iSoul In the beginning is reality

Symmetric laws of physics

Because of the symmetry of space and time, the laws of physics should be symmetric in space and time, or at least show their symmetry. Granted, one must either use the speed (change in position per unit of time) or the pace (change in time per unit of length). But other than such choices, the form of a law of physics should show the symmetry of space and time.

I have written on Galileo revised, in which the symmetry of space and time leads to a modification of the Galilean (or Galilei) transformation. The addendum includes the need to make the transformations for space and time similar. If the spatial displacement is r, the temporal displacement is t, the relative velocity v, and the conversion constant from time to space is c, then the following transformations fulfill those requirements:

= (1 – v/c) r and= (1 – v/c) t.

The same requirements may be applied to the Lorentz transformation as well, with its inclusion of the Lorentz factor, γ:

= (1 – v/c) γr and= (1 – v/c) γt,

where γ2 = c2 / (c2v2) = 1 / (1 – (v/c)2).

The similarity between these transformations is remarkable. Since (v/c)2 approaches zero faster than (v/c), the Lorentz factor approaches one and the Lorentz transformation approaches the revised Galilean transformation for relatively small velocities. Both transformations include a standard conversion between time and space, that is, an absolute speed, contrary to the original Galilean (and Newtonian) assumption of an absolute time.

Diachronic and synchronic physics

Diachronic, 1857, from Greek dia “throughout” + khronos “time” means something happening over time, particularly the historical development of something such as a language through time.

Synchronic, 1775, means “occurring at the same time,” from Late Latin synchronus “simultaneous,” means the analysis of something such as a language over a wide area at a point or period in time.

The terms diachronic and synchronic may be used to distinguish two approaches to the analysis of anything with spatial and temporal aspects. The diachronic approach stays with one place or people and focuses on the development through time. In transportation it is the perspective on a moving vehicle or data gathered from inside moving vehicles. The synchronic approach looks at a wide area or multiple places at a point in time or within a particular time period. In transportation it is the perspective from the side of the road, on the earth.

Diachronically, the pace of each vehicle is measured from as the ratio of its travel time over a road segment. The (arithmetic) average pace or harmonic average speed is the space mean traffic speed for the length of roadway.

Synchronically, vehicle speeds (spot speeds) are measured from sensors at a location on the road over a period of time. The (arithmetic) average is the time mean traffic speed for a given period of time.

Physics normally uses speeds, not paces, combined with the time displacement and so is synchronic. If the pace is used instead of the speed, combined with the length displacement, physics is diachronic. The laws of physics are the same in either case: space and time are symmetric. For example, the Lorentz transformation:

x’ = γ (x – vt), y’ = y, z’ = z, and t’ = γ (t – vx/c2)

may be interpreted as spatial coordinates x, y, and z, time displacement t, and speeds c and v; or as temporal coordinates x, y, and z, length displacement t, and paces c and v.

Measurement of space and time

Here is a roundup of various instruments and methods for measuring space and time that may be stopped or continued indefinitely:

A bematist (from ancient Greek βῆμα bema ‘pace’) was a specialist in ancient Greece who was trained to measure distance by counting their steps.

An odometer for measuring distance was first described by Vitruvius (c. 27 – 23 BC) although the actual inventor may have been Archimedes of Syracuse (c. 287 – 212 BC). The odometer of Vitruvius was based on chariot wheels of 4 feet (1.2 m) diameter turning 400 times in one Roman mile (about 1400 m).

In 1903 Arthur P. and Charles H. Warner from Beloit, Wisconsin, introduced their patented Auto-meter, which used a magnet attached to a rotating shaft to induce a magnetic pull upon a thin metal disk. Measuring this pull provided automobile drivers with accurate measurements of both distance and speed in a single instrument.

A measuring wheel or surveyor’s wheel is a wheel attached to a handle that can be pushed or pulled along by a person walking to measure distance traveled. It is marked in fractional increments of revolution from a reference position. If the wheel is rotated a full turn, the distance traveled is equal to the circumference of the wheel. Otherwise, the distance the wheel traveled is the circumference of the wheel multiplied by the fraction of a full turn.

A trip meter (tripometer) is an odometer that may be reset to record the distance traveled in a particular journey or part of a journey.

