iSoul In the beginning is reality

# Fixed sizes and rates in space and time

A ruler is a measuring device with a fixed size. A ruler with a fixed length marked in standard linear units is a linear ruler. A linear ruler measures linear space, i.e., length or distance, which is the extent of an object or motion that is contiguous to a linear ruler. The units of linear space are metres, millimetres, kilometres, feet, miles, etc.

The common measuring rod marked in units of length is an example of a linear ruler. Other examples are measuring tapes, measuring wheels, odometers, and laser rulers.

A ruler with a fixed size marked in standard angular units is an angular ruler. An angular ruler measures angular space, which is the extent of an object or motion that is contiguous to an angular ruler. The units of angular space are radians, degrees, minutes, seconds, etc.

A common protractor is an example of an angular ruler. Other examples are bevel squares, theodolites, sextants, etc.

Since linear rulers are more common than angular rulers, the unqualified term ruler defaults to a linear ruler unless specified otherwise.

A clock is a measuring device or method with a fixed rate of motion. A clock with a fixed rate of angular motion (or rotation) marked in standard angular units is an angular clock. An angular clock measures angular time, which is the extent of an event or motion that is simultaneous with the motion of an angular clock.

The common circular clock with hands that indicate the angular time is an example of an angular clock. The motion of the sun across the sky functions as an angular clock, or its shadow may be marked with a sundial. Other examples are angular clocks based on oscillations of pendulums, crystals, or electronic circuits.

A clock with a fixed rate of linear motion marked in standard linear units is a linear clock. A linear clock measures linear time, which is the extent of an event or motion that is simultaneous with the motion of a linear clock. The linear time may also be indicated by a (possibly imaginary) device that represents the typical amount of linear time. I’ve written previously about this here.

The location of a regularly scheduled train can be used to measure linear time. It is not unusual for people to speak prospectively of a journey or a drive in terms of the time taken by a typical traveler, which is an example of an imaginary linear clock with a typical rate of travel. The speed of light functions in astronomy as a linear clock with units in light-years.

The units of angular and linear time, i.e., duration, are the same: seconds (the SI base unit), minutes, hours, days, years, etc. Since angular clocks are far more common than linear clocks, the unqualified term clock defaults to an angular clock unless specified otherwise.

Angular rulers and clocks don’t go anywhere, except in circles, as they are only magnitudes, which are one-dimensional. Linear rulers and clocks do go somewhere, and can do so in three dimensions. As there are three dimensions of linear motion, so there are three dimensions of linear rulers and clocks, that is, of linear space and time.

Which dimensions are observed depends on whether one uses angular or linear measures for space and time. Three dimensional linear space and angular time go together, as do three dimensional linear time and angular space.

# Universe of limits

If we accept that the actual infinite does not exist except as an attribute of God, then the universe is finite. And if the universe is finite, then any use of the infinite or infinitesimal in physical science is a reference to an indefinite unknown or a manner of speaking, which at a greater level of precision should be replaced with finite concepts.

If we accept that physical science proceeds through simplifying and downplaying particulars in order to articulate universals, then there is always something missing or not given its due significance. This is a limitation of physical science, which should be represented in any theory of physical science as a limit on its domain of applicability.

Yet it is part of the culture and method of physical science to posit open theories that are universal, even though they will have a domain of applicability that is less than universal. The purpose of this method is to push the concepts and laws of a theory to their maximum extent, and stop only where the theory breaks down in practice.

The ancient approach to science kept science within the confines of deduction but the modern approach allows science to extend into hypothetical domains supported by inductive experience and experiment. But ultimately a theory of science should be fully demonstrable, and so developed as a deductive system that encapsulates inductive experience and experiment.

Therefore, science proceeds from data collection to hypothesis to further data collection to revised hypothesis until a stable theory is developed. Then science further proceeds to push the theory to its limits until its domain of applicability is known. At that point an axiomatic system may be constructed that defines its concepts and attributes, which constitutes a closed theory.

Werner Heisenberg was the first to describe open and closed theories. Closed theories have achieved an axiomatic form and cannot be falsified. The set of closed theories constitute the permanent achievement of science, which open theories endeavor to extend.

The goal for an open theory is to become a closed theory, which occurs as its concepts become clarified and its limits found. One should not give full assent to any open theory because it lacks clarity and its limits are unknown. Full assent should be given to any closed theory but their truth is only axiomatic.

Science progresses from open to closed theory, from universal to limited domain, from induction to deduction, with a growth of knowledge and humility, that is, knowledge of limits and humility at how much is yet to be known.

