iSoul In the beginning is reality.

# 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, 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 legerity

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 legerity. 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.

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 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, legerity and lentation 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.

Tempocity 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 lenticity, legerity, celerity, progressity]

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.

Lentation is the rate of change of legerity with respect to distance moved (trajectory length). A smaller lentation leads to faster movement. A larger lentation leads to slower movement. A negative lentation is called delentation (or expedience). Verb is lentate.

# Three arguments for 3D time

There are three main arguments for duration to have three dimensions:

(1) The speed of light is a conversion factor between length space (distance) and duration (distime). Transportation conversion factors include the maximum, minimum, or typical speeds associated with different travel modes. Since length space is three dimensional, its conversion into duration space is also three-dimensional. The scale of maps may be in units either of length or of duration using a standard conversion speed.

(2) Observation follows the movement of light, which is three dimensional. In astronomy it is often said that observation of the sky is a way of looking into the past. Observation is a form of communication and any type of signal (sound, mail, etc.) will suffice though not as exact as light. As the observation of length space is three-dimensional, so the observation of duration space is also three-dimensional. Maps of duration space are similar to maps of length space: they show observations (signals) in different directions.

(3) Movement in orthogonal directions entails moving in three dimensions. Each dimension of movement has an average speed from the change in length divided by the change in duration . As movement is in three dimensions, so the duration aspect of movement is in three dimensions. Travel shows the same result. Maps of movement may show either stantial positions or temporal positions or both.

Each argument may be illustrated by a map. The conversion factor argument shows how any map may be scaled in units of length or duration. The observation argument shows that maps of observations have units or length or duration. The movement argument shows that maps of movement may show length space or duration space or both (as in a isochrone map).

Note that every argument has an analogue in ordinary travel and so is not unknown to nonspecialists. The arguments are however exact in the case of physics.

# Conversion of space and time

If there exists a constant, characteristic speed, then one may speak of the characteristic conversion of space and time. For example, the speed of light in a vacuum is a defined constant in the SI system of units. So in physical science and its applications one may speak of the characteristic conversion of space into time and vice versa. This means that even if in some sense light curves (as by gravity), then the path of light is a geodesic, that is, equivalent to a straight line.

In other contexts, there may be no such characteristic speed but still there may be a constant speed within a specified context, which serves as a contextual conversion of space and time. This allows a map with a consistent scale, for example this map of the London Tube:

http://www.oskarlin.com/images/timetravel_no_zones_old_colours.pdf

Informally, this is done quite often. When asked how far away something is, we answer with the travel time by car or other mode.

Now the surprising thing is that the Lorentz transformation arises just because there exists such a conversion between space and time. It shows how to transform particular velocities in the context of a conversion speed between space and time. See the previous posts on the Lorentz transformation.

# Homogeneity and isotropy

A circle or sphere are omnidirectional in two or three dimensions, respectively. This is equivalent to isotropy, uniformity in all directions. A straight line is unidirectional but multiple straight lines may require multiple dimensions. This is equivalent to rectilinear homogeneity.

Pure space or average space is homogeneous and isotropic. Then space may be modeled by one dimension, although since the word dimension usually has to do with degrees of freedom or potential directionality, we say it has three dimensions.

It’s the same with time. Pure time or average time is homogeneous and isotropic, and may be modeled by one dimension, though it has three degrees of freedom and so we say it has three dimensions. If time is isotropic, only one dimension is needed to model it. If time is anisotropic, then three dimensions are needed to model it.

This is like the duality of wave and particle in quantum mechanics. Space and time have one or three dimensions depending on the aspect modeled.

Universal simultaneity requires homogeneity: “the transport of an ideal clock without distortion of time-intervals, requires a homogeneous space” (*).

In surface transportation a distinction can be drawn between congestion-type and current-type hindrances to travel. Radial congestion, such as a simple model of a city with a central business district, is isotropic. Travel across or in a river current could be modeled as rectilinearly homogeneous.

The conclusion is that homogeneity and isotropy come with a pure or average conception of space or time and require only one dimension to model. But the particulars of many situations do not exhibit either homogeneity or isotropy and so require three dimensions to model.