A ruler (aka rule or line gauge) is a straightedge with calibrated lines at specified distances from one edge to measure distances or lengths. A tape measure or measuring tape is a flexible ruler, designed to be rolled up for portability.

A clock (or timepiece) is any device for measuring and displaying the time that is designed not to stop. It is one of the oldest human inventions, meeting the need to consistently measure intervals of time shorter than the natural units: the day, the lunar month, and the year.

All oscillating clocks, mechanical and digital and atomic, work similarly and can be divided into analogous parts. They consist of an object that repeats the same motion over and over again, an oscillator, with a precisely constant time interval between each repetition, or ‘beat’. Attached to the oscillator is a controller device, which sustains the oscillator’s motion by replacing the energy it loses to friction, and converts its oscillations into a series of pulses. The pulses are then counted by some type of counter, and the number of counts is converted into convenient units. Finally some kind of indicator displays the result in human readable form.

A stopwatch is a clock that can be started and stopped easily.

Note that the ability of a measuring device to operate continuously is irrelevant to its utility for measuring the dimensions of or to an object or event. Alternatively, an odometer or other rotation-based distance measuring device could be operated continuously. The conclusion is that there is no necessary connection between time or space and continuous change or movement.

A complete explanation

Who, what, when, where – journalists repeat these adverbial questions to find key factors that explain things. That and the four explanatory factors or “causes” of Aristotle are needed to cover all aspects of a complete explanation.

Consider Aristotle’s example of a statue:

The material factor is what it is made from, “that out of which” it is made, e.g., the bronze of a statue.

The formal factor is the form/design that makes it what it is (“what-it-is-to-be”), e.g., the shape of a statue.

The efficient/mechanism factor is what makes it, “the primary source of the change (or rest)”, e.g., the art of bronze-casting the statue.

The final factor: the end/purpose, what is it made for, “that for the sake of which a thing is done”, e.g., beauty as the end of art, health as the end of walking.

There are also adverbial questions to complete the explanation:

The who factor: who made the statue?

The what factor: what is it? A statue.

The when factor: when was it made?

The where factor: where was it made?

All eight of these factor are necessary for a complete explanation.

Modern natural science looks at the efficient/mechanism factors, the material factors, the what, when and where factors. The who, why, and formal factors are excluded. Thus every explanation of modern natural science is incomplete – and so should be treated as input for others to complete them, which could include changing the partial explanations of science if necessary.

Addendum: There are also what might be called causal metafactors. These come after the causal factor and ask, Why? Alternatives likely exist for each factor. Why was this material selected? Why was this mechanism/force used? Why was this design used? Why was this goal sought?

Lorentz with 3D time

Just as three dimensions of space are combined with one dimension of time, so we can combine three dimensions of time with one dimension of space. The place to start is the Lorentz transformation. Let’s take a common approach, that of spherical wavefronts of light but instead of taking three length coordinates and converting time into length via the speed of light, let’s take three duration coordinates and convert space into duration via the speed of light.

Here’s a revision of the Wikipedia text, using Greek letters for time and Latin letters for space:

Consider two inertial frames of reference O and O′, assuming O to be at rest while O′ is moving with a velocity v with respect to O in the positive ξ-direction. The origins of O and O′ initially coincide with each other. A light signal is emitted from the common origin and travels as a spherical wave front. Consider a point P on a spherical wavefront at a distance r and r′ and a duration τ and τ′ from the origins of O and O′ respectively. According to the second postulate of the special theory of relativity the speed of light, c, is the same in both frames, so for the radial coordinates to the point P:

τ = r / c and τ’ = r’ / c.

The equation of a sphere of time (duration) in frame O is given by

ξ2 + η2 + ζ2 = τ2.

For a spherical wavefront that becomes

ξ2 + η2 + ζ2 = r2 / c2.

Similarly, the equation of a sphere in frame O’ is given by

ξ’2 + η’2 + ζ’2 = τ’2.

so the spherical wavefront satisfies

ξ’2 + η’2 + ζ’2 = r’2 / c2.

The origin O’ is moving along the ξ-axis. Therefore,

η’ = η and ζ’ = ζ.

