the post here.

Here is a derivation of the space-time equations of motion, in which acceleration is constant. Let time = t, location = x, initial location = x(t0) = x0, velocity = v, initial velocity = v(t0) = v0, speed = v = |v|, and acceleration = a.

First equation of motion

v = ∫ a dt = v0 + at

Second equation of motion

x = ∫ (v0 + at) dt = x0 + v0t + ½at²

Third equation of motion

From v² = vv = (v0 + at) ∙ (v0 + at) = v0² + 2t(av0) + a²t², and

(2a) ∙ (xx0) = (2a) ∙ (v0t + ½at²) = 2t(av0) + a²t² = v² ‒ v0², it follows that

v² = v0² + 2(a ∙ (xx0)), or

v² − v0² = 2ax, with x0 = 0.

Here is a derivation of the time-space equations of motion, in which retardation is constant. Let stance = x, time (chronation) = t, initial time = t(x0) = t0, lenticity = w, initial lenticity = w(x0) = w0, pace w = |w|, and retardation = b.

First equation of motion

w = ∫ b dx = w0 + bx

Second equation of motion

t = ∫ (w0 + bx) dx = t0 + w0x + ½bx²

Third equation of motion

From w² = ww = (w0 + bx) ∙ (w0 + bx) = w0² + 2x(bw0) + b²x², and

(2b) ∙ (tt0) = (2b) ∙ (w0x + ½bx²) = 2x(bw0) + b²x² = w² ‒ w0², it follows that

w² = w0² + 2(b ∙ (tt0)), or

w² − w0² = 2bt, with t0 = 0.

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Speed of light

Speed is defined as “The time rate of change of position of a body without regard to direction; in other words, the magnitude of the velocity vector.” (Dictionary of Physics, 3rd edition, McGraw-Hill, 2002.

This is ambiguous, however. Consider a light beam reflected off a surface:

light clock at rest(1) Since the light returns to its starting point, the total travel distance is zero, so the overall velocity is zero and the speed is zero.

(2) However, the interest is in each leg of the journey. In that case, in the first leg light travels +L in time t, and in the second leg light travels –L in time t. The mean velocity in the first leg is v1 = +L/t, and the mean velocity in the second leg is v2 = –L/t. The mean velocity for both legs is the harmonic mean of these two velocities because what is fixed and independent is the length, not the duration.

1/((1/v1) + (1/v2)) = 1/((1/L) – (1/L)) = 1/0 = ∞.

Thus the mean velocity is infinite, and the mean speed of light is infinite.

(3) Another approach looks at length of each leg apart from direction. In that case, in the first leg light travels L in time t, and in the second leg light travels L in time t. The speed in each leg is L/t, so the mean speed of light is L/t. This is the best known approach to the speed of light.

It’s interesting that (2) leads to the Galilean transformation, and (3) leads to the Lorentz transformation.

A theory of time

Note: as the research develops this post will be updated.

Speed is the travel distance per unit of travel time. In racing there is a measure of the travel time per unit of travel distance, which is called the pace. These are not exactly inverses since the denominated in different units. Note that a faster motion is indicated by a lower pace since it takes a shorter time to travel the same distance.

Velocity is a vector quantity whose magnitude is a body’s speed and whose direction is the body’s direction of motion. What is the opposite concept, a vector quantity whose magnitude is a body’s pace and whose direction is the body’s direction of motion? The dictionary lacks a word for this concept; I propose calling it lenticity [lentitude] from Latin lentus, slow, since a larger value indicates a slower motion.

Motion is a form of change, and change is characterized by difference. A body at rest does not change. A body in motion changes. But a body is at rest only with respect to another body at rest; they change the same way. A body is in motion only with respect to another body in motion; they change in different ways.

What is a body but something physical with some consistency; some attribute must not change. If something changes completely, it is not a body, or at least not a single body. What is the length of a body? It is the difference between one end and the other end. This difference is a change, a motion, with respect to the body or with respect to an observer at rest with respect to the body.

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History and science balanced

As I’ve noted before (here etc.) history and science have different aims and methods. Mixing them just confuses both of them. There is no genuine “historical science” or “scientific history”. History narrates particulars among unique events. Science theorizes universals among repeatable events. In physics time is homogeneous: an experiment is the same whether conducted today or 100 years in the past or future. That is not true in history. Time is not homogeneous there.

History and science can and should balance one another. The more science expands its universals, the more history can point out particulars that are overlooked or are important in a particular context. The more history focuses on unique particulars, the more science can point out the significance of universals.

The homogeneous and inhomogeneous aspects of time can both be known only by balancing history and science. One could say something similar about all universals and particulars. The universal and particular aspects of reality can both be known only by balancing history and science.

Abstract and concrete movements

Abstraction in Western culture has increased over time, so much so that Hegel made this the engine of history: his dialectic is a progression from the concrete to the less concrete, the abstract to the more abstract. Certainly, the history of natural science shows this progression. Modern physics is more abstract than classical physics. Every science becomes more abstract over time.

