iSoul In the beginning is reality

Pace of light

Excerpts from How is the speed of light measured?

Before the seventeenth century, it was generally thought that light is transmitted instantaneously.  This was supported by the observation that there is no noticeable lag in the position of the Earth’s shadow on the Moon during a lunar eclipse, which would otherwise be expected if c were finite.

Instantaneously means a pace of zero. This is a basis for the Galilean transformation.

The first successful measurement of c was made by Olaus Roemer in 1676.  He noticed that, depending on the Earth–Sun–Jupiter geometry, there could be a difference of up to 1000 seconds between the predicted times of the eclipses of Jupiter’s moons, and the actual times that these eclipses were observed.  He correctly surmised that this is due to the varying length of time it takes for light to travel from Jupiter to Earth as the distance between these two planets varies.

This was actually the pace of light since the measured travel time was dependent on the known distance.

In 1728 James Bradley made another estimate by observing stellar aberration, being the apparent displacement of stars due to the motion of the Earth around the Sun.  He observed a star in Draco and found that its apparent position changed throughout the year.  All stellar positions are affected equally in this way.  … Bradley measured this angle for starlight, and knowing the speed of the Earth around the Sun, he found a value for the speed of light of 301,000 km/s.

The distance that the stars appear to move is proportional to the speed that the Earth moves divided by the speed of light, which is equivalent to the pace of light divided by the Earth’s orbital pace.

The first measurement of c that didn’t make use of the heavens was by Armand Fizeau in 1849.  He used a beam of light reflected from a mirror 8 km away.  The beam was aimed at the teeth of a rapidly spinning wheel.  The speed of the wheel was increased until its motion was such that the light’s two-way passage coincided with a movement of the wheel’s circumference by one tooth.  This gave a value for c of 315,000 km/s.  Leon Foucault improved on this result a year later using rotating mirrors, which gave the much more accurate value of 298,000 km/s.

Again, this was actually the pace of light since the measured travel time was dependent on the known distance.

According to the conventionality of simultaneity, the speed of light is actually the harmonic mean speed of the two-way speed of light. The harmonic mean is used because the length of the trips is the same. The pace of light is actually the arithmetic mean pace of the two-way pace of light.

The speed of light adopted as an international standard in 1983 is 299 792.458 km/s, which is equivalent to a pace of light that is 3.335 640 952 s/Gm.

Terminology for space and time, part 3

This is part 3 of an open series of posts on terminology; see part 1 and part 2.

As another way of indicating the dimensions of space and time, let space-time mean 3D space + 1D time, and let time-space mean 3D time + 1D space.

Consequently, Galilean time-space means what I called co-Galilean here. And Lorentz time-space means what I called co-Lorentz here.

The appropriate way of measuring the rate of movement in space-time is speed and velocity, since time is the independent variable. Pace is equivalent to the inverse of speed but their independent variables differ.

The appropriate way of measuring the rate of movement in time-space is pace and lenticity, since space (length or distance) is the independent variable. Speed is equivalent to the inverse of pace but their independent variables differ.

The term now includes all events in space-time with the same time coordinate. The term here includes all events in time-space with the same space coordinate.

This political moment

It’s difficult not to say something about the political moment of the U.S. Presidential election. Earlier this year I wrote briefly about its symmetry.

The media coverage has been mostly fantasies about what a king or queen would do rather than an actual president with enumerated powers. The low-information voter has almost nothing but fantasy to go on.

People are eager not only for the election to be over (the political ads, the mudslinging, the exaggerations and falsehoods) but to undo this year, as if it has turned out all wrong and needs a retake.

There are few bumper stickers this year. People are not excited about their candidate, even if they have a candidate. Others aren’t saying, and maybe don’t know, and perhaps won’t know until they enter the voting booth.

It’s worth thinking some about how this happened. It’s easy to blame them — the politicians, the parties, the mass media, etc. But it comes back to We the people. We have met the enemy and they are us.

An election with bad candidates concentrates the mind. The choice between Bad A and Bad B requires more wisdom than the choice between Sorta Ok A and Sorta Not Ok B.

It doesn’t makes sense to ask which candidate people like. It’s not about like anymore. People wouldn’t want to spend time alone with either candidate.