Other than these equations the derivation follows as before. A slightly different Lorentz factor is found (call it g):

g2 = v2 / (v2 – c2).

Compare that with the usual Lorentz factor, gamma:

γ2 = c2 / (c2 – v2).

Note that γ2 + g2 = 1. Also note that γ is real only if v < c and g is real only if v > c.

The 3D time Lorentz transformation is then

r’ = g (r – ξ c2 / v)

ξ’ = g (ξ – r/ v)

η’ = η and ζ’ = ζ.

In this Lorentz transformation length is the independent variable, whereas in the usual Lorentz transformation time is the independent variable.

Time defined anew

“Time is that which is measured by a clock” wrote Hermann Bondi in Relativity and Common Sense (p.65), though the idea goes back to Albert Einstein, and ultimately to Aristotle.

“A space is that which is measured by a ruler; time is that which is measured by a clock.” (George Lundberg, quoted in Abrahamson, 1981: p.256)

I think the truth of the matter is somewhere between a ruler and a clock. Let’s start with what a clock is.

“Almost any clock consists of three main parts: (1) a pendulum or other nearly periodic device, which determines the rate of the clock; (2) a counting mechanism, which accumulates the number of cycles of the periodic phenomenon; and (3) a display mechanism do indicate the accumulated count (i.e., time).” (Clocks, Atomic and Molecular)

The key component is a periodic movement, that is, a cycle. What is measured is a number of these cycles but notice that our clock nomenclature includes part of a cycle, too. The standard clock cycle is one hour, subdivided into minutes and seconds. These parts of a cycle are naturally associated with angles; they are angular divisions of a cycle. A clock is essentially a way of measuring a constant angular velocity.

Consider a vehicle; it travels on wheels and the turning of the wheels (or the axle mechanism) allows it to measure its distance traveled. If the wheels are turning at a constant angular velocity, they are a kind of clock — with this important difference: it is the movement of the circumference that is significant for the distance, not the angular movement.

A ruler is a tool to measure length. We do not associate movement with a ruler but in fact a ruler must be moved into position to measure anything. A measuring wheel shows that a ruler need not be a linear object, though length is a linear measure. It is the circumference of the measuring wheel that is significant; its angular movement is a means to linear movement. The odometer on a vehicle uses the circular movement of the wheels, but the result is a linear measure because the vehicle is moving linearly.

All this shows that movement is required for measuring length and time. The only difference is that we associate time with a continually moving system, a clock; whereas we associate length (and space) with a temporarily moving system. When the measuring wheel stops, the length ends; but time goes on, so we say.

This is so confused! The measuring wheel that measures length can also measure time (duration) if it is going at a constant rate. Where the wheel ends, the length ends; when the wheel ends, the duration (time) ends. The length (space) and the duration (time) go together.

If we wanted to have devices that measure length, and keep on going without stopping, we could do that. We could use a vehicle moving at constant speed around a racetrack. “What length is it?” would mean how far has the vehicle traveled. It would be exactly like “what time is it?” The only difference is whether the linear measure or the angular measure is taken.

The conclusion is that time (duration) is just like length: a device with constant cyclic movement is related to what is being measured; for the length, take the circumferential movement and for the duration, take the angular movement. After the measure is taken the device may be turned off or it may be left running. If it is left running, it is called a clock; otherwise not.

What’s the difference between a stopwatch and a clock? One stops and the other doesn’t. What’s the difference between an express train and a local train? One stops (more) and the other doesn’t. Movement can be measured by angular devices or linear devices. Either way, it’s still movement but we distinguish between them.

There are various reasons for a standard reference movement, as clocks provide. But the standard movement could be linear; it could be measured in units of length. It could be like a satellite circling the earth: its position over the earth could be taken as a measure of its movement; or if it were in a polar orbit, the corresponding longitude could be the measure. It doesn’t stop so it’s like a clock but it could just as well measure distance as an angle.

I could go around and around about this but the bottom line is that time is a difference in angular movement and space is a difference in linear movement. In the case of length, we’re only concerned with the difference; with time we’re more concerned with the movement. But it could just as well be the other way around. Clocks could cease and constant linear movements could be kept going. Time is that which is measured by a constant angular movement that stops when the measurement is complete.