Increased abstraction in society and politics requires larger collections of people. Equality with increased abstraction requires equality within larger groups of people. For example, pan-European equality is less abstract than equality within global equality. Increased abstraction requires loyalty to ever larger groups.

History does seem to progress toward greater abstraction. Tribal cultures gave way to city-states, then to nations, then to globalism. In the U.S., there has been a progression from an English culture to a European culture, to a Euro-Afro-Latin culture, to an increasingly global culture. Those who promote this movement are called “progressives”. Those who resist it or support caution about it are called “conservatives”.

In sub-cultures of the West and in some non-Western societies there are movements in the opposite direction, toward more concreteness. They are often called “regressive”, which assumes a prior progressive movement. They could simply be called “concretive” (or “introgressive”) since they prefer the more concrete to the more abstract.

Those who prefer more concrete or at least a less abstract culture are considered traditional, old-fashioned, or backwards. In order to engage their opponents, traditionalists need to justify their preference for the concrete in more abstract ways, which they may find difficult. But the concrete has its advantages as much as the abstract does.

One danger of greater abstraction is that one loses touch with concrete reality. After all, human beings are concretely embodied. Concrete food, shelter, and much more are necessary for human life. Traditional social and political structures have much experience and stability behind them and so “should not be changed for light and transient causes” (the U.S. Declaration of Independence). And the new global human who ignores the local culture where they happen to be is looking for misunderstanding and worse.

In fact, there is no global, pan-religious, pan-racial, pan-sexual, pan-economic, pan-linguistic culture. Is such a culture even possible? In this world, that is highly doubtful. People are both concrete and abstract, body and spirit.

Concrete and abstract movements both have their place. Cultures will lean more toward one than the other, but both are legitimate.

What is a clock?

What is a clock? it is a device that measures time, but what are the essentials of a clock? I submit these are the essentials of a clock:

(1) A clock requires a uniform motion. Because only the kinematics (not the dynamics) are significant, a uniform rotation is acceptable. But because the result will be represented as a line – a timeline or time axis – a linear uniform motion has a more direct connection with what is measured, so let us take the first essential as a uniform linear motion.

(2) In order for clocks to be measuring alike, it is necessary that there be a standard rate for all clocks. In addition, clocks should have a standardized beginning point, so that clocks are interchangeable.

(3) A clock requires a pointer which indicates the present time on a time scale as it moves at the standard uniform rate. This would be the hands and dial on a common analogue clock. On a linear clock it is a part whose position in motion is interpreted as the present value of time. The pointer and scale are essentials of a clock.

Furthermore, a clock must be interpreted as showing the present time of the observer’s rest frame.

Figure 1

All the essentials of a clock can be represented by a frame in standard uniform motion relative to the observer’s rest frame. In that case, a clock should be definable in terms of frames of reference: one rest frame and one frame in uniform motion relative to the observer’s rest frame, as in the following.

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Space, time, and dimension

The post continues the ones here, here, and here.

There are three dimensions of motion. The extent of motion in each dimension may be measured by either length or time (duration). There are three dimensions of length and three dimensions of time (duration) for a total of six dimensions.

But there is no six-dimensional metric. Why? Because a metric requires all dimensions to have the same units, which requires a ratio to convert one unit into the other unit. The denominator on a ratio is a one-dimensional quantity, which means either the length or time dimensions need to be reduced by two dimensions.

This ratio is a conversion factor that is either a speed, which multiplied by a time equals a length, or a pace, which multiplied by a length equals a time. In general a speed is the ratio Δdr²/|Δdt²| = Δdr²/(Δt1² + Δt2² + Δt3²)1/2, and a pace is the ratio Δdt²/|Δdr²| = Δdt²/(Δx² + Δy² + Δz²)1/2. The denominator is a distance or distime, which is a linear measure of length or time (duration).

The conversion factors required are the speed of light in a vacuum, c, or its inverse, the pace of light in a vacuum, k. The resulting four-dimensional metric is either c²dt² − dx² − dy² − dz² (with time reduced to one dimension) or dt1² − dt2² − dt3² − k²dr² (with space reduced to one dimension).

These metrics are often simplified by taking c = 1 and k = 1 so that symbolically they are the same. Their units are not the same, however.

Each metric may be further reduced by separating space and time, as in classical physics. Then the space metric is |Δdr²| = (Δx² + Δy² + Δz²)1/2 and the time metric is |Δdt²| = (Δt1² + Δt2² + Δt3²)1/2. In the classical (3+1) of three space dimensions and one time dimension, time is replaced by its metric, and in the classical (1+3) of one space dimension and three time dimensions, space is replaced by its metric.

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. Trime (3D time) is the geometry of times and durations in R3. A time point (or timepoint) is a point in trime. The time origin is a reference time point in trime. The chronation of a time point is the trime 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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