iSoul In the beginning is reality.

# Paceometer

We’ve all seen a vehicle speedometer, scaled in km/h or mph like this:

A paceometer is made by adding another scale, as with the outer scale of minutes per 10 miles below (image source here):

Fitness apps on smartphones can display distance traveled, travel time, and pace. In the article Pace yourself: Improving time-saving judgments when increasing activity speed, the authors conclude:

In this paper, we tried to highlight a general cognitive phenomenon in which people misestimate time-savings when increasing speed and suggested that this bias is due to the commonly used speed measures that extenuate people’s inability to recognize the curvilinear relationship between these speed measures and the reduction of activity time. We offer a simple remedy for this bias: converting speed values to pace data, which simplifies the cognitive task of judging time-savings because the data conforms to people’s lay perceptions. We believe that this simple solution can be applied to other contexts in which individuals’ judgments might be biased.

Here’s a table to convert miles per hour to minutes per 10 miles, or kilometers per hour to minutes per 10 kilometers – note it’s the same conversion, relatively speaking:

 Speed Pace mph min/10mi kph min/10km 0 0 5 120 10 60 15 40 20 30 25 24 30 20 35 17 40 15 45 13 50 12 55 11 60 10 65 9.2 70 8.6 75 8.0 80 7.5 85 7.1 90 6.7 95 6.3 100 6.0 105 5.7 110 5.5 115 5.2 120 5.0

# Lorentz transformation for time-space

It is worth returning to my post on Lorentz and Dual Lorentz transformations in order to make the point that the Lorentz transformation is sufficient for 3D time. Superluminal speeds are not required for 3D time, contrary to what others have said, such as here.

What is required for time-space (1+3) is the use of measures relating measured changes in duration to independent changes in length. Pace, lenticity, and retardation are needed instead of speed, velocity, and acceleration. See glossary above for definitions of terms.

The conception of length space is then simplified to a one-dimensional distance from some conventional origin point, the stance. This conception of stance is analogous to the common conception of time, as something that flows on independently of us. That is what it means to be an independent variable: it’s out of our control.

# 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 timepoint 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; it is a constant, c.

What would Galileo do? To preserve the Galilean transformation as much as possible, the speed of light either outgoing or incoming could be made effectively infinite. That would require a value of ε that is 0 (infinite speed outgoing) or 1 (infinite speed incoming). 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 (ASC).

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.

# Lorentz transformations and dimensions

In recent post on Lorentz and dual 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 – rç²/u), with γ = (1 – ç²/u²)–1/2 ,
which applies only if |v| < |c| or |u| > |ç|.

The dual Lorentz transformation is
speed: t′ = λ (t − r/v) and r′ = λ (r − c² t/v), with λ = (1 − c²/v²)–1/2, or
pace: t′ = λ (t – ur) and r′ = λ (r – tu/ç²), with λ = (1 – u²/ç²)–1/2,
which applies only if |v| > |c| or |u| < |ç|.

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 multiple dimensions of time but not of space.

It has been noted that tachyons are possible with multiple dimensions of time but not space, which is consistent with the dual 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 (modal) rate can experience this from their own perspective.

As the Lorentz transformation arises from Minkowski geometry so the dual Lorentz transformation arises from dual Minkowski geometry. Relativity without tears (p.23-4) notes that the points of Minkowski geometry correspond to the lines of dual Minkowski geometry (and vice versa), and the distance between points in Minkowski geometry correspond to the angle between lines in the dual 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 dual 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 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 (surveyor’s 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.

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.

# Lorentz and dual 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 dual 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).

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

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

The measurement of movement in which duration has multiple dimensions but length does not requires that instead of speed and velocity, one must use pace and lenticity. That is, movement is measured by the change in time (duration) per unit of movement in 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 stance 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.

Lenticity 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 lenticity.

Zero lenticity 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 lenticity 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 length has an absolute meaning, independent of the observer. That is, either duration or length 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 stance 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.

# Relativity at any speed

This is a summary of posts (such as here and here) about the application of relativity theory to transportation. This is different from applying theories of physics to other subjects such as economics since here it is real relativity, not some analogy. However, the application is an approximation, but that is the nature of transportation, which has both physical and social aspects.

Transportation (or transport) modes are means of transporting goods and/or people. The common modes are roads (including motorized and non-motorized modes), railroad (passenger and freight), pipeline, maritime (ferry and shipping modes), air (aviation), etc. One can consider information a good and the telecommunication of information as a form of transportation. In that sense any signal or high-speed particle can be considered a mode of transportation.

Transportation is subject to a variety of obstacles, including excess volume per available capacity (called congestion) and stoppages caused by crashes, storms, construction, or other disruptions (which may also cause congestion). There may be complicating factors such as the mixing of modes (e.g., bike and pedestrian traffic).

The minimum and maximum (free-flow) rates of a transportation mode are characteristic of the transportation mode as a system, rather than the speed of a particular vehicle or particle (even though a particular vehicle or particle might have this value in a particular context). As such, these characteristic or modal speeds are constants within the transportation system under consideration.

The modal speed plays a role similar to the speed of light in a vacuum for high-speed physics. Such characteristic speeds are constants that are independent of the speed of particular objects (vehicles) in that mode.