# Lorentz and co-Lorentz transformations

I’ve written several related posts, such as one on the Complete Lorentz transformation. This post extends the previous post on the Galilean transformation to the Lorentz transformation, and what I’m now calling the co-Lorentz transformation, in order to show their similarities and differences.

There are many expositions of a Lorentz transformation, such as here. It is standard to describe them in terms of two reference frames and their coordinate systems in uniform relative motion along the x-axis. Here we take the spatial axis to be the r-axis, which is parallel to the spatial axis of motion. Similarly, the temporal axis is taken to be the t-axis, which is parallel to the temporal axis of motion.

One aspect of the exposition here is that the notation is indifferent as to the existence of other dimensions. If they exist, they are orthogonal to the direction of motion, whether spatial or temporal, and their corresponding values are the same for both frames.

The two frames are differentiated by primed and unprimed letters. Their relative speed is v, and their relative pace is u = 1/v. The key difference between speed and pace is their independent unit of measure: speed is measured per unit of time (duration), whereas pace is measured per unit of space (length).

A Lorentz transformation requires what I’m calling a characteristic (modal) rate, in units of speed or pace, which is the same for all observers within a context such as physics or a mode of travel. The characteristic speed, c, or pace, ç (c-cedilla), may take any positive value, and may represent a maximum, a minimum, or a typical value, depending on the context. In the context of physics the characteristic rate is that of light traveling in a vacuum.

Note that the trajectory of a reference particle (or probe vehicle) that travels at the characteristic rate follows these equations in the two frames:

speed: r = ct or r/c = t and r′ = ct′, or r′/c = t′,

pace: çr = t or r = t/ç and çr′ = t′ or r′ = t′/ç.

Lorentz transformation

This starts with the Galilean transformation and includes a factor, γ, in the transformation equation for the direction of motion, along with the characteristic rate:

speed (–): r′ = γ (rvt) = γr (1 – v/c) = ct′ = γ (ctrv/c) = γt (cv),

pace (–): rγ (rt/u) = γr (1 – ç/u) = t/çγ (t/çrç/u) = γt (1/ç – 1/u),

with equal values for the other corresponding primed and unprimed coordinates. The inverse transformations are then:

speed (+): r = γ (r′ + vt′) = γr′ (1 + v/c) = ct = γ (ct′ + r′v/c) = γt′ (c + v),

pace (+): rγ (r + t/u) = γr′ (1 + ç/u) = t/bγ (t/ç + rç/u) = γt′ (1/ç + 1/u).

Multiply each corresponding pair together to get:

speed: rr′ = γ²rr′ (1 – v²/c²) = c²tt′ = γ²tt′ (),

pace: rr´ = γ²rr′ (1 – ç²/u²) = tt/ç² = γ²tt′ (1/ç² – 1/).

Dividing out rr′ yields:

speed: 1 = γ2 (1 – v2/c2),

pace: ç² = γ2 (1 – ç2/u2).

Or dividing out tt′ yields:

speed: c2 = γ2 (c2v2),

pace: 1/ç² = γ2 (1/ç² – 1/).

Either way, solving for γ leads to:

speed: γ = (1 – v2/c2)–1/2,

pace: γ = (1 – ç2/u2)–1/2.

which is the standard Lorentz transformation and applies only if |v| < |c| or |u| > |ç|.

Co-Lorentz transformation

Now start with the co-Galilean transformation and include a factor, κ, in the transformation equation for the direction of motion, along with the characteristic rate:

speed (–): tκ (tr/v) = κt (1 – c/v) = r′/cκ (r/c – tc/v) = κr (1/c – 1/v),

pace (–): t′κ (tur) = κt (1 – u/ç) = çr′ κ (çrtu/ç) = κr (çu),

with equal values for the other corresponding primed and unprimed coordinates. The inverse transformations are then:

speed (+): tκ (t + r/v) = κt′ (1 + c/v) = r/cκ (r/c + tc/v) = κr′ (1/c + 1/v),

pace (+): tκ (t′ + ur′) = κt′ (1 + u/ç) = çr κ (çr + tu/ç) = κr′ (ç + u).

Multiply each pair together to get:

speed: tt = κ² tt (1 – c²/v²) = rr′/c² = κ² rr (1/ – 1/),

pace: tt = κ² tt′ (1 – u²/ç²) = ç²rr = κ² rr′ (ç²).

Dividing out tt′ yields:

speed: 1 = κ² (1 – /),

pace: 1 = κ2 (1 – u²/ç²).

Or dividing out rr′ yields:

speed: 1/ = κ² (1/ – 1/),

pace: ç² = κ2 (ç²).

