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Conventions and properties

Everything in science is a combination of conventions and properties. For example, frames of reference have certain conventions in common and particular properties that each individual frame has. The definition of a frame of reference is the first convention. Every frame of reference has an origin and at least the possibility of one or more coordinate axes. But the particular origin of a frame need not be in common with other frames; it is a particular property of one frame.

Definitions and postulates are conventions. Stipulations and measurements are properties. Physical laws are conventions with the appropriate supporting definitions and postulates. Interpretations of events become conventions when they are widely accepted.

The SI metric system is the international convention for measurement (i.e., metrology). Individual measurements are properties of things. Kinematics and dynamics have a convention for simultaneity (as well as simulbaseity). The orientation of orthogonal axes follows a convention for the order of the axes and the direction of positivity.

Two principles of velocity reciprocity

Velocity reciprocity in relativity theory is the relation between two observers, each associated with a frame of reference and moving at different, but constant, velocities. That is, an observer-frame S observes another observer-frame traveling with velocity +v relative to observer-frame S. A velocity reciprocity relation concerns the velocity of S that is observed by . Einstein’s principle of velocity reciprocity states that each velocity is the same magnitude (speed) but is in the opposite direction. That is, the velocity of S observed by  is –v.

Two frames with same orientation

Einstein’s principle of velocity reciprocity reads

We postulate that the relation between the coordinates of the two systems is linear. Then the inverse transformation is also linear and the complete non-preference of the one or the other system demands that the transformation shall be identical with the original one, except for a change of v to −v. Ref.

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Length and duration

Although it has seemed natural to speak of “space and time”, that is a confused designation of length and duration, as well as their metrics, base and time (see glossary here). Space is the space of motion, so length space is only one side; the other side is duration space. Speaking consistently is challenging but a must in order to bring clarity to a confused subject.

Here is a diagram of the combinations of length and duration, base and time (click to enlarge):

length-duration diamond

The extent of motion is measured by length and duration. The space of motion has three dimensions. Length space and duration space therefore have three dimensions as well. The space of motion is  represented by an ordered pair of length space and duration space coordinates. If there is absolute inter-convertibility between length and duration, then the space of motion is a six-dimensional space of either length units or duration units.

The metric of length is distance and the metric of duration is distime (though often called time). Base variable is the distance of a point from the origin in length space. Time variable is the distime of an instant from the origin in duration space. Events are located in length space and chronated in duration-space.

As one can speak of distance in three-dimensional length space, one can speak of distime in three-dimensional duration space. In this sense, time is a three-dimensional concept. There are three dimensions of duration rather than three dimensions of time per se, but time has a three-dimensional aspect.

Two measures of motion

By common experience, we know there are three dimensions of motion. That is, space, which is the space of motion, is three dimensional. To measure the extent of motion requires comparing one motion with another, of which there are two ways: length and duration. The length of a motion is measured by comparing it with symbasal but not necessarily synchronous motion. The duration of a motion is measured by comparing it with synchronous but not necessarily symbasal motion.

Length of motion considered by itself forms a length space, which is space with a metric of length. Duration of motion considered by itself forms a duration space, which is space with a metric of duration. Since there are three dimensions of motion, length space and duration space are both three dimensional metric spaces. By convention, both are Euclidean. The length metric is called distance. The duration metric may be called distime.

Each point in length space has a length position (LP) vector that begins with the length origin. Each point in duration space has a duration position (DP) vector that begins with the duration origin. The magnitude of a length position vector is called the base. Every point in length space that is equidistant from the origin has the same base. The magnitude of a duration position vector is called the time. Every point in duration space that is an equal distime from the origin has the same time.

Stance and time are vector magnitudes, with their direction ignored. Base is a radius from the origin of length space. A unit of length is the absolute value difference between two bases, that is, between the radii of two length vectors with unit difference. Time is a radius from the origin of duration space. A unit of duration is the absolute value difference between two times, that is, between the radii of two duration vectors with unit difference.

The rate of motion measured by the length of motion per unit of duration is called speed. The rate of motion measured by the duration of motion per unit of length is called pace. Note that a faster speed is a larger ratio, whereas a faster pace is a smaller ratio. Also, the ratio of a slower speed to a faster speed is less than one but the ratio of a faster pace to a slower pace is less than one.

The vector rate of change in the length vector per unit of duration is called velocity. The vector rate of change in the duration vector per unit of length is called legerity. The vector rate of change in velocity per unit of duration is called acceleration. The vector rate of change in legerity per unit of length is called expedience.

The length position vector of a trajectory evolves as a function of the time. The duration position vector of a trajectory evolves as a function of the base. These functions are inverses of one another.

Newton’s laws and their duals

The following is based on Classical Mechanics by Kibble and Berkshire, 5th ed., Imperial College Press, 2004, with the dual version indented and changes italicized.

p.2 The most fundamental assumptions of physics are probably those concerned with the concepts of space and time. We assume that space and time are continuous, that it is meaningful to say that an event occurred at a specific point in space and a specific instant of time, and that there are universal standards of length and time (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements).