What issues there are are mostly about party ideology, so individual candidates matter less. The right to be born. Religious freedom. The role of  government. Health care. Jobs. Foreign affairs.

It’s becoming clear that the election won’t do what elections are supposed to do: dissipate public tension. Polarization will continue, even get worse after the election.

It’s hard to be optimistic about America today. People with a variety of opinions have been remarking about the moral decline of the nation for some time. Now it’s too obvious to think otherwise.

Creation of ubiquitous light

The first chapter of the first book of the Bible, Genesis 1, has attracted many commentators over the centuries. Recent scholarly work attempts to place it in the context of ancient Near East writings. (Near East is the European moniker for what Americans call the Middle East.) That however undervalues the unique, nuanced text of Genesis.

Creation ex nihilo is analogous in some ways to the creation of an axiomatic system such as Euclid’s Elements of Geometry. Before the first postulate (“A straight line segment can be drawn joining any two points.”) one should not assume that any such straight lines exist. “Let there be a line such that …” is the act of creating a line.

Similarly, in reading Genesis 1 we should not assume that before something was created, it existed or it existed in the way that we know it. Things we take for granted today, such as light, had to be created. This requires a close reading of Genesis 1 as a step by step process in which as little as possible is assumed to exist before there is some indication that it does exist.

Genesis 1 begins with some of the most famous words ever written:

1 In the beginning, God created the heavens and the earth. 2 The earth was without form and void, and darkness was over the face of the deep. And the Spirit of God was hovering over the face of the waters.

3 And God said, “Let there be light,” and there was light. 4 And God saw that the light was good. And God separated the light from the darkness. 5 God called the light Day, and the darkness he called Night. And there was evening and there was morning, the first day.

In regards to light, the second verse says there was darkness but no light, at least in the earthly world (we’re not told about the heavens of verse 1). Light is created in verse 3: “And God said, ‘Let there be light,’ and there was light.”

Where was the light shining from that was created in verse 3? And what time was the light shining? The text answers the second question first, in verses 4 and 5: “And God saw that the light was good. And God separated the light from the darkness. God called the light Day, and the darkness he called Night. And there was evening and there was morning, the first day.”

The light of verse 3 was separated from darkness to produce daylight, that is, a time of light. Before that separation, light and darkness were commingled in time. That is, at first light was ubiquitous in time. After the separation, light was concentrated in time, which is what constituted Day, that is, daylight.

Several verses later the text reads about the fourth day (Gen. 1:14-18):

14 And God said, “Let there be lights in the expanse of the heavens to separate the day from the night. And let them be for signs and for seasons, and for days and years, 15 and let them be lights in the expanse of the heavens to give light upon the earth.” And it was so. 16 And God made the two great lights—the greater light to rule the day and the lesser light to rule the night—and the stars. 17 And God set them in the expanse of the heavens to give light on the earth, 18 to rule over the day and over the night, and to separate the light from the darkness. And God saw that it was good. 19 And there was evening and there was morning, the fourth day.

For centuries people have found this passage perplexing. How could there be light on earth without the sun? Why was the sun needed if there was already light on earth? To start with, there was light on earth before the sun; that’s what the text says about day one. There was also evening and morning, nighttime and daytime without the sun.

Again, where was the light shining from that was created in verse 3? The answer is given in verse 18, which says why the sun, moon, and stars were created: to separate the light from the darkness. Prior to this light and darkness were commingled in space. That is, at first light was ubiquitous in space.

The image is that of the creation of ubiquitous light, which is then separated from darkness in time, and later separated from darkness in space. The separation of light and darkness on the fourth day produced stars, including the sun. The stars were not created from nothing at that time but were made by concentrating the light in space. Stars are a concentration of light that was already there.

This answers another perplexing question, which is asked since the speed of light is known to be finite, and some stars are many light-years away: How did the light get from the stars to the earth so quickly? The answer is that the light was already on the earth because light was ubiquitous in space before the stars were made. Concentrated darkness was lacking, too, before the light and darkness were separated.