Either way, solving for κ leads to:

speed: κ = (1 − c2/v2)–1/2,

pace: κ = (1 – u2/ç2)–1/2,

which is the co-Lorentz transformation and applies only if |v| > |c| or |u| < |ç|.

Complete Lorentz transformations

The Lorentz transformation is then

speed: r′ = γ (r − vt) and t′ = γ (trv/c²), with γ = (1 – v2/c2)–1/2, or

pace: rγ (rt/u) and tγ (trç²/u), with γ = (1 – ç2/u2)–1/2.

which applies only if |v| < |c| or |u| > |ç|.

The co-Lorentz transformation is then

speed: t′κ (t − r/v) and r′κ (rc2 t/v), with κ = (1 − c2/v2)–1/2, or

pace: t′κ (tur) and r′ κ (rtu/ç²), with κ = (1 – u2/ç2)–1/2,

which applies only if |v| > |c| or |u| < |ç|.

If |v| = |c|, then r′ = r and t′ = t.

Note that in each case γ is an even function of v or u, as it needs to be (see here).

# Galilean and co-Galilean transformations

This is a topic I have addressed before, such as here. In this post I want to show some similarities and differences between the Galilean (Galilei) transformation and what I’m now calling the co-Galilean transformation.

There are many expositions of the Galilean transformation, such as here. It is standard to describe them in terms of two reference frames and their coordinate systems in uniform relative motion along the x-axis. Here we take the spatial axis to be the r-axis, which represents the spatial direction of motion, whatever that is.

Similarly, the temporal axis is taken to be the t-axis, which represents the temporal direction of motion, whatever that is.

One aspect of the exposition here is that the notation is indifferent as to the existence of other dimensions. If they exist, they are orthogonal to the direction of motion, whether spatial or temporal, and their corresponding values are the same for both frames.

The two frames are differentiated by primed and unprimed letters. Their relative speed is v. Their relative pace is u = 1/v. The key difference between speed and pace is their independent unit of measure: speed is per unit of time (duration), whereas pace is per unit of space (length).

In the Galilean transformation, the measurement of time is the same for all observers, which is often called absolute time, whereas the measurement of space is relative to the motion of each observer. So the relationship between the coordinates is as follows:

speed: = rvt,

pace: = rt/u,

and for all other coordinates the primed and unprimed values are equal. Note that the reason the other coordinates are equal may be either because there is no motion in their direction or other directions are not known to exist.

In the co-Galilean transformation, the measurement of space is the same for all observers, which is called absolute space, whereas the measurement of time is relative to the motion of each observer. So the relationship between the coordinates is as follows:

speed: = tr/v,

pace: = tur,

and for all other coordinates the primed and unprimed values are equal. Again, the reason the other coordinates are equal may be either because there is no motion in their direction or other directions are not known to exist.

# Relativity of time at any speed

It is not well known that the Theory of Relativity is almost misnamed. Relativity was well known in physics since Galileo Galilei. That is, the relativity of space was well known. With Albert Einstein’s derivation of the Lorentz transform, the relativity of time was introduced. But the relativity of time was not of the same order as the relativity of space since it required speeds approaching that of the speed of light.

Nicholas Copernicus wrote in his 1543 book, The Revolutions of the Heavenly Spheres: “Every observed change of place is caused by a motion of either the observed object or the observer or, of course, by an unequal displacement of each. For when things move with equal speed in the same direction, the motion is not perceived, as between the observed object and the observer.”

Galileo in his 1632 book Dialogue Concerning the Two Chief World Systems noted that a person below the deck of a uniformly moving ship has no sense of movement and so objects dropped fall directly toward the feet. He articulated a principle of relativity: “any two observers moving at constant speed and direction with respect to one another will obtain the same results for all mechanical experiments.”

That is, a Galilean inertial frame of reference is in relative uniform motion to absolute space. This leads to the notion of absolute time and relative space: although different observers on different reference frames see space differently, they observe time the same way.

Einstein derived the Lorentz transformation from this foundation, allowing measures of time and space to affect one another. This solved the problem of electromagnetism and the result of the Michelson–Morley experiment.

One could say that time in the theory of relativity is relative in a minor way, whereas space is relative in a major way. The relativity of time requires speeds approaching that of the speed of light in a vacuum. The relativity of space is true at any speed.

But note one could modify Galileo’s statement to say that any two observers moving at constant pace and direction with respect to one another will obtain the same results for all mechanical experiments. In that case, time is relative at any speed, too. The key is to measure motion with pace, rather than speed, so that the independent variable is space. That allows time to be relative and multidimensional.