So, for example, a free-flow highway speed may be considered a constant over a region or transportation network. Then in this context such a constant speed would play the role of c, the speed of light in a vacuum. This speed would relate space and time. The Lorentz transform would be needed to determine relative speeds.

A transportation mode may have a minimum speed, for example the minimum speed required to keep an airplane airborne. In this context speeds greater than the minimum will exist, which is a dual situation of a maximum speed in relativity. In any case, relativity theory can cover these cases, too.

# Direction with three dimensions of time, part 1

I

Multiple dimensions of time are easier to see if we look at transportation. Consider the time table of a railway or subway system. Directions are typically shown by the station at the end of each line. The time table lists arrivals and departures – events in space and time. A railway station some distance away is in a certain direction both in space and in time within a system. For example, a famous distance-time graph by Etienne-Jules Marey shows the Paris-Lyons line in the 1885.

If we know something about the geography of the area, that is likely to be in our minds when reading a railway time table. But if we don’t know much about the geography, travel times provide a way to map the railway system. The “time scale” may even be more useful than the distance scale. Here are two examples of time scale maps for the Boston-area MBTA: Time-Scale Commuter Rail Map and Time-Scale Subway Map.

Another way to see time directions is simply to take a time-space or isochron map and remove indications of spatial directions and distances. See for example Travel Times on Commuter Rail. The point is that time directionality is real. To see it requires separating time directions from geographic directions.

II

A degree of space is an angular distance equal to 1/360th of a circle. An arc minute of space is an angular distance equal to 1/60th of one degree of space. An arc second of space is an angular distance equal to 1/60th of one minute of space.

A minute of time is an angular duration equal to 1/60th of a full rotation or cycle at a rate of one rotation per hour. A second of time is an angular duration equal to 1/60th of a full rotation or cycle at a rate of one rotation per minute.

Angular time is measured by a duration of angular movement. For example, if a motor turns clockwise at a rate of one rotation per hour for five minutes (like a minute hand), then turns at a rate of one rotation per minute for ten seconds (like a second hand), the angular duration of motion will be 5:10 minutes but the angular distance of motion will be 90:00 degrees.

III

A direction is in the context of a geometry. A moving object such as a vehicle has a travel distance and a travel time that are both scalars. The odometer is increasing no matter which direction the vehicle is moving. It is only relative to a space and time beyond the vehicle and its movement that one can speak of its direction.

This direction may be conceived spatially and/or temporally. Directions in space and time will be the same if space and time are proportional, that is, distance and duration are proportional. In that case, we might say either there is no time or there is no space, though either statement would not be not strictly correct. There is always both space and time but they may be equivalent, and one may be hidden behind the other so to speak.

# Terminology for space and time, part 2

I’ve written about terminology and used new terms before, for example, Movement and dimensions. I want to take another look at coining new terms needed for studying movement and the symmetry of space and time. In what follows, the terms pace, lenticity and retardation have new senses.

Consider these parallel terms, defined with respect to a frame of reference:

Speed is the time rate of change of position of a body without regard to direction (The McGraw-Hill Dictionary of Physics, Third Edition). By position is meant the spatial position. By time rate of change is meant the rate of change per unit of travel time.

Pace is the space rate of change of temporal position of a body without regard to temporal direction. By space rate of change is meant the rate of change per unit of travel distance (trajectory length). Pace comes from Latin passus, a step or stride, which relates to the unit of length in the denominator, as when walking or running.

For example: “The Centers for Disease Control and Prevention (CDC) defines brisk walking as being at a pace of three miles per hour or more (but not racewalking) or roughly 20 minutes per mile. That equates to about five kilometres per hour or 12 minutes per kilometre.” (verywell)

The time mean speed is the arithmetic average speed of multiple objects passing a point in space during a particular period of time (spot speeds). Time-mean speeds are used in reference to objects passing a point in space, averaged over a common time period.

The space mean speed is the arithmetic average speed of multiple objects over a common length of space, i.e., the average travel time divided by the particular length. If speeds are constant, it equals the harmonic average of multiple objects passing a point in space during a period of time (spot speeds).

Velocity (ve∙loc′∙i∙ty) is the time rate of change of position of a body; it is a vector quantity having direction as well as magnitude (The McGraw-Hill Dictionary of Physics, Third Edition). By position is meant the spatial position. By time rate of change is meant the rate of change in space (length) per unit of elapsed time. By direction is meant spatial direction. It is equivalent to a specification of the speed and direction of motion (e.g. 60 km/h toward the east, i.e., toward eastern places). Velocity is from Latin velocitas, speed, swiftness, rapidity.

Lenticity is the space rate of change of temporal position of a body; it is a vector quantity having temporal direction as well as magnitude. By space rate of change is meant the rate of change in time per unit of trajectory length. It is equivalent to a specification of the pace and direction of motion (e.g. 60 min/km toward magnetic north). [was legerity, celerity, progressity, tempocity]

Acceleration is the rate of change of velocity with respect to time (The McGraw-Hill Dictionary of Physics, Third Edition). Time means the travel time. Acceleration is from Latin acceleratus, past participle of accelerare “to hasten, quicken,” from ad– “to” [toward] + celerare “hasten”. Negative acceleration is deceleration.

Retardation is the rate of change of lenticity with respect to distance moved (trajectory length). A smaller retardation leads to faster movement. A larger retardation leads to slower movement. A negative rapidation is called deretardation. Verb is retardate.