The most fundamental assumptions of physics are probably those concerned with the concepts of length and duration. We assume that length and duration are continuous, that it is meaningful to say that an event occurred at a specific point in length space and a specific instant of duration space, and that there are universal standards of length and duration (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements).

In ‘classical’ physics, we assume further that there is a universal time scale (in the sense that two observers who have synchronized their clocks will always agree about the time of any event), that the geometry of space is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all positions and velocities.

In dual ‘classical’ physics, we assume further that there is a universal length scale (in the sense that two observers who have symbasalized their clocks will always agree about the base of any event), that the geometry of time is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all chronations and legerities.

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Interchange of length and duration spaces

In geometry, a spherical coordinate system specifies points by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle, which is the horizontal angle between the origin to the point of interest.

A spherical coordinate system is implicitly behind length space (space-time) (3+1), with the temporal directional ignored for scalar time. That is, every instant in duration space (3D time) is projected onto a temporal sphere centered on the origin instant. The scalar time is the radial distime of each instant.

Let the spherical coordinates of length space be (r, θ, φ) with r representing the radial distance, and θ and φ representing the zenith and azimuth angles, respectively. Let the spherical coordinates of duration space (3D time) be (t, χ, ψ) with t representing the radial distime, and χ and ψ representing the temporal zenith and azimuth angles, respectively.

To represent the fullness of length space and duration space requires six dimensions (3+3), three for length and three for duration: ((r, θ, φ); (t, χ, ψ)) or (r, θ, φ; t, χ, ψ). Then length-duration (space-time) (3+1) can be represented by the coordinates [r, θ, φ; t] and duration-length (time-space) (1+3) by the coordinates 〈r; t, χ, ψ〉.

If rectilinear coordinates are used for duration space, say (ξ, η, ζ), then the time, i.e., radial distime, t = √(ξ² + η² + ζ²). The corresponding length space concept, base, is the radial distance. If rectilinear coordinates are used for length space, say (x, y, z), then the base r = √(x² + y² + z²). For 2D applications such as mapping, polar coordinates would be used instead of spherical, in which case r = √(x² + y²) and t = √(ξ² + η²).

To convert (3+1) function to a (1+3) function requires expansion to its (3+3) function, inversion by interchange of length and duration, and then contraction to (1+3). In symbols with ↑ as expand, ↓ as contract, and ↔ as interchange: (3+1) ↑ (3+3) ↓ (1+3), or (r, θ, φ; t) ↑ (r, θ, φ; t, χ, ψ) ↓ (r; t, χ, ψ). In this way length and duration, time and base are interchanged.

In more detail, a parametric length space function is converted to a parametric time function: r = [r, θ, φ] = r(t) = [r(t), θ(t), φ(t)] ↑ [(t´, χ´, ψ´), θ´(t´, χ´, ψ´), φ´(t´, χ´, ψ´)] ↔ ((r´, θ´, φ´), χ´(r´, θ´, φ´), ψ´(r´, θ´, φ´)) ↓ [t(r), χ(r), ψ(r)] = t(r) = [t, χ, ψ] = t.

Take for example the definition v = dr/dt. We have: v = dr/dt = [dr/dt, dθ/dt, dφ/dt] = [r(t), θ(t), φ(t)]) ↑ [(t´, χ´, ψ´), θ´(t´, χ´, ψ´), φ´(t´, χ´, ψ´)] ↔ ((r´, θ´, φ´), χ´(r´, θ´, φ´), ψ´(r´, θ´, φ´)) ↓ [t(r), χ(r), ψ(r)] = [dt/dr, dχ/dr, dψ/dt] = dt/dr = u. The result is that length space and duration space are interchanged, with length space vectors becoming bases and times becoming duration space vectors.

Functions may be converted by differentiation, then inversion, then integration. Take for example, the function s(t) = s0 + v0t + ½at². Differentiating twice leads to s(t)´´ = a = dv/dt. Expanding to spherical coordinates, inverting, and contracting to the radial component results in t(s)´´ = du/ds = b. Integrating twice produces t(s) = t0 + u0s + ½bs², which has the same form as the original function.

Length and duration

Let us begin with (1) the motion of a body between two events and (2) two ways of measuring the extent of that motion: length and duration (or time). The measurement of length and duration is coordinated so that both measures are of the same motion. Length and duration are measured by a rigid rod and a stopwatch, respectively. A smooth manifold of length is called space (or 3D space), and a smooth manifold of duration is called time (or 3D time).

The length and duration of a motion are commonly measured along the trajectory (or arc) of the motion. The length along the trajectory of motion is the arc length (or proper length or simply length). The duration along this trajectory is the arc time (or proper time or simply time).

Once the length and duration are in hand, the next step is to form their ratio. The ratio with arc time as the independent variable and arc length as the dependent variable is the speed. Which is to say, speed is the time rate of arc length change.