# Fourfold history and cosmology

As a generalist I tend to think of the big picture and push global conceptions, which can get speculative, but should provide insight in some way. There are many ways of slicing up history that show a pattern, but we crave meaning and so expect patterns. For example, it is helpful to adopt a rather conventional division of history into periods of primeval, ancient, medieval, modern, and post-modern (for lack of a better term). At least this gives us something to start with and modify or clarify later on.

I have written before briefly about the fourfold Church. Here is a division of Christian history and cosmology that corresponds to the fourfold Gospel and the fourfold Church:

Patristic period – ca. first through fifth century, which is championed by the (Eastern) Orthodox Church. Their authoritative writings are the Bible and the seven ecumenical councils. This corresponds to a cosmology of the seven celestial bodies visible to the naked eye (Sun, Moon, Mars, Mercury, Jupiter, Venus, and Saturn).

Medieval period – ca. fifth through fifteenth centuries, which is championed by the (Roman) Catholic Church. Their authoritative writings are the Bible and that of the Magisterium centered in Rome. This corresponds to a geocentric cosmology in which space and time are absolute.

Modern period – ca. fifteenth through twentieth centuries, which is championed by the churches of the Reformation. Their authoritative writings are the Bible and the various confessions or statements of faith. This corresponds to a heliocentric cosmology in which time is absolute.

Post-modern period – ca. from the twentieth century, which is championed by the Pentecostal and Charismatic churches. Their authoritative writings are the Bible and the writings of the various spirit-led teachers. This corresponds to a relativistic cosmology in which space and time are relative.

When we learn about history, we should learn the importance of the change from a geocentric to a heliocentric cosmology. Changes in cosmology go beyond theories of physics or astronomy. They correspond to spiritual changes as well.

# Motion science basics

Motion science is variously called mechanics, dynamics, kinetics, or kinematics. This post will be concerned with the study of motion apart from its causes or consequences.

Kinematics is the branch of classical mechanics which describes the motion of points (alternatively “particles”), bodies (objects), and systems of bodies without consideration of the masses of those objects nor the forces that may have caused the motion. (Wikipedia)

What is a motion? Let’s define motion initially as a continuous change of position of a body in space versus time. Movement is the act or process of moving.

There are two types of simple motion: translation (linear) and rotation (angular). Translation means motion in a straight line. Rotation means motion around a fixed point or axis (line). In linear motion all parts of an object move in the same direction and each part moves an equal distance. In angular motion some parts of an object move further or faster than other parts.

How is motion measured? A motion is measured by comparing it with a standard motion, called a clock, or a standard object such as a marked rod or protractor. A clock has two parts: a standard movement and markings for measurement.

A movement is measured in two ways. One way is synchronously, by matching of the beginning and ending of the movement to be measured with the moving part of a clock and noting the corresponding marking. The result is a number of units of the clock.

The other way to measure a movement is asynchronously, by matching of the beginning and ending of the movement to be measured with the markings on a measurement device such as a marked rod or protractor. The result is a number of units of the device. It is possible for the markings on a clock to be used for asynchronous measurement, too.

The units of a standard movement depend on the type of clock and how it is marked. An angular clock may be marked by its angles or its circumferential distances (as a distance wheel). A linear clock may be marked by the distance moved.

For example, the hands of a circular clock are the moving part, and the circumferential numbers from one to twelve are the marked part. Examples of linear clocks are this animation and this diagram of a light clock:

Linear motion is measured by a ratio and a direction. Conceptually, the ratio is between the asynchronous measurement of motion and the synchronous measurement of motion. In practice, a motion is measured by (1) fixing an independent, standard movement and measuring the asynchronous, dependent, standard movement, or by (2) fixing an independent, standard movement and measuring the synchronous, dependent standard movement. (1) is the speed in length per unit of time. (2) is the pace in time (or duration) per unit of length.

# Flow of motion

Note: previous posts on this topic are here and here.

Motion flows. That is, there is always motion independent of us. We can also make standard motions that are effectively independent of us. They are called clocks. They can be used as standards of comparison to measure other motions.

Clocks are needed for synchronous measurement of motion. They can also be used for asynchronous measurement of motion, but simpler devices can be used for that, too. For example, a circular clock provides a standard angular motion to compare synchronously with another motion. The marked angles or circumferential lengths could be used for asynchronous measurement. So could protractors and rigid rods.

Clocks can have various units of measure. A population clock estimates current population growth (or its decline). Mechanical clocks use an escapement to count periods of standard motion. A water clock measures the flow of water in units of volume. An hourglass measures the flow of sand. A clock can be made from any regular motion that can be associated with or marked in units.