In order to explain how starlight got to earth in a short time, it is sometimes asserted that God created light in transit. That is a different view than the one presented here, and one that lacks support in the text of Genesis 1. There are those who say Genesis 1 is just poetry and so can be interpreted any way you want. I have no patience for such a low view of poetry or anyone who plays fast and loose with the text. The close reading above shows that the text of Genesis 1 makes sense on its own terms.

Synchrony conventions

Reichenbach and Grünbaum noted “that the relation of simultaneity within each inertial reference frame contains an ineradicable element of convention which reveals itself in our ability to select (within certain limits) the value to be assigned to the one-way speed of light in that inertial frame.” (John A. Winnie, “Special Relativity without One-Way Velocity Assumptions: Part I,” Philosophy of Science, Vol. 37, No. 1, Mar., 1970, p. 81.)

Because the speed of light is measured as a round-trip speed, the one-way speed of light is unknown. It is a convention. This thesis is called the conventionality of simultaneity. Winnie writes (p. 82):

At time t1, by the clock at A, a light beam is sent to B and reflected back to A. Suppose that the return beam arrives at A at time t3 by the A-clock. The question now arises: at what time t2 (by the A-clock) did the beam arrive at B? Were we to “postulate” that the back and forth travel-times of the light beam were equal, clearly the answer is:

(1-1 ) t2 = t1 + (t3 – t1)/2

But should we fail to postulate the equality of the back-and-forth travel times, the best that we could do in this case would be to maintain that t2 is some instant between t1 and t3. Thus (using Reichenbach’s notation) we have the claim:

(1-2) t2 = t1 + ε(t3 – t1), (0 < ε < 1).

The simplest convention is one in which ε=½, which is what Einstein suggested, by synchronizing clocks this way and slowly transporting them to other locations. It is called Einstein synchronization or synchrony convention (ESC). That way, the speed of light has the same value going and coming (called c).

What would Galileo do? To preserve the Galilean transformation as much as possible, the speed of light either going or coming could be made effectively infinite. That would require a value of ε that is 0 (infinite speed going) or 1 (infinite speed coming). Such a Galilean synchrony convention (GSC) would need the speed of light in the other direction to equal c/2, that is, half of the ESC speed of light. That is because the harmonic mean of infinity and c/2 equals c (sequential speeds are averaged by the harmonic mean, not the arithmetic mean). Jason Lisle proposed such an anisotropic synchrony convention.

Synchrony is also required for those who use the diurnal movement of the sun and stars to tell time, i.e., the apparent or mean solar time. This applies to travel on or near the surface of the earth. The round trip time should not be affected by latitude or longitude, but without a correction for longitude, the travel time east or west would not match the return trip because of the direction of the sun’s motion. If one did not correct for latitude, travel north or south could also not match the return trip. It would be possible for a airplane to fly west with the sun or stars in the same position, so that no time elapsed in this sense.

Consciousness of space and time

If we are blindfolded or are in a room with no light, we still have an awareness of where we are relative to our memory of where we were and our imagination of where we might be. We are always here in relation to there. That is our consciousness of space.

Our consciousness of time is similar. We are always now in relation to then. If we don’t have a clock or other marker of time, we still have a memory of past events and an imagination of future events. We still sense motion and change.

We live in real-time, which is the actual time during which events take place. Time is measured in real-time, unless there is a recording that reproduces what happened and the rate that it happened. The measurement of space does not need real-time. It may be measured at leisure, or whenever. We live in what could be called real-space, which is the actual space in which things exist.

Our consciousness of space and time are often confused with space and time. That is why we need objective definitions and measurements of space and time to avoid confusion. We might also coin different words for the consciousness of space and time, such as cognitime and cognispace, respectively.

The flow of time is a property of the consciousness of time, which seems to flow on like a river. To a lesser extent, the measurement of time by a clock has a kind of flow in that it continues indefinitely, though we can stop it or it can stop on us at any time.

Lorentz transformations and dimensions

In recent post on Lorentz and co-Lorentz Transformations, I derived a complete set of Lorentzian transformations:

The Lorentz transformation is
speed: r′ = γ (r − vt) and t′ = γ (t – rv/c²), with γ = (1 – v²/c²)–1/2, or
pace: r′ = γ (r – t/u) and t′ = γ (t – rb²/u), with γ = (1 – b²/u²)–1/2 ,
which applies only if |v| < |c| or |u| > |b|.