When a measuring wheel stops, does space stop? No. When a clock stops, does time stop? No. Is it possible to have a zero length while a stopwatch is going? Yes, it’s called stationary. Is it possible to have a zero time (duration) while a measuring wheel is going? Yes, and that is also stationary.

We see both linear and angular movement with light. Frequency and wavelength, and the distance it travels and the time it takes to travel are proportional. The special properties of light make it an ideal standard for relating angular and linear movement, space and time.

Lorentz interpreted

The question is how to interpret the Lorentz transformation. In a previous post, Lorentz generalized, a modest generalization of the Lorentz transform was derived. Absolute reference speeds were combined with a relative actual speed.

Let’s step back and look at a map of space and time:

Interstate Drive Times & Distances Sample

Interstate Drive Times & Distances Sample

This map of nodes and links on the U.S. interstate highway system displays travel distances and driving times between cities. If you look closer, you can see that it is based on a standard travel speed of about 50 mph (with some local variations). So each point on the network represents a travel distance and a travel time: in other words, space and time are in sync.

Now compare this map with some actual travel experience, say, one traveler going at 40 mph and another at 60 mph. If they start together, after one hour of travel they will have gone 40 and 60 miles respectively, compared to the standard of 50 miles. After one hour, the standard “map” distance is 50 miles but the actual distances are 40 and 60, so space and time are not in sync with these travelers.

The problem is space and time can no longer be mapped together: either the distance traveled or the travel time can be mapped but not both. At most all the distances for one travel time or all the travel times for one distance can be mapped.

A physicist approaching this situation might ask, is there some function of space and time that can still be mapped? Is there a quantity that is invariant no matter what the travel speed is? Can an alternate map be constructed?

The answer is yes and the key is the Lorentz transformation. Note that this is for an alternate map: if travel speeds equal the standard speed, no new map is needed. So we’re looking at speeds u and u’ that differ from the standard speed, c.

The alternate map has one limitation: it’s from the point of view of one traveler. But an alternate map can be constructed for any traveler and the principles of its construction are the same for all travelers. That’s the best that can be done.



Art and science

Aristotle’s four explanatory factors (aka four causes) provide a template for a full explanation of anything. However in attempting to explain the natural world (and many other things) it is impossible for us to know all the final causes involved. We may have an idea about some purpose or function of some aspect of things, but since we were not “present at the creation” and even the Bible gives only a general idea about such things, it is best for a science of nature to lay aside claim to final causes and focus on finding the rest of the explanation.

But an artist who “imitates nature” can take all four factors into account because they can know their purpose, their design, their means, and the matter used, as Aristotle showed with his example of the sculptor (in his Physics II 3 and Metaphysics V 2). But the artist constructs an alternate reality, not something from nothing as the Creator has done. So the artist is working with illusion rather than knowledge of reality.

Fine craft is closer to what Aristotle meant by art — techne is the Greek word for art or craft. The craftsman has a particular purpose in mind, for example, when making a bowl of pottery and can aim at the best quality with the other factors. This is the greatest fulfillment of art, that is, making things for a specific purpose. The ambiguous purposes of art in the modern world do not fit this as well.

So the artist has purpose but lacks reality and the scientist has reality but lacks purpose. As science expands, it can lose sight of what its purpose is, and as art expands it can lose sight of reality.

There is a parallel to the world of politics, too. The “Left” inclines toward the artist, having a vision of what the world should be but lacking a connection with reality. The “Right” inclines toward the scientist, having a vision of what the world is but lacking a connection with what the world should be. The political give and take can result in a joint effort toward a better world — or a lurching back and forth accomplishing nothing.

But is it not better to attempt to do the right thing rather than to do things right without the right goal?

Marriage explained

The four explanatory factors (aka four causes) Aristotle described can be used to explain marriage in a time in which people have forgotten what marriage is. The ancient book of Genesis provides the explanation, so it’s not a recent attempt to promote an agenda. Societies through the ages have implicitly followed the explanation, which may have led to complacency concerning what marriage is all about.