Thus clocks, or “flowkeeping devices,” are independent, standard movements to compare with other motions, either synchronously or asynchronously.

From common experience we know there are three dimensions of motion. These three dimensions of motion can be measured synchronously or asynchronously. Synchronous measurements measure time; asynchronous measurements measure space.

For every motion one can associate six measurements: three synchronous measurements and three asynchronous measurements. That is, there are three dimensions of time and three dimensions of space.

To associate motion with position or location one must sum or integrate motions. One takes an arbitrary starting point, an origin, and integrates motion in different dimensions to construct a coordinate system for positions. Asynchronous integration leads to spatial coordinates. Synchronous integration leads to temporal coordinates.

# Uniformity and naturalism

My previous posts on this topic are here, here, and here. I am indebted to John P. McCaskey’s writings on the subject of induction (see here). In this post I want to make the connection between the principle of the uniformity of nature with naturalism.

In the 18th century there was a decline in understanding induction and an increase in skepticism about it, notably from David Hume and Immanuel Kant. The “problem of induction” in philosophy stems from this time and is often considered insoluble. The simplest solution is to accept as a principle that nature is uniform. J. S. Mill endorsed this solution, which became widely accepted in the 19th century. Today many consider science impossible without it.

The possibility of scientific history arises with a principle of uniformity. Uniformitarianism is “uniformity in time” (James Hutton), which was taken as the foundation of historical geology. Similarly, evolutionism is uniformity in time, taken as the foundation of historical biology. Among other things, that confuses history and science. Genuine induction encapsulates history into observation-based definitions of changes and processes, which keeps science and history distinct.

Modern inferential induction is less concerned with justification than with quickly finding new discoveries. It takes generalizations with some data behind it (possibly cherry-picked) and applies the uniformity of nature so that future observations will show the same result. But that requires similarity and what counts as similar? That is the key question that real induction answers with evidence-based universal concepts and definitions.

What would geology for example look like with no principle of uniformity? It would mean that geologists defined terms and processes based on observations, then went out looking for things that matched those definitions. The result would not be a pseudo-history of the earth but a real science of the earth that is systematic, consistent, and as complete as geologists can make it. Such a science could be used by historians and archaeologists with their documents and artifacts to develop a history of the earth. That’s how science and history should work together.

Naturalism is the application of uniformity to the whole search for truth. Previous posts tell how it became the dominant philosophy of science in the late 19th century (see here). Naturalism is based on an absolute uniformity of nature. It makes nature an autonomous, all-encompassing substitute for God.

John McCaskey has good news for us. Science doesn’t need naturalism or a principle of uniformity. Evidence and intuitions of uniformity can be embedded into definitions instead. Science can be demonstrably true.

# 3D time + 1D space, pace, and legerity

Although there are three dimensions of space and three dimensions of time, I have pointed out before that we measure movement as either 3D space + 1D time (3+1) or 1D space + 3D time (1+3) or 1D space + 1D time (1+1). The (1+3) perspective is the focus of this post.

The measurement of movement in which time is multidimensional but space is not requires that instead of speed and velocity, one must use pace and legerity. That is, movement is measured by the change in time (duration) per unit of movement in space (length). Pace is the directionless version of this.

For example, instead of speed in metres per second, one would use pace in seconds per metre or the like. This is not exactly the inverse of speed because the dependent units are different. Speed normally means the space speed, that is, the distance traveled in a fixed period of time. The time speed is a fixed travel distance per the corresponding travel time (which is strange because the independent variable is in the numerator). The pace is the time speed inverted, which puts the independent variable back in the denominator.

Legerity is the directional version of pace. An inertial system is a frame of reference that is at rest (zero velocity) or moves with a constant linear velocity. This can be expanded to include a frame of reference that is at zero or constant linear legerity.

Zero legerity means there is no change in time (duration) per unit of distance moved. We easily understand no change in distance per unit of time but this is strange. We have to remember that here the independent unit of motion is distance, not duration. In this context the distance measures the flow of movement (misleadingly called the flow of time).

So zero legerity means there is no change in time (duration) while a unit of distance passes, as by a “distance clock” like the odometer of an automobile moving at a constant rate. I have written about this here.

In classical (3+1) physics, time has an absolute meaning, independent of an observer. For a classical version of (1+3) physics space has an absolute meaning, independent of the observer. That is, either time or space continue indefinitely, and always serve as an independent variable, never as a dependent variable.

So there is always available information about an independent, inertial movement that provides a standard reference to measure any other movement. For absolute time this is called a clock or watch. For absolute space it could be called a distance clock (discussed here, here, and here). Then movement could always be measured by reference to this independent, standard movement.