Note that the ratio could just as well be formed in the opposite way, with the arc length as the independent variable and the arc time as the dependent variable. This ratio is called the pace from its use in racing, in which an arc length is first set and then the racer’s arc time is measured. As another way to state this, pace is the space rate of arc time change.

The reference trajectory for measuring the length of a motion is the minimum length trajectory between two event points. The length along this trajectory is the distance between the two event points, which forms the metric of space. Distance is represented as a straight line on a length-scale map.

The reference trajectory for measuring the duration of a motion is the minimum time trajectory between two event instants. The duration along this trajectory is the distime between the two event instants, which forms the metric of time. On a map, two isochrons are separated by a constant distime. Distime is represented as a straight line on an time-scale map.

Motion has direction as well as extent, and direction may also be measured in two ways. Consider the motion of rotation, which can be measured as a proportion of a circle and as a proportion of a cycle. For example, in an analogue clock a minute hand that moves the length of a right angle correspondingly moves a duration of 15 minutes and vice versa. The direction of motion may be measured by either length or duration.

A motion measured with length direction and distance comprises a vector displacement. A motion measured with time direction and distime comprises a vector dischronment. The ratio with time as the independent variable and displacement as the dependent variable is called the velocity. The magnitude of the velocity vector is the speed. The ratio with distance as the independent variable and dischronment as the dependent variable may be called the legerity. The magnitude of the legerity vector is the pace.

There are three dimensions of motion, and correspondingly three dimensions of length and duration. The three dimensions of length comprise space. The three dimensions of duration comprise time. The three dimensions of time come as a surprise, since the distime is often a parameter for ordering events. But the scalar distime should not be confused with the vector dischronment, which has three dimensions of motion measured by duration.

Space and time as opposites

A theme of this blog is that space and time are dual concepts, which means they are two ways of understanding the same thing. But in what ways are space and time opposite concepts?

Space is oriented toward its origin, the place that motion begins. Time is oriented toward its destination, the time that motion ends. Both length and duration are measured from an “origin,” a reference point, which is a zero point for each, but zero speed leaves a body in space at the beginning, whereas zero pace puts the body in time at its destination.

Length in the denominator of speed is a measure of the progress from the origin to the current location in space, whereas time in the denominator of pace is a measure of the lag from the destination to the current location in time. A body at zero speed will remain at its origin and never reach its destination, whereas a body at zero pace will arrive at its destination in literally no time. A body with a small speed will take a long time to reach its destination, whereas a body with a small pace will reach its destination quickly.

Large quantities in space correspond to small quantities in time. Large quantities in time correspond to small quantities in space. A high speed is fast, and a small speed is slow. A small pace is fast, and a large pace is slow. Mass and vass are inverses, as are energy and lethargy.

The origin in space corresponds to the destination in time. Time in space flows from the past toward the future. Space (or base) in time flows from the future toward the past.

Arcloge

An arcloge (arc’-loje) is a continuous, independent measurement of length. That is, it measures an ever-increasing length, which is the base, similar to how a clock shows the time. The term is a combination of arc (as in arc length) and loge (as in horologe, a clock).

What does an arcloge look like? Start with a sector, which is a geometric figure fixed to the center of a circle that sweeps out an angle and a curved edge:

sector and arc of a circle Read more →

Basic definitions

Independent variable is a quantity that is not dependent on another quantity, which is either (a) a quantity chosen before an experiment or race, or (b) an ever-increasing quantity. Dependent variable is a quantity that whose value is a function of another variable.

Length is measured by a rigid rod. 3D length space is length measured in three directions of motion.

Duration is measured by a stopwatch, timer, or clock. 3D duration is duration measured in three directions of motion.

Length space is (1) 3D differentiable manifold of length; (2) the order of events on a baseline; (3) the base, the reading on an arcloge.

Duration space is (1) a 3D differentiable manifold of duration; (2) the order of events on a time line; (3) the time, the reading on a clock.

Length-duration is the 6D manifold formed from 3D length space and 3D duration space. Worldline is the path in length-duration traced out by an object in motion.

Displacement is the vector between two points (events) on a worldline. Distance is the magnitude of a displacement.

Dischronment is the vector between two points (events) on a worldline. Distime is the magnitude of a dischronment.

Reference frame (or frame) is an abstract coordinate system and set of reference points in 3D length space that uniquely fix the coordinate system and standardize measurements. Rest frame of a body is the reference frame in which the body is moving at zero speed, which is the time conversion pace.

Reference timeframe (or timeframe) is an abstract coordinate system and set of reference points in 3D duration space that uniquely fix the coordinate system and standardize measurements. Freeflow frame of a body is the reference timeframe in which the body is moving at zero pace, which is the base conversion speed.

Proper length is the length of a body measured by a rigid rod moving with it. Proper time is the time of a body measured by a clock moving with it.

Lorentz transformation is a set of equations that relate length space and duration space coordinates of reference frames moving at a constant velocity relative to each other.

Dual Lorentz transformation is a set of equations that relate length space and duration space coordinates of reference timeframes moving at a constant legerity relative to each other.