Genesis 2
18 The Lord God said, “It is not good for the man to be alone. I will make a companion for him who corresponds to him.” 19 The Lord God formed out of the ground every living animal of the field and every bird of the air. He brought them to the man to see what he would name them, and whatever the man called each living creature, that was its name. 20 So the man named all the animals, the birds of the air, and the living creatures of the field, but for Adam no companion who corresponded to him was found. 21 So the Lord God caused the man to fall into a deep sleep, and while he was asleep, he took part of the man’s side and closed up the place with flesh. 22 Then the Lord God made a woman from the part he had taken out of the man, and he brought her to the man. 23 Then the man said,

“This one at last is bone of my bones
and flesh of my flesh;
this one will be called ‘woman,’
for she was taken out of man.”

24 That is why a man leaves his father and mother and unites with his wife, and they become a new family. 25 The man and his wife were both naked, but they were not ashamed.

The material factor is male and female: no companion who corresponded to Adam was found, unlike the animals who already had males and females. Without male and female sexes, there is no marriage.

The efficient factor is a voluntary union “of the flesh” (as the more literal translations have it), i.e., sexual intercourse. Without such a “conjugal act” the marriage is unconsummated. Without the possibility of such a union, there is no marriage.

The formal factor is a lifelong commitment, “until death do us part”. Although there is the possibility of divorce for cause (e.g., unfaithfulness), without an intention for a lifelong relationship, there is no marriage.

The final factor is the various purposes for marriage, including companionship, fulfillment of sexual desire, generation of offspring, nurturing of a family, and stability for society. The listing of these purposes does not limit the purposes of marriage; it only indicates some of what marriage can mean. Those who argue that a particular purpose could (or could possibly) be met in other ways are missing all the purposes of marriage.

Defenders of marriage have put too much stock in nailing down the purpose of marriage – that is only one factor of marriage; there are three other factors to consider. Until those who wish to redefine marriage meet the other explanatory factors, they have failed and marriage, real marriage, is the same as it always was. Legal fictions do not change reality. Marriage is still a voluntary union of one man and one woman for life.

Lorentz generalized

In some ways transportation is more general than physics, which is surprising. In terms of extent, from the microscopic to the astronomical, from extremes of temperature, etc., physics is the more general subject. But because transportation includes people, there are some additional possibilities. Let’s look at one transportation situation in which this is the case. (Note: we are not talking about transport theory here.)

Consider transportation in terms of positions in space and time, directions and speeds plus the expectations people have for a trip — in particular, what they see as their typical or expected travel speeds. The point is that people use a particular speed for trip planning and forecasting purposes, which may reflect general travel conditions or their personal travel experience, or simply their driving style. Call this the reference speed to distinguish it from their actual speed(s).

Let there be observer-travelers going in the same direction but in different vehicles (or trains, boats, etc.). Distinguish them by their frame of reference, unprimed or primed. Call their frames S and S’, their positions in space r and r’, in time t and t’, the actual speed of the second frame relative to the first v, and their reference travel speeds b and c respectively. Allowing different reference speeds is more general than the Lorentz transformation.

To make it more general we could say they may begin at different positions or their units of measure are different, but we’ll leave these as an exercise for the reader. The actual speeds could also vary over time but we’ll consider them constant.

Consider only the path/trajectory followed, i.e., one dimension of space and time each. Then we have: r = bt and r’ = ct’ as time-space conversions for each frame. We will follow the derivation of the Lorentz transformation (wavefront approach). A general linear transformation between (r, t) and (r’, t’) can be written as: r’ = ex + ft and t’ = gr + ht where the constants e, f, g, and h depend only on b, c, and v. The derivation is an exercise in algebraic manipulation with the following result:

e = 1 / sqrt(1 – v2/ b2) = γb,

f = -v e = -v / sqrt(1 – v2/ b2) = – v γb,

g = – (v / (bc)) γc,

h = (b/c) / sqrt(1 – v2/ c2) = (b/c) γc,

where γc = 1 / sqrt(1 – v2/ c2).

So the general Lorentz transformation is:

r’ = γb (x – vt),

ct’ = γc (b t – vx / b).

If b = c, there is only one reference speed for both traveler-observers, which is the requirement of the Lorentz transformation.

r’ = γ (x – vt),

t’ = γ (t – vx / c2).

This is the case with the speed of light, which acts as a reference speed to which all speeds can be compared.