The co-Lorentz transformation is
speed: t′ = γ (t − r/v) and r′ = γ (r − c² t/v), with γ = (1 − c²/v²)–1/2, or
pace: t′ = γ (t – ur) and r′ = γ (r – tu/b²), with γ = (1 – u²/b²)–1/2,
which applies only if |v| > |c| or |u| < |b|.

If |v| = |c|, then r′ = r and t′ = t.

Other dimensions were not discussed since the notation didn’t specify whether or not there were other dimensions. Here I note that speed requires that the denominator be a scalar of time, so in that case time is either one-dimensional or resolves itself into one dimension. Similarly, pace requires that the denominator be a scalar of space, so in that case space is either one-dimensional or resolves itself into one dimension. Another way to look at it is that speed allows multidimensional space but not time, whereas pace allows multidimensional time but not space.

It has been noted that tachyons are possible with multidimensional time but not space, which is consistent with the co-Lorentz transformation except that there is also the possibility of space and time resolving themselves into one dimension each. Anyone who travels faster than the characteristic rate can experience this from their own perspective.

As the Lorentz transformation arises from Minkowski geometry so the co-Lorentz transformation arises from co-Minkowski geometry. Relativity without tears (p.23-4) notes that the points of Minkowski geometry correspond to the lines of co-Minkowski geometry (and vice versa), and the distance between points in Minkowski geometry correspond to the angle between lines in the co-Minkowski geometry (and vice versa). This is consistent with the previous post: Linear space and angular time in Minkowski geometry corresponds to angular space and linear time in co-Minkowski geometry.

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 meters, millimeters, kilometers, 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 (distance).

A Lorentz transformation requires what I’m calling a characteristic 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, b, 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: br = t or r = t/b and br′ = t′ or r′ = t′/b.

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 – b/u) = t/bγ (t/brb/u) = γt (1/b – 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 + b/u) = t/bγ (t/b + rb/u) = γt′ (1/b + 1/u).

Multiply each corresponding pair together to get:

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

pace: rr´ = γ²rr′ (1 – b²/u²) = tt/b² = γ²tt′ (1/ – 1/).

Dividing out rr′ yields:

speed: 1 = γ2 (1 – v2/c2),

pace: = γ2 (1 – b2/u2).

Or dividing out tt′ yields:

speed: c2 = γ2 (c2v2),

pace: 1/b² = γ2 (1/ – 1/).

Either way, solving for γ leads to:

speed: γ = (1 – v2/c2)–1/2,

pace: γ = (1 – b2/u2)–1/2.

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

Co-Lorentz transformation

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

speed (–): t = γ (tr/v) = γt (1 – c/v) = r′/cγ (r/c – tc/v) = γr (1/c – 1/v),

pace (–): t′ = γ (tur) = γt (1 – u/b) = br′ = γ (brtu/b) = γr (bu),

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

speed (+): t = γ (t + r/v) = γt′ (1 + c/v) = r/cγ (r/c + tc/v) = γr′ (1/c + 1/v),

pace (+): t = γ (t′ + ur′) = γt′ (1 + u/b) = br = γ (br + tu/b) = γr′ (b + u).

Multiply each pair together to get:

speed: tt = γ² tt (1 – c²/v²) = rr′/c² = γ² rr (1/ – 1/),

pace: tt = γ² tt′ (1 – u²/b²) = b²rr = γ² rr′ ().

Dividing out tt′ yields:

speed: 1 = γ² (1 – /),

pace: 1 = γ2 (1 – u²/b²).

Or dividing out rr′ yields:

speed: 1/ = γ² (1/ – 1/),

pace: = γ2 ().

Either way, solving for γ leads to:

speed: γ = (1 − c2/v2)–1/2,

pace: γ = (1 – u2/b2)–1/2,

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

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 (trb²/u), with γ = (1 – b2/u2)–1/2.

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

The co-Lorentz transformation is then

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

pace: t′ = γ (tur) and r′ = γ (rtu/b²), with γ = (1 – u2/b2)–1/2,

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

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).